基于鲁棒优化的城市交通网络设计模型与算法研究
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摘要
城市交通网络设计问题是城市综合规划的核心问题,也是关系到城市经济长期、快速、和谐和稳定发展的基本问题。当前,随着城市的高速发展,城市交通拥堵现象日益严重,交通供需矛盾日益突出,缓解和预防交通拥堵已经成为城市发展当务之急。另一方面,城市交通网络中存在着大量的不确定因素,如果在交通网络设计中忽视这些不确定性因素,可能会导致交通网络更加严重的拥堵。因此,不确定的交通网络设计问题的研究是必不可少的。当前,不确定城市交通网络设计的研究方法主要有随机规划和鲁棒优化两种,其中随机规划的方法需要事先假定不确定参数满足某种概率分布。然而,在现实中,由于缺少大量数据去校准这种概率分布,这种假定的概率分布可能不能用。而鲁棒优化的方法则不需要事先假定不确定参数满足某种概率分布。因此,应用鲁棒优化的方法研究不确定交通网络设计问题具有更加实际的意义。
本论文基于鲁棒优化的方法,研究不确定的城市交通网络设计问题,探讨不确定交通网络设计问题的建模和求解算法。具体来讲,本论文研究工作主要有以下几个方面：
(1)运用鲁棒非线性优化方法研究了基于用户均衡下不确定需求的连续交通网络设计问题,其中不确定需求属于一个椭球集合。通过运用鲁棒优化的思想和灵敏度分析的方法,我们将连续交通网络设计问题的鲁棒对应(Robust Counterpart,RC)模型转化为一系列带互补约束的数学规划问题(Mathematical Programms with Complementarity Problem, MPCC),并运用一种松弛算法求解这一系列的MPCC。另外,我们将它和Yin和Lawphongpanich[1]提出的鲁棒对应模型进行了比较。数值实验的结果表明,我们提出的鲁棒对应模型比Yin和Lawphongpanich[1]的鲁棒对应模型更加灵活,没那么保守。
(2)探讨了不确定需求下的鲁棒可靠性用户均衡模型,其中模型并不要求知道不确定需求的准确的概率分布,而仅需知道它的前m阶矩。基于最坏风险值(Worst-case Value-at-Risk, WVaR)和最坏条件风险值(Worst-Case Conditional Value-at-Risk, WCVaR)[2],我们定义了鲁棒分位走行时问和鲁棒均值-超量走行时问,并证明了两种走行时间在需求一般分布情况下是等价的。基于这种等价的走行时问提出了鲁棒分位用户均衡(鲁棒均值-超量交通均衡)模型,模型被表示为一个非线性互补问题(Nonlinear Complementarity Problem, NCP),并证明了模型的等价性和解的存在性。然后一种基于间隙函数的方法被用来求解这个非线性互补问题。基于提出的均衡模型,我们进一步研究了带分布式鲁棒联合机会约束的连续交通网络设计模型,通过利用Bonferroni不等式,模型中的分布式鲁棒联合机会约束被近似为非线性约束,我们应用积极集的算法求解近似后的模型,数值实验的结果验证了提出的模型和算法的有效性。
(3)运用可调整的鲁棒优化方法研究了基于元胞传输模型(Cell Transimission Model, CTM)[3-4]的单层动态交通网络设计模型,其中不确定需求被假定属于一个多面体集合。通过运用仿射决策准则和线性规划的对偶,我们构建了相应的仿射可调整的鲁棒对应模型,同时将它与传统的鲁棒对应模型进行了比较,数值算例的结果显示,可调整的鲁棒对应模型比传统的鲁棒对应模型更加灵活。
(4)基于元胞传输模型,研究了单层动态交通网络设计问题的分布式鲁棒联合机会约束模型,模型假定OD需求的概率分布是未知的,仅知道它的期望和方差。首先,我们将模型中的分布式鲁棒联合机会约束近似为最坏条件风险值约束,然后,利用锥对偶原理,将最坏条件风险值约束等价的转化为半定规划约束。另外,这种基于半定规划的近似被用来和基于Bonferroni不等式的近似以及基于二阶锥优化(Second-Order Cone Programming,SOCP)的近似进行比较。数值算例的结果证实了基于半定规划(Semidefinite Programming, SDP)的近似方法更加灵活,没那么保守,比基于Bonferroni不等式和基于SOCP的近似有更优的目标函数值
(5)基于元胞传输模型,通过利用最坏条件风险值,我们建立了不确定需求下的双层动态交通网络设计模型,其中不确定需求的概率分布被假定属于由几种已知概率分布所组成的多面体集合。基于下层的用户最优的最优性条件,我们将双层动态交通网络设计模型等价转化为带互补约束的数学规划模型。一种松弛的算法被用来求解转化后的模型,通过数值实验的结果证实了模型和算法的有效性。
ABSTRACT:Network design problem is at the core of one of comprehensive urban planning, and is also a basic problem that contributes to the long-term, rapid, harmonious and stable development of the urban economics. Currently, with the rapid development of cities, urban traffic congestion is becoming increasingly serious, the contradiction of supplying and demand of traffic has risen, and congestion alleviation and prevention have become urgent tasks for urban development. On the other hand, a lot of uncertainties surround the urban traffic system. If these uncertainties are ignored in the network design problem, it may lead to the more serious traffic congestion. Therefore, the network design problem under uncertainty research is essential. Research methods in the network design problem under uncertainty are stochastic programming and robust optimization. The stochastic programming approach needs to assume the probability distribution of the uncertain parameter in advance. However, in reality, this assumptive distribution may be unavailable(inaccurate) as we may have no insufficient data to calibrate this distribution, and the robust optimization approach doesn't require known probability distribution of the uncertain parameter. Therefore, it has more practical meaning by using the robust optimization approach to study network design problem under uncertainty.
The aim of this dissertation is to adopt the robust optimization approach to study network design problem under uncertainty, and develops the models and algorithms of the network design problem under uncertainty. The main contents of the dissertation are summarized as follows:
(1) We consider the continuous network design problem under demand uncertainty based on user equilibrium by using the nolinear robust optimization approach, in which the uncertain demand is assumed to belong to an ellipsoid set. We reformulate the robust counterpart (RC) of continuous network design problem under demand uncertainty as a series of mathemtical progrms with complementarity problem(MPCC) by adopting the idea of robust opitmization and the sensitivity analysis, and applies a relax algorithm to solve this series of MPCC. In addition, we compare it with the robust counterpart (RC) by proposed Yin and Lawphonganich[1]. The number example results show that the our RC model is more flexible than, and no less conservative than the robust solution obtained by Yin and Lawphonganich [1].
(2) We develop the robust reliable traffic equilibrium problem under demand uncertainty, in which the model doesn't require known the probability distribution of the uncertain demand, while only needs to know its first-m moments. The robust percentile travel time (RPTT) and robust mean-excess travel time (RMETT) are defined by adopting the Worst-case Value-at-Risk (WVaR) and Worst-case Conditional Value-at-Risk measures (WCVaR)[2], and then the RPTT and RMETT are equal under general distribution are demonstrated. According to the two equivalent travel time, the robust percentile user equilibrium (RPUE) or robust mean-excess traffic equilibrium (RMETE) is proposed. The RPUE or RMETE model is formulated as a nonlinear complementarity problem (NCP) and the equivalence and existence of the equilibrium solution are demonstarted, and then is solved by a solution method based on the gap function. Furthermore, the network design problem with the distributionally robust joint chance constraints is developed based on the proposed equilibrium model. The distributionally robust joint chance constraints of network design problem are approximated as the nonlinear constraints based on the Bonferroni's inequalitiy. The active set algorithm are developed to solve the approximated model. The numerical example is depicted to examine the proposed network design model and the solution algorithm.
(3) The cell transmission model-based (CTM)[3-4]single level dynamic network design problem under demand uncertainty is developed by applying the adjustable robust optimization approach. The model assumes that the demand belongs to a polyhedral set. The affinely adjustable robust counterpart (AARC) of the CTM-based single level dynamic network design problem under demand uncertainty is formulated through the affine decision rule and the duality of liner programming, and is compared with the robust counterpart of it. Numerical example shows that the adjustable robust solution of the CTM-based single level dynamic network design problem under demand uncertainty is more flexible than robust solution, and no less conservative than robust solution.(4) We propose a distributionally robust joint chance-constrained optimization model for the CTM-based single level dynamic network design problem under demand uncertainty, where the probability distribution of uncertain demand is unknown and only the mean and variance of uncertain demand are known. The distributionally robust joint chance constraints of the model are approximated as the Worst-Case Conditional Value-at-Risk (WCVaR) constraints, and then the Worst-Case Conditional Value-at-Risk (WCVaR) constraints are equivalently reformulated as the semidefinite programming (SDP) constraints. The SDP-based approximation are compared with the two other approximation approches based on the Bonferroni's inequalities and second order cone programming (SOCP). Numerical experiment is conducted to demonstrate that the SDP-based approximation is more flexible, no less conservative, superior to Bonferroni's inequality-based approximation and SOCP-based approximation.
(5) The CTM-based two-level dynamic network design model under demand uncertainty is formulated by applying the WCVaR, in which the probability distribution of uncertain demand is assumed to belong to a polyhedral set that the known mean and variance of possible probability distribution composed. The CTM-based two-level dynamic network design model is equivalently reformulated as the mathematical programs with complementarity constraint (MPCC) based on the KKT condition of the low-level user optimal dynamic network design problem. A relaxed algorithm is developed to solve the MPCC. Numerical example is depicted to demonstrate the proposed model and the solution algorithm.
本论文基于鲁棒优化的方法,研究不确定的城市交通网络设计问题,探讨不确定交通网络设计问题的建模和求解算法。具体来讲,本论文研究工作主要有以下几个方面：
(1)运用鲁棒非线性优化方法研究了基于用户均衡下不确定需求的连续交通网络设计问题,其中不确定需求属于一个椭球集合。通过运用鲁棒优化的思想和灵敏度分析的方法,我们将连续交通网络设计问题的鲁棒对应(Robust Counterpart,RC)模型转化为一系列带互补约束的数学规划问题(Mathematical Programms with Complementarity Problem, MPCC),并运用一种松弛算法求解这一系列的MPCC。另外,我们将它和Yin和Lawphongpanich[1]提出的鲁棒对应模型进行了比较。数值实验的结果表明,我们提出的鲁棒对应模型比Yin和Lawphongpanich[1]的鲁棒对应模型更加灵活,没那么保守。
(2)探讨了不确定需求下的鲁棒可靠性用户均衡模型,其中模型并不要求知道不确定需求的准确的概率分布,而仅需知道它的前m阶矩。基于最坏风险值(Worst-case Value-at-Risk, WVaR)和最坏条件风险值(Worst-Case Conditional Value-at-Risk, WCVaR)[2],我们定义了鲁棒分位走行时问和鲁棒均值-超量走行时问,并证明了两种走行时间在需求一般分布情况下是等价的。基于这种等价的走行时问提出了鲁棒分位用户均衡(鲁棒均值-超量交通均衡)模型,模型被表示为一个非线性互补问题(Nonlinear Complementarity Problem, NCP),并证明了模型的等价性和解的存在性。然后一种基于间隙函数的方法被用来求解这个非线性互补问题。基于提出的均衡模型,我们进一步研究了带分布式鲁棒联合机会约束的连续交通网络设计模型,通过利用Bonferroni不等式,模型中的分布式鲁棒联合机会约束被近似为非线性约束,我们应用积极集的算法求解近似后的模型,数值实验的结果验证了提出的模型和算法的有效性。
(3)运用可调整的鲁棒优化方法研究了基于元胞传输模型(Cell Transimission Model, CTM)[3-4]的单层动态交通网络设计模型,其中不确定需求被假定属于一个多面体集合。通过运用仿射决策准则和线性规划的对偶,我们构建了相应的仿射可调整的鲁棒对应模型,同时将它与传统的鲁棒对应模型进行了比较,数值算例的结果显示,可调整的鲁棒对应模型比传统的鲁棒对应模型更加灵活。
(4)基于元胞传输模型,研究了单层动态交通网络设计问题的分布式鲁棒联合机会约束模型,模型假定OD需求的概率分布是未知的,仅知道它的期望和方差。首先,我们将模型中的分布式鲁棒联合机会约束近似为最坏条件风险值约束,然后,利用锥对偶原理,将最坏条件风险值约束等价的转化为半定规划约束。另外,这种基于半定规划的近似被用来和基于Bonferroni不等式的近似以及基于二阶锥优化(Second-Order Cone Programming,SOCP)的近似进行比较。数值算例的结果证实了基于半定规划(Semidefinite Programming, SDP)的近似方法更加灵活,没那么保守,比基于Bonferroni不等式和基于SOCP的近似有更优的目标函数值
(5)基于元胞传输模型,通过利用最坏条件风险值,我们建立了不确定需求下的双层动态交通网络设计模型,其中不确定需求的概率分布被假定属于由几种已知概率分布所组成的多面体集合。基于下层的用户最优的最优性条件,我们将双层动态交通网络设计模型等价转化为带互补约束的数学规划模型。一种松弛的算法被用来求解转化后的模型,通过数值实验的结果证实了模型和算法的有效性。
ABSTRACT:Network design problem is at the core of one of comprehensive urban planning, and is also a basic problem that contributes to the long-term, rapid, harmonious and stable development of the urban economics. Currently, with the rapid development of cities, urban traffic congestion is becoming increasingly serious, the contradiction of supplying and demand of traffic has risen, and congestion alleviation and prevention have become urgent tasks for urban development. On the other hand, a lot of uncertainties surround the urban traffic system. If these uncertainties are ignored in the network design problem, it may lead to the more serious traffic congestion. Therefore, the network design problem under uncertainty research is essential. Research methods in the network design problem under uncertainty are stochastic programming and robust optimization. The stochastic programming approach needs to assume the probability distribution of the uncertain parameter in advance. However, in reality, this assumptive distribution may be unavailable(inaccurate) as we may have no insufficient data to calibrate this distribution, and the robust optimization approach doesn't require known probability distribution of the uncertain parameter. Therefore, it has more practical meaning by using the robust optimization approach to study network design problem under uncertainty.
The aim of this dissertation is to adopt the robust optimization approach to study network design problem under uncertainty, and develops the models and algorithms of the network design problem under uncertainty. The main contents of the dissertation are summarized as follows:
(1) We consider the continuous network design problem under demand uncertainty based on user equilibrium by using the nolinear robust optimization approach, in which the uncertain demand is assumed to belong to an ellipsoid set. We reformulate the robust counterpart (RC) of continuous network design problem under demand uncertainty as a series of mathemtical progrms with complementarity problem(MPCC) by adopting the idea of robust opitmization and the sensitivity analysis, and applies a relax algorithm to solve this series of MPCC. In addition, we compare it with the robust counterpart (RC) by proposed Yin and Lawphonganich[1]. The number example results show that the our RC model is more flexible than, and no less conservative than the robust solution obtained by Yin and Lawphonganich [1].
(2) We develop the robust reliable traffic equilibrium problem under demand uncertainty, in which the model doesn't require known the probability distribution of the uncertain demand, while only needs to know its first-m moments. The robust percentile travel time (RPTT) and robust mean-excess travel time (RMETT) are defined by adopting the Worst-case Value-at-Risk (WVaR) and Worst-case Conditional Value-at-Risk measures (WCVaR)[2], and then the RPTT and RMETT are equal under general distribution are demonstrated. According to the two equivalent travel time, the robust percentile user equilibrium (RPUE) or robust mean-excess traffic equilibrium (RMETE) is proposed. The RPUE or RMETE model is formulated as a nonlinear complementarity problem (NCP) and the equivalence and existence of the equilibrium solution are demonstarted, and then is solved by a solution method based on the gap function. Furthermore, the network design problem with the distributionally robust joint chance constraints is developed based on the proposed equilibrium model. The distributionally robust joint chance constraints of network design problem are approximated as the nonlinear constraints based on the Bonferroni's inequalitiy. The active set algorithm are developed to solve the approximated model. The numerical example is depicted to examine the proposed network design model and the solution algorithm.
(3) The cell transmission model-based (CTM)[3-4]single level dynamic network design problem under demand uncertainty is developed by applying the adjustable robust optimization approach. The model assumes that the demand belongs to a polyhedral set. The affinely adjustable robust counterpart (AARC) of the CTM-based single level dynamic network design problem under demand uncertainty is formulated through the affine decision rule and the duality of liner programming, and is compared with the robust counterpart of it. Numerical example shows that the adjustable robust solution of the CTM-based single level dynamic network design problem under demand uncertainty is more flexible than robust solution, and no less conservative than robust solution.(4) We propose a distributionally robust joint chance-constrained optimization model for the CTM-based single level dynamic network design problem under demand uncertainty, where the probability distribution of uncertain demand is unknown and only the mean and variance of uncertain demand are known. The distributionally robust joint chance constraints of the model are approximated as the Worst-Case Conditional Value-at-Risk (WCVaR) constraints, and then the Worst-Case Conditional Value-at-Risk (WCVaR) constraints are equivalently reformulated as the semidefinite programming (SDP) constraints. The SDP-based approximation are compared with the two other approximation approches based on the Bonferroni's inequalities and second order cone programming (SOCP). Numerical experiment is conducted to demonstrate that the SDP-based approximation is more flexible, no less conservative, superior to Bonferroni's inequality-based approximation and SOCP-based approximation.
(5) The CTM-based two-level dynamic network design model under demand uncertainty is formulated by applying the WCVaR, in which the probability distribution of uncertain demand is assumed to belong to a polyhedral set that the known mean and variance of possible probability distribution composed. The CTM-based two-level dynamic network design model is equivalently reformulated as the mathematical programs with complementarity constraint (MPCC) based on the KKT condition of the low-level user optimal dynamic network design problem. A relaxed algorithm is developed to solve the MPCC. Numerical example is depicted to demonstrate the proposed model and the solution algorithm.
引文
[1]Yin, Y., Lawphongpanich, S. A robust approach to continuous network design problems with demand uncertainty [C]. Proceedings of 17th International Symposium of Transportation and Traffic Theory (R.E. Allsop, M.G.H. Bell and B.G. Heydecker, Editors),2007:111-126.
[2]Cerbakova, J. Worst-Case VaR and CVaR [C]. Operations Research Proceedings,2006, 2005(22):817-822.
[3]Daganzo, C.F. The cell transmission model:a dynamic representation of highway traffic consistent with the hydrodynamic theory[J]. Transportation Research Part B,1994,28(4):269-287.
[4]Daganzo, C.F. The cell transmission model:part Ⅱ:network traffic[J]. Transportation Research Part B,1995,29(2):79-93.
[5]高自友.现代城市综合交通网络设计的优化模型、求解算法及程序设计[R].北京：北京交通大学系统科学研究所技术报告,2002.
[6]Yang, H., Bell, M.G.H. Models and algorithms for road network design:a review and some new developments [J]. Transportation Reviews,199818(3):257-278.
[7]Magnanti, T.L., Wong, R.T. Network design and transportation planning:models and algorithms [J]. Transportation Science,1984,18(1):1-15.
[8]Boyce, D.E. Urban transportation network equilibrium and design models:recent achievements and future prospects [J]. Environment and Planning A,1984,16(11):1445-1474.
[9]Friesz, T.L., Harker, P.T. Properties of the iterative optimization-equilibrium algorithm [J]. Civil Engineering Systems,1985,2 (3):142-154.
[10]Minoux, M. Network synthesis and optimum network design problem:models, solution methods and application [J]. Network,1989,19(3):313-360.
[11]Askura, Y. Road network reliability caused by daily fluctuation of traffic flow [J]. European Transport, Highway& Planning,1991,19:73-84.
[12]Clark, S.D., Watling, D. Modeling network travel time reliability under stochastic demand [J]. Transportation Research Part B,2005,39(2):119-140.
[13]Chen, A., Yang, H., Lo, H.K., Tang, W.H. Capacity reliability of a road network:an assessment methodology and numerical results. Transportation Research Part B,2002,36(3): 225-252.
[14]Lo, H.K., Luo, X.W., Siu, B.W.Y. Degradable transportation network:travel time budget of travelers with heterogeneous risk aversion[J]. Transportation Research Part B,2006,40(9):792-806.
[15]Al-Deek, H., Eman, E.B. New methodology for estimating reliability in transportation networks with degraded link capacities [J]. Journal of Intelligent Transportation System,2006,10(3): 117-129.
[16]Waller, S.T., Schofer, J.L., Ziliaskopoulos, A.K.2001. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record,2001,1771:69-74.
[17]Chen, A., Zhou, Z., Chootinan, P., Ryu, S., Yang, C, Wong, S.C. Transportation network design problem under uncertainty:a review and new developments [J]. Transportation Review, 2011b,31(6):743-768.
[18]Yin, Y., Idea, H. Optimal improvement scheme for network reliability [J]. Transportation Research Record,2002,1783:1-6.
[19]Chen, A., Yang, C. Stochastic transportation network design problem with spatial equity constraint [J]. Transportation Research Record,2004,1882:97-104.
[20]Chen, A., Subprasom, K. Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty [J]. Transportation Research Part A,2007, 41(6):537-558.
[21]Chen, A., Kim, J., Lee, S., Kim, Y. Stochastic multi-objective models for network design problems [J]. Expert systems and Application,2010,37(2):1608-1619.
[22]Chow, J.Y.J., Regan, A.C., Network-based real option models [J]. Transportation Research Part B,2011,45(4):682-695.
[23]Xu, H., Lou, Y., Yin, Y, Zhou, J. A prospect-based user equilibrium model with endogenous with reference points and its application in congestion pricing [J]. Transportation Research Part B, 2011,45(2):311-328.
[24]许良,基于可靠性分析的城市道路交通网络设计问题研究[D].北京：北京交通大学硕十学位论文,2006.
[25]孙强,王庆云,高咏玲.不确定需求下的多阶段区域综合交通网络设计的双层规划模型[J].交通运输系统工程与信息,2011,11(6)：111-116.
[26]蒋洋,基于结构和随机特性的交通网络设计问题的研究[D].北京：北京交通大学硕十学位论文,2011.
[27]Chen, A., Xu, X.D. Goal programming approach to solving network design problem with multiple objectives and demand uncertainty [J]. Expert systems and Application,2012,39(4): 4160-4170.
[28]Waller, S.T., Ziliaskopoulos, A.K. Stochastic dynamic network design problem[J]. Transportation Research Record,2001,1771:106-113.
[29]Ukkusuri, S.V, Karoonsoontawong, A., Waller, S.T. A stochastic dynamic user network design model accounting for demand uncertainty [C]. Proceedings of the Interational Conference of Transportation System Planning and Operations(TRANSPO), Madras, India, Feb,2004,18-20.
[30]Ukkusuri, S.V., Waller, S.T. Linear programming models for the user and system optimal dynamic network design problem:formulations, comparisons and extensions [J]. Networks and Spatial Economics,2008,8(4):383-406.
[31]Karoonsoontawong, A., Waller, S.T. Robust dynamics continuous network design problem [J]. Transportation Research Record,2007,2029:58-71.
[32]Ukkusuri, S.V., Patil, G. Multi-period transportation network design under demand uncertainty [J]. Transportation Research Part B,2009,43(6):625-642.
[33]Markowtiz, H. Mean-variance analysis in portfolio choice and capital markets[M]. (Pennsylvania:New Hope),1927.
[34]Chen, A., Subprasom, K., Ji, Z. Mean-variance model for the build-operate-transfer scheme under demand uncertainty [J]. Transportation Research Record,2003,1857:93-101.
[35]Chen, A., Subprasom, K., Ji, Z. A simulation-based multi-objective genetic algorithm (SMOGA) for build-operate-transfer network design problem [J]. Optimization and Engineering Journal,2006,7(3):225-247.
[36]Ukkusuri, S.V., Mathew, T.V., Waller, S.T. Robust transportation network design under demand uncertainty [J]. Computer-Aided Civil and Infrastructure Engineering,2007,22(1):6-18.
[37]Li, H., Bliemer, M.C.J., Bovy, P.H.L. Network reliability-based optimal toll design [J]. Journal of Advanced Transportation,2008,42(3):311-332.
[38]Gardner, L.M., Unnikrishnan, A., Waller, S.T. Robust pricing of transportation networks under demand uncertainty [J]. Transportation Research Record,2008,2085:21-30
[39]Sharma, S., Ukkusui, S.V., Mathew, T.V., A Pareto optimal multi-objective optimization for the robust transportation network design problem [J]. Transportation Research Record,2009,2090: 95-104
[40]潘艳荣,邓卫,2008.考虑需求不确定性的交通网络设计[J].交通运输工程学报,8(6)：82-87.
[41]卞长志.需求不确定的离散交通网络设计模型与算法[D].北京：清华大学博士学位论文,2009.
[42]Yin, Y., Madanat, S.M., Lu, X. Robust improvement schemes for road networks under demand uncertainty [J]. European Journal of Operational Research,2009,198(2):470-479.
[43]Ng, M.W., Waller, S.T. Reliable system optimal network design:a convex mean-variance type model with implicit chance constraints [J]. Transportation Research Record,2009,2090:68-74.
[44]Sumalee, A., Luathep, P., Lam, W.H.K., Connors, R.D. Evaluation and design of transport network capacity under demand uncertainty [J]. Transportation Research Record,2009,2090:17-28.
[45]Karoonsoontawong, A., Waller, S.T. Integrated network capacity expansion and traffic signal optimization problem:robust bi-level dynamic formulation [J]. Networks and Spatial Economics, 2010,10(4):525-550.
[46]Charnes, A., Cooper, W.W., Symonds, G.H. Cost horizons and certainty equivalents:an approach to stochastic programming of heating oil [J]. Management Science,1958,4(3):235-263
[47]Lo, H.K., Tung, Y.K. Network with degradable links:capacity analysis and design [J]. Transportation Research Part B,2003,37(4):345-363.
[48]许良,高自友.基于路段能力可靠性的城市交通网络设计[J].中国公路学学报,2006,19(2)：86-90.
[49]Dimitriou, L., Stahopoulos, A. Reliable stochastic design of road network systems [J]. International Journal of Industrial and System Engineering,2008,3(5):549-574.
[50]Ji, Z., Kim, Y.S., Chen, A. Multi-objective a-reliable path finding in stochastic networks with correlated link costs:A simulation-based multi-objective genetic algorithm approach (SMOGA) [J]. Experts Systems and Applications,2011,38(3):1515-1528.
[51]许项东.需求不确定环境下的道路网络均衡模型与系统优化[D].南京：东南大学博士学位论文,2012.
[52]Chootinan, P., Wong, S.C., Chen, A. A reliability-based network design problem [J]. Journal of Advanced Transportation,2005,39(3):247-270
[53]Yim, K.K.W., Wong, S.C., Chen, A., Wong, C.K., Lam, W.H.K. A reliability-based land use and transportation optimization model [J]. Transportation Research Part C,2011,19(2):351-362
[54]Chen, A., Chootinan, P., Wong, S. C. New reserve capacity model of a signaled-controlled road network [J]. Transportation Research Record,2006,1964:35-41
[55]Sumalee, A., Watling, D. P., Nakayama, S. Reliable network design problem:the case with uncertain demand and total travel time reliability [J]. Transportation Research Record,2006,1964: 81-90.
[56]Larsen, N., Mausser, H., Uryasev, S. Algorithms for optimization Value-at-Risk[C]. In. Pardolos P, Tsitsiringos. (Ed.) Financial Engineering, E-commerce and Supply Chain,2002, pp.19-46.
[57]Rockafellar, R.T., Uryasev, S. Optimization of Conditional Value-at-Risk. Journal of RiskfJ], 2000,2(3):21-41.
[58]Rockafellar, R.T., Uryasev, S. Conditional Value-at-Risk for general loss distributions [J]. Journal of Banking and Finance,2002,26(7):1443-1471.
[59]Chen, A., Kim, J., Zhou, Z., Chootinan, P. An alpha reliable network design problem [J]. Transportation Research Record,2007,2029:49-57.
[60]Chen, A., Ryu, S., Yang, C., Wong, S.C. Alpha reliable network design with multiple objectives and demand uncertainty[C]. Paper presented at the 89th Annual Meeting of the Transportation Research Board, Washington, DC, USA,.2010.
[61]Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Thinking coherently [J]. Risk,1997,10 (11): 68-71.
[62]Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Coherent measures of risk [J]. Mathematical Finance,1999,9(3):203-228.
[63]Hellman, F. Towards the solution of large-scale and stochastic traffic network design problems [D]. Master's Thesis, Uppsala University, Uppsala, Sweden,2010.
[64]Bazaraa, M.S., Sherali, H.D., Shetty, C.M. Nonlinear Programming:Theory and Algorithms, Second Edition[M]. John Wiley & Sons, New York, NY,1993.
[65]Yin, Y., Lawphongpanich, S., Lou, Y. Estimating investment requirement for maintaining and improving highway systems[J]. Transportation Research A,2008,16(2):199-211
[66]Yin, Y. Robust optimal signal timing. Transportation Research Part B,2008,42 (10): 911-924.
[67]Lu, Y. Robust transportation network design under user equilibrium[D]. Master's Thesis MIT, Cambridge, MA,2007.
[68]Lou, Y., Yin, Y., Lawphongpainch, S. A robust approach to discrete network design with demand uncertainty [J]. Transportation Research Board,2009,2090:86-94.
[69]Zhang, L.H., Lawphongpanich, S., Yin, Y. An active set algorithm for discrete network design problems[C]. The 18th International Symposium on Transportation and Traffic Theory, Hong Kong, Springer. In:Lam, W.H.K., Wong, S.C., Lo, H.K.,2009. Transportation and Traffic Theory 2009:Golden Jubilee, pp.283-300.
[70]Ban, X., Lu, S., Ferris, M., Liu, H. Risk-averse second-best toll pricing[C]. Proceedings 18th International Symposium Transportation and Traffic System,2009,197-218.
[71]Lou, Y., Yin, Y., Lawphongpainch, S. Robust congesting pricing under boundedly rational user equilibrium [J]. Transportation Research Part B,2010,44(1):15-28.
[72]Chung, B.D., Yao, T, Xie, C., Thorsen, A. Robust optimization model for a dynamic design problem under demand uncertainty [J]. Networks and Spatial Economics,2011,11(2):371-389
[73]Yao, T., Mandala, R.S., Chung, B.D. Evacuation transportation planning under uncertainty:a robust optimization approach[J]. Networks and Spatial Economics,2009,9(2):171-189.
[74]Ben-Tal, A., Chung, B.D., Mandala, S.R., Yao, T. Robust optimization for emergency logistic planning:Risk mitigation in humanitarian relief supply chains [J]. Transportation Research Part B, 2011,45(8):1177-1189.
[75]Chung, B.D., Yao, T., Friesz, T.L., Liu H. Dynamic congestion pricing with demand uncertainty:A robust optimization approach [J]. Transportation Research Part B,2012,46(10): 1504-1518
[76]Houska, B., Diehl, M. Nonlinear robust optimization via sequential convex bilevel programming [J]. Mathematical Programming,2013,142(1-2),539-577.
[77]Ben-Tal, A., Goryashko, E., Guslitzer, A., Nemirovski, A. Adjustable robust solutions of uncertain linear programs [J]. Mathematical Programming,2004,99 (2):351-376.
[78]Zymler, S., Kuhn, D., Rustem, B. Distributionally robust joint chance constraints with second-order moment information [J]. Mathematical Programming,2013,137(1-2):167-198.
[79]Abdulaal, M.S., LeBlanc, L.J. Continuous equilibrium network design models[J]. Transportation Research Part B,1979,13(1):19-32.
[80]Davis, G. A. Exact local solution of the continuous network design problem via stochastic user equilibrium assignment [J]. Transportation Research Part B,1994,28(1):61-75
[81]Maher, M.J., Zhang, X., Van Vliet, D. A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows [J]. Transportation Research Part B,2001,35(1):23-40.
[82]Meng, Q., Yang, H., Bell, M.G.H.,2001. An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem [J]. Transportation Research Part B,2001,35(1):83-105.
[83]Meng, Q., Yang, H. Benefit distribution and equality in road network design [J]. Transportation Research Part B,2002,36(1):19-35.
[84]Gao, Z.Y., Sun, H.J., Shan, L.L. A continuous equilibrium network design model and algorithm for transit system[J]. Transportation Research Part B,2004,38(3):235-250.
[85]Chen, M., Bernstein, D.H. Solving the toll design problem with multiple user groups [J]. Transportation Research Part B,2004,38(1):61-79.
[86]Wang, D.Z.W., Lo, H.K. Global optimum of the linearized network design problem with equilibrium flows[J]. Transportation Research Part B,2010,44(4):482-492.
[87]Luathep, P., Sumalee, A., Lam, W.H.K., Li, Z.C., Lo, H.K. Global optimization method for mixed transportation network design problem:A mixed-integer linear programming approach[J]. Transportation Research Part B,2011,45(5):808-827.
[88]Wong, S.C., Yang, H, Reserve capacity of a signal-controlled road network [J]. Transportation Research Part B,1997,31(5):397-402.
[89]Gao, Z.Y., Song, Y.F. A reserve capacity model of optimal signal control with user-equilibrium route choice[J]. Transportation Research Part B,2002,36(4):313-323.
[90]Ceylan, H., Bell, M.G.H. Reserve capacity for a road network under optimized fixed time traffic signal control [J]. Journal of Intelligent Transportation Systems:Technology, Planning, and Operations,2004,8(2):87-99
[91]Chiou, S.W. Reserve capacity of signal-controlled road network. Applied Mathematics and Computation,2007,190(2):1602-1611.
[92]Chen, A., Kasikitwiwat, P. Modeling capacity flexibility of transportation networks [J]. Transportation Research Part A,2011,45(2):105-117.
[93]Lam, W.H.K., Tam, M.L., Bell, M.G.H. Optimal road tolls and parking charges for balancing the demand and supply of road transport facilities [J]. In:Taylor, M.A.P. (Ed.), Transportation and Traffic Theory in the 21st Century:Proceedings of the 15th International Symposium on Transportation and Traffic Theory. Elsevier Science, New York,2002,561-582.
[94]Szeto, W.Y., Lo, H. Strategies for road network design over time:robustness under uncertainty[J]. Transportmetrica,2005,1(1):47-63.
[95]Szeto, W.Y., Lo, H. Time-dependent transport network improvement and tolling strategies [J]. Transportation Research Part A,2008,42(2):376-391.
[96]Yang, H., Xiao, F., Huang, H.J. Private road competition and equilibrium with traffic equilibrium constraints [J]. Journal of Advanced Transportation,2009,43(1):21-45.
[97]Lo, H.K., Szeto, W.Y. Time-dependent transport network design under cost-recovery [J]. Transportation Research Part B,2009,43(1):142-158.
[98]Heydecker, B.G. Dynamic equilibrium network design[C]. In:Taylor, M.A.P. (Ed.), Transportation and Traffic Theory in the 21st Century:Proceedings of the 15th International Symposium on Transportation and Traffic Theory. Elsevier Science, New York,2002, pp.349-370.
[99]Szeto, W.Y., Lo, H. Transportation network improvement and tolling strategies:The issue of intergeneration equity[J]. Transportation Research Part A,2006,40(3):227-243
[100]Wu, D., Yin, Y., Lawphongpanich, S. Optimal selection of build-operate-transfer projects on transportation networks[J]. Transportation Research PartJB,2011,45(10):1699-1709.
[101]Friesz, T.L., Harker, P.T. Multicriteria spatial price equilibrium network design:theory and computational results[J]. Transportation Research Part B,1983,17(5):411-426
[102]Friesz, T.L., Anandalingam, G., Mehta, N.J., Nam, K., Shah, S.J., Tobin, R.L. The multiobjective equilibrium network design problem revisited:a simulated annealing approach. European Journal of Operational Research,1993,65(1):44-57.
[103]Gartner, N.H., Stamatiadis, C. Combining traffic assignment and adaptive control in a dynamic traffic management system[C]. In:Becker, U., Schnieder, E. (Eds.), Control in Transportation Systems 2000:Proceedings Volume from the 9th IFAC Symposium, vol.1. Elsevier Science, UK,2001, pp.281-286.
[104]Sohn, K. Multi-objective optimization of a road diet network design [J]. Transportation Research Part A,2011,45(6):499-511.
[105]Wardrop, J.G. Some theoretical aspects of road traffic research[J]. Proceedings of the Institution of Civil Engineers,1952,1(2):325-378.
[106]Sheffi, Y. Transportation Network:Equilibrium analysis with mathematical programming methods [M]. Prentice-Hall, Incorporated, Englewood Cliffs, NJ,1985.
[107]Friesz, T.L. Transportation network equilibrium, design and aggregation:key development and research opportunities [J]. Transportation Research Part A,1985,19(5-6):413-427.
[108]Suwansirikul, C, Friesz, T.L. Equilibrium decomposed optimization:a heuristic for the continuous equilibrium network design problem [J]. Transportation Science,1987,21(4):254-263
[109]Ferrari, P. Road pricing and network equilibrium [J]. Transportation Research Part B,1995, 29(5):357-372.
[110]Ferrari, P. Road network toll pricing and social welfare. Transportation Research Part B,2002, 36(5):471-483.
[111]Yang, H., Lam, W.H.K. Optimal road tolls under conditions of queuing and congestion [J]. Transportation Research Part A,1996,30(5):319-332.
[112]Yang, H., Bell, M.G.H. Traffic restraint, road pricing and network equilibrium [J]. Transportation Research Part B,1997,31(4):303-314.
[113]Allosp, R.E. Some possibility for using traffic control to influence trip distribution and route choice[C]. Proceedings of the Sixteenth International Symposium on Transportation and Traflc Theory. Elsevier, Amsterdam,1974,345-374.
[114]Allsop, R.E., Charlesworth, J.A. Traffic in a signal-controlled road network:an example of different signal timings inducing different routes [J]. Traffic Engineering and Control,1977,18(5): 262-264.
[115]Cantarella, G.E., Improta, G., Sforza, A. Iterative procedure for equilibrium network traffic signal setting [J]. Transportation Research Part A,1991,25(5):241-249.
[116]Wong, S.C., Yang. C. An iterative group-based signal optimization scheme for traffic equilibrium networks [J]. Journal of Advanced Transportation,1999,33(2):201-217.
[117]Chiou, S.W. Optimization of area traffic control for equilibrium network flows. Transportation Science,1999,33(3):279-289.
[118]Yang, H., Yagar, S., Iida, Y., Asakura, Y. An algorithm for the inflow control problems on urban freeway networks with user optimal flows [J]. Transportation Research Part B,1994,28(2): 123-139.
[119]Yang, H., Yagar, S. Traffic assignment and traffic control in general freeway-arterial corridor systems [J]. Transportation Research Part B,1994,28(6):463-486
[120]Cho, H.J. Sensitivity analysis of equilibrium network flows and its application to development of solution methods for equilibrium network design problem [D]. PhD Dissertation. University of Pennsylvania, Philadelphia,1988.
[121]Friesz, T.L., Tobin, R.L., Cho, H.L., Metha, N.J. Sensitivity analysis based heuristic algorithms for mathematical programs with equilibrium constraints[J]. Mathematical Programming, 1990,48(2):265-284.
[122]Yang, H., Yagar, S. Traffic assignment and signal control in saturated road networks [J]. Transportation Research Part A,1995,29(2):125-139
[123]Yang, H. Sensitivity analysis for queuing equilibrium network flow and its application to traffic control [J]. Mathematical and Computer Modeling,1995,22(4-7):247-258.
[124]Yang, H. Sensitivity analysis for the elastic demand network equilibrium problem with application [J]. Transportation Research Part B,1997,31(1):55-70.
[125]Connors, R.D., Sumalee, A., Waiting, D.P. Sensitivity analysis of the variable demand probit stochastic user equilibrium with multiple user-classes [J]. Transportation Research Part B,2007, 41(6):593-615.
[126]Chiou, S.W. Bilevel programming for the continuous network design problem [J]. Transportation Research Part B,2005,39(4):361-383.
[127]Lawphongpanich, S., Yin, Y. Solving the Parteo-improving toll problem via manifold suboptimziation [J]. Transportation Research Part C,2010,18(2):234-246.
[128]Wu, D., Yin, Y., Lawphongpanich, S. Parteo-improve congestion pricing on multimodal transportation network [J]. European Journal of Operation Research,2011,210(3):660-669.
[129]Chiou, S.W. A subgradient optimization model for the continuous road network design problem [J]. Applied Mathematical Modeling,2009,33(3):1386-1396
[130]Yang, H., Zhang, X., Meng, Q. Modeling private highways with entry-exit based toll charges[J]. Transportation Research Part B,2004,38(3):463-486
[131]Gao, Z.Y., Sun, H.J., Zhang, H.Z. A globally convergent algorithm for transportation continuous network design problem [J]. Optimization and Enginnering,2007,8(3):241-257.
[132]Ban, X., Liu, H.X., Ferris, M., Ran, B. A general MPCC model and its solution algorithm for continuous network design problem [J]. Mathematical and Computer Modeling,2006,43(5-6): 493-505.
[133]Ban, X., Liu, H.X., Lu, J.G., Ferris, M. A decomposition scheme for continuous network design problem with asymmetric user equilibria[J]. Transportation Research Record,2006,1964: 185-192
[134]Asakura, Y., Sasaki, T. Formulation and feasibility test of optimal road network design model with endogenously determined travel demand[C]. Proceedings of the 5th World Conference on Transport Research, Yokohama, Japan, July,1990,351-365.
[135]Van Vuren, T., Van Vliet, D. Route Choice and Signal Control [M]. Newcastle upon Tyne, U.K.:Athenaeum Press,1992.
[136]Smith, M.J., Van Vuren, T. Traffic equilibrium with responsive traffic control [J]. Transportation Science,1993,27(2):118-132
[137]Koh, A. An adaptive differential evolution alogithm applied to highway network capacity optimization[C]. In:Avineri, E., Koppen, M., Dahal, K., Sunitiyoso, Y., Roy, R.,(Eds.), Application of Soft Computing:Update the state of Art,2009,211-220.
[138]Sun, Y., Turnquist, M.A. Investment in transportation network design under demand uncertainty:simulated annealing approach [J]. Transportation Research Record,2007,2039:67-74.
[139]Cree, N.D., Maher, M.J., Paechter, B. The continuous equilibrium optimal network design problem:a genetic approach [C]. Transportation Networks:Recent Methodological Advances, Selected Proceedings of the 4th EURO Transportation Meeting, Newcastle, UK,1998,163-174
[140]Yin, Y. Genetic-algorithm-based approach for bilevel programming models[J]. Journal of Transportation Engineering,2000,126(2):115-120.
[141]Sumalee, A. Multi-concentric optimal charging cordon design [J]. Transportmetrica,2007, 3(1):41-71.
[142]Mousko, K. A tabu-serch based approach for network design [C]. Ph.D. Dissertation. The University of Texas at Austin, Austin, TX,1992.
[143]Xu, T.Z, Wei, H., Hu, G.H Study on continuous network design problem using simulated annealing and genetic algorithm [J]. Expert system and Application,2009,36(2):1322-1328
[144]Friesz, T.L., Hsunjung, H.L., Tobin, R.L.., Metha, N.J., Anandalingam, G. A simlated annealing appraoch to the network design problem with variational inequility constaints[J]. Transporatation Science,1992,26(1):18-26.
[145]Li, C.M., Yang, H., Zhu, D.L., Meng, Q. A global opitmization for continuous network design problems [J]. Transportation Reserach Part B,2012,46(9):1144-1158.
[146]Marcotte, P.A. Network optimization with continuous control parameters [J]. Transportation Science,1983,17(2):181-197.
[147]Lawphongpanish, S., Hearn, D.W. An MPEC approach to second-best toll pricing [J]. Mathematical Programming,2004,101(1):33-55.
[148]Joseph, Y.J., Chow, P.E., Regan, A.C., Arkhipov, D.I.. A faster converging global heuristic for continuous network design using radial basis functions [C]. The 89th Transportation Research Board annual meeting,2010.
[149]LeBlanc, L.J. An algorithm for the discrete network design problem [J]. Transportation Science,1975,9(3):183-199.
[150]Poorzahedy, H., Turnquist, M.A. Approximate algorithms for the discrete network design problem [J]. Transportation Research Part B,1982,16(1):45-55.
[151]Chen, M, Alfa, A.S. A network design algorithm using a stochastic incremental traffic assignment problem [J]. Transportation Science,1991,25(3):215-224.
[152]Drezner, Z., Salhi, S. Using hybid metaheurisitics for the one-way and two-way network design problem [J]. Naval Research Logistic,2002,49(5):449-463.
[153]Drezner, Z., Wesolowsky, G.O. Selecting an optimum configuration of one-way and two-way routes [J]. Transportation Science,1997,31(4):386-394.
[154]Drener, Z., Wesolowsky, G.O. Network design:selecting and design of links and facility location [J]. Transportation Research Part A,2003,37(3):241-256.
[155]Farvareshhe, H., Sepehri, M.M. A branch and bound algorithm for bi-level discrete network design [J]. Networks and Spatial Economics,2013,13(1):67-106.
[156]Solanki, R.S., Gorti, J.K., Southworth, F. Using decomposition in large-scale highway network design with a quasi-optimization heuristic [J]. Transportation Research Part B,1998,32(2): 127-140.
[157]Farvareshhe, H., Sepehri, M.M. A single-level mixed integer linear formulation for a bi-level discrete problem [J]. Transportation Research Part E,2011,47(5):623-640.
[158]Gao, Z.Y., Wu, J.J., Sun, H.J. Solution algorithm for the bi-level discrete network design problem [J]. Transportation Research Part B,2005,39(6):479-495.
[159]Wang, S., Meng, Q., Yang, H. Global optimization methods for the discrete network design problem. Transportation Research Part B,2013,50:42-60.
[160]Poorzahedy, H., Rouhanni, O.M. Hybrid meta-heuristic algorithm for solving network design problem [J]. European Journal of Operations Research,2007,182(2):578-596.
[161]Poorzahedy, H., Abulghasemi. Application of ant system to network design problem [J]. Transportation,2005,32(3):251-273.
[162]Kim, B.J., Kim, W. An equilibrium network design model with a social cost function for multimodal networks [J]. Annual Regional Science,2006,40(3):473-491.
[163]Xiong, Y., Schneider, J.B., Transportation network design using a cumulative genetic algorithm and neutral network [J]. Transportation Research Record,1995,1364:37-44.
[164]Lee, C.K., Yang, K.I. Network design of one-way street with simulated annealing [J]. Papers in Regional Science,1994,73(2):119-134
[165]Ekstrom, J., Sumalee, A., Lo, H.K. Optimizing toll location and levels using a mixed integer linear approximation approach [J]. Transportation Research Part B,2012,46(7):834-854.
[166]Zhang, H.Z., Gao, Z.Y. Bilevel programming model and solution methods for mixed network design problem[J]. Journal of Systems Science and Complexity,2009,22(3):446-459
[167]Cantarella, G.E., Vitetta, G. The multi-criteria road network design problem in an urban area. Transportation,2006,33(6):567-588.
[168]Sun, Y., Song, R., He, S., Chen, Q. Mixed transportation network design based immune clone anneal algorithm [J]. Journal of Transportation System Engineering and Information Technology, 2009,9(3):103-108
[169]Lin, D.Y., Karoonsoontawong, A., Waller S.T.. A Dantzig-Wolfe decomposition based heuristic scheme for bi-level dynamic network problem [J]. Networks and Spatial Economics,2011, 11(1):101-126.
[170]Ziliaskopoulos, A.K. A linear programming model for the single destination system optimum dynamic traffic assignment problem [J]. Transportation Science,2000,34(1):37-49.
[171]Lighthill, M.H., Whitham, G.B. On kinematics wave:II A theory of traffic flow on long crowed roads [J]. Proceedings of the Royal Society, London, Series A,,1955,229(1178):317-345.
[172]Waller, S.T., Mousokos, K.C., Kamaryiannis, D., Ziliaskopoloulos, A.K., A linear model for the continuous network design problem [J]. Computer-Aided Civil Infrastructure Engineering,2006, 21(5):334-345.
[173]Ukkusuri, S.V. Linear programs for the user optimum dynamic traffic assignment problem [D]. Master Thesis, University of Illinois at Urbana-Champaign,2002.
[174]Karoonsoontawong, A., Waller, S.T. Dynamic network design problem:linear bi-level programming and metaheuristic approaches [J]. Transportation Research Record,2006,1964: 104-117.
[175]Lin, D.Y., Unnikrishnan, A., Waller S.T. A genteic algorithm for bi-level linear programming dynamic network design problem [J]. Transportation Letters:the international journal of transportation research,2009,1(4):281-294.
[176]Lin, D.Y. A dual variable approximation-based descent method for a bi-level continuous dynamic network design problem [J]. Computer-Aided Civil Infrastructure Engineering,2011,26(8): 581-594.
[177]Mulevy, J.M., Vanderbei, R.J., Zenios, S.A. Robust optimization of large-scale systems [J]. Operations Research,1995,43(2):264-281.
[178]Soyster, L.A. Convex programming with set inclusive constraints and applications to inexact linear programming [J]. Operations Research,1973,21(5):1154-1157.
[179]Bertsimas, D., Sim, M. Robust discrete optimization and network flow [J]. Mathematical Programming,2003,98(1/3):49-71
[180]Bertsimas, D., Sim, M. The price of robustness [J]. Operations Research,,2004,52(1):35-53
[181]Atamturk, A. Strong formulation of robust mixed 0-1 programming [J]. Mathematical Programming,2006,108(2):235-250
[182]Ben-Tal, A., Nemirovski, A.,1998. Robust convex optimization [J]. Mathematics of Operations Research,23(4):769-805
[183]Ben-Tal, A., Nemirovski, A.,1999. Robust solutions of uncertain linear programs [J]. Operation Research Letter,25(1):1-13
[184]Ben-Tal, A., Nemirovski, A.,2000. Robust solutions of linear programming problems contaminated with uncertain data [J]. Mathematical Programming,88(3):411-424
[185]EL Ghaoui, L., Oustry, F., Lebert, H. Robust solutions to uncertain semiefinite programming [J]. SIAM Journal on Optimization,1998,9(1):35-52.
[186]Ben-Tal, A., EL Ghaoui, L., Nemirovski, A. Robust semidefinite programming[C]. In: Wolkowicz, H., Saignal, R., Vandeberghe, L., eds., Handbook on semidefinte programming, Kluwer Academic Publishers,2000,303-328
[187]Ben-Tal, A., Nemirovski, A. Robust optimization-methodology and application [J]. Mathematical Programming,2002,92(3):453-480.
[188]Diehl, M., Bock, H.G., Kostina, E. An approximation technique for robust nonlinear optimization [J]. Mathematical Programming,2006,107(1-2):213-230
[189]Zhang, Y. General robust-optimization formulation for nonlinear programming [J]. Journal of Optimization Theory and Applications,2007,132(1):111-124
[190]Takeda, A., Taguchi, S., Tutiincu, R.H. Adjustable Robust Optimization Models for a nonlinear two-period system [J]. Journal of Optimization Theory and Application,2008,136(2):275-295.
[191]Demyanov, A.V., Demyanov,V.F., Malozemov, V.N. Minmaxmin problem revisited [J]. Optimization Methods and Software,2002,17(5):783-804.
[192]Ben-Tal, A., EL Ghaoui, L., Nemirovski, A. Robust Optimization [M]. Princeton Series in Applied Mathematics, Princeton University Press,2009.
[193]Bertsimas, D., Brown, D.B., Caramanis, C. Theory and application of robust optimization [J]. SIAM Review,2011,53(3):464-501.
[194]Popescu, I. A semidefinite programming approach to optimal-moment bounds for convex classes of distributions [J]. Mathematics of Operations Research,2005,30(3):632-657.
[195]Delage, E., Ye, Y. Distributionally robust optimization under uncertain moment application to data-driven problem [J]. Operations Research,2010,58(3):595-612.
[196]Ben-Tal, A., Bertsimas, D., Brown, D.B. A soft robust model for optimization under ambiguity [J]. Operations Research,2010,58 (4):1220-1234.
[197]Chen, W., Sim, M., Sun, J., Teo, C.P. From CVaR to uncertainty set:implications in the joint chance-constrained optimization [J]. Operations Research,2010,58(2):470-48.
[198]EL Ghaoui, L., Oks, M., Oustry, F. Worst-case Value-at-Risk and robust portfolio optimization:a conic programming approach [J]. Operations Research,2003,51(4):543-556.
[199]Bertsimas, D., Doan, X., Natarajan, K., Teo, C.P. Models for minmax stochastic linear optimization problems with risk aversion [J]. Mathematics of Operations Research,2010,35(3): 580-602.
[200]Delage, E.H. Distributionally robust optimization in context of data-driven problems [J]. Ph.D.thesis, Standford University, Palo Alto, CA,2009.
[201]Gabrel,V., Murant, C, Thiele, A. Recent advances in robust optimization and robustness:an overview[R]. Technical report, LAMSADE, Universite Paris-Dauphine, Paris, France,2012.
[202]Ukkusuri, S.V., Ramadurai, G, Patil, G.,2010. A robust transportation signal control problem accounting for traffic dynamics. Computers & Operations Research,37(5):869-879
[203]Mudchanatongsuk, S., Ordonez, F., Liu, J. Robust solution for network design under transportation cost and demand uncertainty [J]. Journal of the Operations Research Society,2008, 59(5):652-662.
[204]Chiou, S.W. Optimization of robust area traffic control with equilibrium flow under demand uncertainty [J]. Computers & Operations Research,2014,41:399-411.
[205]Lemarechal, C., Ouorou, A., Petrou, G. Robust network design in telecommunications under polytope demand uncertainty [J]. European Journal of Operations Research,2010,206(3):634-641.
[206]Falk, J.E., Soland, R.M. An algorithm for separable nonconvex programming problems. Management Science,1969,15 (9):550-569.
[207]Scheel, H., Scholte, S. Mathematical programs with complementarity constraints:stationarity, optimality, and sensitivity [J]. Mathematics of Operations Research,2000,22(1):1-22
[208]Brooke, A., Kendirick, D., Meeraus A. GAMS:A User's Guide [M]. The Scientific Press, 1992, South San Franciso, California,1992.
[209]Durd, A.S. CONOPT-A Large-Scale GRG Code [J]. Journal on Computing,1994,6(2): 207-216.
[210]Hearn, D.W., Ramana, M.V., Solving congestion toll pricing models [C]. In:Marcotte, P., Nguyen, S. (Eds.), Equilibrium and Advanced Transportation Modeling. Kluwer Academic Publishers, Nowell, MA,1998,109-124.
[211]Daganzo, C.F., Sheffi, Y. On stochastic models of traffic assignment [J]. Transportation Science,1977,11(3):253-274.
[212]Abdel-Aty, M., Kitamura, R., Jovanis, P. Exploring route choice behavior using geographical information system-based alternative paths and hypothetical travel time information input [J]. Transportation Research Record,1995,1493:74-80
[213]Small, K.A., Noland, R., Chu, X., Lewis, D.,1999. Valuation of travel-time savings and predictability in congested conditions for highway user-cost estimation. NCHRP Report No.431, Transportation Research Board, National Research Council, USA
[214]Brownstone, D., Gabrei, S.A., Golob, T.F., Kazimi, C., Amelsfort, D.V. Drivers' willingness-to-pay to reduce travel time:evidence from San Diego 1-15 congestion pricing project [J]. Transportation Research Part A,2003,37(4):373-387.
[215]Liu, H., Recker, W., Chen, A. Uncovering the contribution of travel time reliability to dynamic route choice using real-time loop data[J]. Transportation Research Part A,2004,38(6): 435-453
[216]de Plama, A., Picard, A. Route choice decision under travel time reliability [J]. Transportation Research Part A,2005,39 (4):295-32.
[217]Fosgerau, M., Karlstrom, A. The value of reliability [J]. Transportation Research Part B,2011, 44(l):38-49.
[218]Fosgerau, M., Engelson, L. The value of travel time variance [J]. Transportation Research Part B,2007,45(1):1-8.
[219]Mirchandani, P., Soroush, H. Generalized traffic equilibrium with probabilistic travel times and perceptions [J]. Transportation Science,1987,21(3):133-152.
[220]Yin, Y, Idea, H. Assessing performance reliability of road network under non-recurrent congestion [J]. Transportation Research Record,2001,1771:148-155.
[221]Chen, A., Ji, Z., Recker, W. Travel time reliability with risk sensitive travelers [J]. Transportation Research Record,2002,1783:27-33.
[222]Yin, F., Lam, W.H.K., Idea, H. New technology and the modeling of risk-taking behavior in congested road networks [J]. Transportation Research Part C,2004,12(3/4):171-192
[223]Watling, D. User equilibrium traffic network assignment with stochastic travel times and late arrival penalty [J]. European Journal of Operational Research,2006,175(3):1539-1556.
[224]Di, S., Pan, C., Ran, B. Stochastic multiclass traffic assignment with consideration of risk-taking behaviors [J]. Transportation Research Record,2008,2085:111-123.
[225]Bell, M.G.H. A game theory approach to measuring the performance reliability of transportation networks [J]. Transportation Research Part B,2000,34(6):533-554.
[226]Bell, M.G.H., Cassir, C., Risk-averse user equilibrium traffic assignment:an application of game theory [J]. Transportation Research Part B,2002,36(8):671-681.
[227]Szeto, W.Y., O'Brien, L., O'Mathony, M., Risk-averse traffic assignment with elastic demands:NCP formulation and solution method for assessing performance reliability [J]. Networks and Spatial Economics,2006,6(3/4):313-332.
[228]Sumalee, A., Connors, R.D., Luthep, P. Network equilibrium under cumulative prospect theory and endogenous stochastic demand and supply [C]. In:Lam WHK, Wong SC, Lo HK (Eds.). Proceedings of 18th International Symposium of Transportation and Traffic Theory,2009, pp.19-38.
[229]Xu, H., Zhou, J., Xu, W. A direction-making rule for modeling travelers'route choice behavior based cumulative prospect theory [J]. Transportation Research Part C,2011,19 (2): 218-228.
[230]Zhang, C., Chen, X., Sumalee, A. Robust Wardrop's user equilibrium assignment under stochastic demand and supply:expected residual minimization approach [J]. Transportation Research Part B,2011,45(3):534-552.
[231]Uchida, T., Iida, Y. Risk assignment:a new traffic assignment model considering the risk travel time[C]. In:Daganzo CF(Ed.). Proceedings of 12th International Symposium on Transportation and Traffic Theory, Amsterdam,1993,89-105.
[232]Shao, H., Lam, W.H.K., Tam, M.L. Demand driven traffic assignment problem based on travel time reliability [J]. Transportation Research Record,2006,1985:220-230.
[233]Shao, H., Lam, W.H.K., Tam, M.L. A reliability-based stochastic traffic assignment model for network with multiple user classes under uncertain in demand [J]. Network and Spatial Economics, 2006,6(3/4):313-332.
[234]Shao, H., Lam, W.H.K., Tam, M.L., Yuan, X.M. Modeling rain effects on risk-taking behaviors of multi-user classes in road networks with uncertainty [J]. Journal of Advanced Transportation,2008,42(3):265-290.
[235]Siu, B.W.Y., Lo, H.K. Doubly uncertain transport network:degradable link capacity and perception variations in traffic conditions [J]. Transportation Research Record,2006,1964:59-69
[236]Siu, B.W.Y., Lo, H.K. Doubly uncertain transport network:Degradable capacity and stochastic demand [J]. European Journal of Operational Research,2008,191(1):166-181
[237]Lam, W.H.K., Shao, H., Sumalee, A. Modeling impacts of adverse weather conditions on a road network with uncertainties in demand and supply [J]. Transportation Research Part B,2008,42 (10):890-910.
[238]Zhou, Z., Chen, A. Comparative analysis of the three user equilibrium models under stochastic demand [J]. Journal of Advanced Transportation,2008,42(3):239-263.
[239]Chen, A., Zhou, Z. The alpha-reliable mean-excess traffic equilibrium model with stochastic travel times [J]. Transportation Research Part B,2010,44(4):493-513.
[240]Chen, A., Zhou, Z., Lam, W.H.K. Modeling stochastic perception error in the mean-excess traffic equilibrium model [J]. Transportation Research Part B,2011,45(10):1619-1640.
[241]Nie, Y. Multi-class percentile user equilibrium with flow-dependent stochasticy. Transportation Research Part B,2011,45(10):1641-1659.
[242]Jorion, P. Value-at-Risk:the new benchmark for controlling market risk. Irwin Professional Publishing, Chicago,1996.
[243]Hall, P.R. Travel outcome and performance:the effect of uncertainty on accessibility [J]. Transportation Research Part B,1983,17(4):275-290.
[244]Yamai, Y., Yoshiba, Y. On the validity of value-at-risk:comparative analysis with expected shortfall [J]. Monetary and Economic Studies,2002,20(1):57-85
[245]Wu, X., Nie, Y. Modeling heterogeneous risk-taking behavior in route choice:a stochastic dominance approach [J]. Transportation Research Part A,2011,45 (9):896-915
[246]Xu, X., Chen, A., Zhou, Z., Cheng, L. A multi-class mean-excess traffic equilibrium model with elastic demand [J]. Journal of Advanced Transportation,2012, DOI:10.1002/atr.205.
[247]Zhu, S.S., Fukushima, M. Worst-Case Conditional Value-at-Risk with application to robust portfolio management [J]. Operations Research,2009,57(5):1155-1168
[248]Ordonez, F., Stier-Moses, N.E. Wardrop equilibria with risk-averse users [J]. Transportation Science,2010,44(1):63-86
[249]Facchinei, F., Pang, J.S. Finite-dimensional variational inequalities and complementarity problem[M], Springer-Verlag, New York,2003.
[250]Bernstein, D., Gabriel, S.A. The traffic equilibrium problem with nonadditive path costs [J]. Transportation Science,1997,31(4):337-348
[251]Xu, M., Chen, A., Qu, Y.C., Gao, Z.Y. A semismooth Newton method for traffic equilibrium problem with a general nonadditive route cost [J]. Applied Mathematical Modeling,2011,35(6): 3048-3062
[252]Chen, A., Zhou, Z., Xu, X.D.,2012. A self-adaptive gradient projection algorithm for the nonadditive traffic equilibrium problem [J]. Computers & Operations Research,2012,39(2): 127-138
[253]Panicucci, B., Pappalardo, M., Passacantando, M.,2007. A path-based double projection method for solving the asymmetric traffic network equilibrium problem [J]. Optimization Letters, 1(2):171-185
[254]Lo, H.K., Chen, A. Traffic equilibrium problem with route-specific costs:formulation and algorithms [J]. Transportation Research Part B,2000,34 (6):493-513
[255]Nguyen, S., Dupuis, C. An efficient of method for computing traffic equilibria in network with asymmetric transportation costs [J]. Transportation Science,1984,18(2):185-202
[256]Nemirovski, A., Shapiro, A. Convex approximations of chance constrained programs [J]. SIAM Journal on Optimization,,2006,17(4):969-996
[257]Janson, B.N. Network design effects of dynamic traffic assignment. Journal of Transportation Engineering [J],1995,121(1):18-26.
[258]Waller, S.T. Optimization and control of stochastic dynamic Transportation system: formulations, solution methodologies, and Computational experience [D], Northwestern University, 2000.
[259]Jeon, K., Ukkusuri, S.V., Waller, S.T. Heuristic approach for discrete network design problem accounting for dynamic traffic assignment conditions:formulation, solution methodologies, implementations and computation experiences [C]. The 84th Annual Meeting of the Transportation Research Board, Washington, D.C.,1995.
[260]Waller, S.T., Ziliaskopoulos, A.K. A chance-constrained based stochastic dynamic traffic assignment model:analysis, formulation and solution algorithms [J]. Transportation Research Part C, 2006,14(6):418-427
[261]Yazici, M.A., Ozbay, K. Evacuation network modeling via dynamic traffic assignment with probabilistic demand and capacity constraints [J]. Transportation Research Record,2010, 2196:11-20
[262]Li, Y., Waller, S.T., Ziliaskopoulos, T. A decomposition scheme for system optimal dynamic traffic assignment models [J]. Networks and Spatial Economics,2003,3(4):441-455
[263]Zhu, F., Ukkusuri, S.V. A cell based dynamic system optimum model with non-holding back flows [J]. Transportation Research Part C,2013,36,367-380
[264]Ben-Tal, A., Boyd, S., Nemirovski, A. Extending scope of robust optimization:robust counterparts of uncertain problems [J]. Mathematical Programming,2006,107(1):63-89
[265]Chung, B.D., Yao, T., Zhang, B. Dynamic traffic assignment under uncertainty:A distributional robust joint chance-constrained Approach [J]. Networks and Spatial Economics,2012, 12(1):167-181
[266]Alizadeh, F., Goldfrb, D. Second-order Cone Programming. [J] Mathematical Programming, 2003,95(1):3-51
[267]Calafiore, G., EL Ghaoui, L. On distributionally robust chance-constrained linear programs with applications [J]. Journal of Optimization Theory and Application,2006,130(1):1-22
[268]Prekopa, A. On probabilistic constrained programming [C]. In:Proceedings of the Princeton Symposium on Mathematical Programming,113-138. Princeton University Press, Princeton,1970.
[269]Shapiro, A., Kleywegt, A.J. Minimax analysis of stochastic problems [J]. Optimization Methods and Software,2002,17(3):523-542.
[270]Chen, W., Sim, M. Goal-driven optimization. Operations Research,2009,57(2):342-357
[271]LOfberg, J.YALMIP:a toolbox for modeling and optimization in MATLAB [C]. In: Proceedings of the CACSD conference, Taipei,2004.
[272]Sturn, J.S., Using Sedumi 1.02, a Matlab toolbox for optimization over symmetric cones [J]. Optimization Methods and Software,1999,11(1-4):625-653.
[273]Luo, Z.Q., Pang, J.S., Ralph, D. Mathematical programs with Equilibrium Constraints [M], Cambridge University Press, Cambridge, England,1996.
[274]Huang, D.S., Zhu, S.S., Fabozzi, F.J., Fukushima, M. Portfolio selection undet distributional uncertainty:a relative robust CvaR approach [J]. European Journal of Operations Research,2010, 203(1):185-194.
[275]Ralph, D., Wright, S.J. Some properties of regularization and penalization schemes for MPECs [J]. Optimization Method for Software,2004,19(5):527-556
[276]Hoheisel, T., Kanzow, C., Schwartz, A. Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints [J]. Mathematical Programming,2013,137(1-2),257-288.
[277]Scholte, S.,2001. Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM Journal on Optimization,11(4):918-936
[278]Chen, X., Zhang Y.H., Uncertain Linear programs:Extended affinely adjustable robust counterparts. Operations Research,2009,57(6):1469-1482.
[279]Isii, K. The extrema of probability determined by generalized moments (i) bounded random variables. Annals of the Institue of Statistical Mathemtics,1960,12(2):119-134.
[2]Cerbakova, J. Worst-Case VaR and CVaR [C]. Operations Research Proceedings,2006, 2005(22):817-822.
[3]Daganzo, C.F. The cell transmission model:a dynamic representation of highway traffic consistent with the hydrodynamic theory[J]. Transportation Research Part B,1994,28(4):269-287.
[4]Daganzo, C.F. The cell transmission model:part Ⅱ:network traffic[J]. Transportation Research Part B,1995,29(2):79-93.
[5]高自友.现代城市综合交通网络设计的优化模型、求解算法及程序设计[R].北京：北京交通大学系统科学研究所技术报告,2002.
[6]Yang, H., Bell, M.G.H. Models and algorithms for road network design:a review and some new developments [J]. Transportation Reviews,199818(3):257-278.
[7]Magnanti, T.L., Wong, R.T. Network design and transportation planning:models and algorithms [J]. Transportation Science,1984,18(1):1-15.
[8]Boyce, D.E. Urban transportation network equilibrium and design models:recent achievements and future prospects [J]. Environment and Planning A,1984,16(11):1445-1474.
[9]Friesz, T.L., Harker, P.T. Properties of the iterative optimization-equilibrium algorithm [J]. Civil Engineering Systems,1985,2 (3):142-154.
[10]Minoux, M. Network synthesis and optimum network design problem:models, solution methods and application [J]. Network,1989,19(3):313-360.
[11]Askura, Y. Road network reliability caused by daily fluctuation of traffic flow [J]. European Transport, Highway& Planning,1991,19:73-84.
[12]Clark, S.D., Watling, D. Modeling network travel time reliability under stochastic demand [J]. Transportation Research Part B,2005,39(2):119-140.
[13]Chen, A., Yang, H., Lo, H.K., Tang, W.H. Capacity reliability of a road network:an assessment methodology and numerical results. Transportation Research Part B,2002,36(3): 225-252.
[14]Lo, H.K., Luo, X.W., Siu, B.W.Y. Degradable transportation network:travel time budget of travelers with heterogeneous risk aversion[J]. Transportation Research Part B,2006,40(9):792-806.
[15]Al-Deek, H., Eman, E.B. New methodology for estimating reliability in transportation networks with degraded link capacities [J]. Journal of Intelligent Transportation System,2006,10(3): 117-129.
[16]Waller, S.T., Schofer, J.L., Ziliaskopoulos, A.K.2001. Evaluation with traffic assignment under demand uncertainty[J]. Transportation Research Record,2001,1771:69-74.
[17]Chen, A., Zhou, Z., Chootinan, P., Ryu, S., Yang, C, Wong, S.C. Transportation network design problem under uncertainty:a review and new developments [J]. Transportation Review, 2011b,31(6):743-768.
[18]Yin, Y., Idea, H. Optimal improvement scheme for network reliability [J]. Transportation Research Record,2002,1783:1-6.
[19]Chen, A., Yang, C. Stochastic transportation network design problem with spatial equity constraint [J]. Transportation Research Record,2004,1882:97-104.
[20]Chen, A., Subprasom, K. Analysis of regulation and policy of private toll roads in a build-operate-transfer scheme under demand uncertainty [J]. Transportation Research Part A,2007, 41(6):537-558.
[21]Chen, A., Kim, J., Lee, S., Kim, Y. Stochastic multi-objective models for network design problems [J]. Expert systems and Application,2010,37(2):1608-1619.
[22]Chow, J.Y.J., Regan, A.C., Network-based real option models [J]. Transportation Research Part B,2011,45(4):682-695.
[23]Xu, H., Lou, Y., Yin, Y, Zhou, J. A prospect-based user equilibrium model with endogenous with reference points and its application in congestion pricing [J]. Transportation Research Part B, 2011,45(2):311-328.
[24]许良,基于可靠性分析的城市道路交通网络设计问题研究[D].北京：北京交通大学硕十学位论文,2006.
[25]孙强,王庆云,高咏玲.不确定需求下的多阶段区域综合交通网络设计的双层规划模型[J].交通运输系统工程与信息,2011,11(6)：111-116.
[26]蒋洋,基于结构和随机特性的交通网络设计问题的研究[D].北京：北京交通大学硕十学位论文,2011.
[27]Chen, A., Xu, X.D. Goal programming approach to solving network design problem with multiple objectives and demand uncertainty [J]. Expert systems and Application,2012,39(4): 4160-4170.
[28]Waller, S.T., Ziliaskopoulos, A.K. Stochastic dynamic network design problem[J]. Transportation Research Record,2001,1771:106-113.
[29]Ukkusuri, S.V, Karoonsoontawong, A., Waller, S.T. A stochastic dynamic user network design model accounting for demand uncertainty [C]. Proceedings of the Interational Conference of Transportation System Planning and Operations(TRANSPO), Madras, India, Feb,2004,18-20.
[30]Ukkusuri, S.V., Waller, S.T. Linear programming models for the user and system optimal dynamic network design problem:formulations, comparisons and extensions [J]. Networks and Spatial Economics,2008,8(4):383-406.
[31]Karoonsoontawong, A., Waller, S.T. Robust dynamics continuous network design problem [J]. Transportation Research Record,2007,2029:58-71.
[32]Ukkusuri, S.V., Patil, G. Multi-period transportation network design under demand uncertainty [J]. Transportation Research Part B,2009,43(6):625-642.
[33]Markowtiz, H. Mean-variance analysis in portfolio choice and capital markets[M]. (Pennsylvania:New Hope),1927.
[34]Chen, A., Subprasom, K., Ji, Z. Mean-variance model for the build-operate-transfer scheme under demand uncertainty [J]. Transportation Research Record,2003,1857:93-101.
[35]Chen, A., Subprasom, K., Ji, Z. A simulation-based multi-objective genetic algorithm (SMOGA) for build-operate-transfer network design problem [J]. Optimization and Engineering Journal,2006,7(3):225-247.
[36]Ukkusuri, S.V., Mathew, T.V., Waller, S.T. Robust transportation network design under demand uncertainty [J]. Computer-Aided Civil and Infrastructure Engineering,2007,22(1):6-18.
[37]Li, H., Bliemer, M.C.J., Bovy, P.H.L. Network reliability-based optimal toll design [J]. Journal of Advanced Transportation,2008,42(3):311-332.
[38]Gardner, L.M., Unnikrishnan, A., Waller, S.T. Robust pricing of transportation networks under demand uncertainty [J]. Transportation Research Record,2008,2085:21-30
[39]Sharma, S., Ukkusui, S.V., Mathew, T.V., A Pareto optimal multi-objective optimization for the robust transportation network design problem [J]. Transportation Research Record,2009,2090: 95-104
[40]潘艳荣,邓卫,2008.考虑需求不确定性的交通网络设计[J].交通运输工程学报,8(6)：82-87.
[41]卞长志.需求不确定的离散交通网络设计模型与算法[D].北京：清华大学博士学位论文,2009.
[42]Yin, Y., Madanat, S.M., Lu, X. Robust improvement schemes for road networks under demand uncertainty [J]. European Journal of Operational Research,2009,198(2):470-479.
[43]Ng, M.W., Waller, S.T. Reliable system optimal network design:a convex mean-variance type model with implicit chance constraints [J]. Transportation Research Record,2009,2090:68-74.
[44]Sumalee, A., Luathep, P., Lam, W.H.K., Connors, R.D. Evaluation and design of transport network capacity under demand uncertainty [J]. Transportation Research Record,2009,2090:17-28.
[45]Karoonsoontawong, A., Waller, S.T. Integrated network capacity expansion and traffic signal optimization problem:robust bi-level dynamic formulation [J]. Networks and Spatial Economics, 2010,10(4):525-550.
[46]Charnes, A., Cooper, W.W., Symonds, G.H. Cost horizons and certainty equivalents:an approach to stochastic programming of heating oil [J]. Management Science,1958,4(3):235-263
[47]Lo, H.K., Tung, Y.K. Network with degradable links:capacity analysis and design [J]. Transportation Research Part B,2003,37(4):345-363.
[48]许良,高自友.基于路段能力可靠性的城市交通网络设计[J].中国公路学学报,2006,19(2)：86-90.
[49]Dimitriou, L., Stahopoulos, A. Reliable stochastic design of road network systems [J]. International Journal of Industrial and System Engineering,2008,3(5):549-574.
[50]Ji, Z., Kim, Y.S., Chen, A. Multi-objective a-reliable path finding in stochastic networks with correlated link costs:A simulation-based multi-objective genetic algorithm approach (SMOGA) [J]. Experts Systems and Applications,2011,38(3):1515-1528.
[51]许项东.需求不确定环境下的道路网络均衡模型与系统优化[D].南京：东南大学博士学位论文,2012.
[52]Chootinan, P., Wong, S.C., Chen, A. A reliability-based network design problem [J]. Journal of Advanced Transportation,2005,39(3):247-270
[53]Yim, K.K.W., Wong, S.C., Chen, A., Wong, C.K., Lam, W.H.K. A reliability-based land use and transportation optimization model [J]. Transportation Research Part C,2011,19(2):351-362
[54]Chen, A., Chootinan, P., Wong, S. C. New reserve capacity model of a signaled-controlled road network [J]. Transportation Research Record,2006,1964:35-41
[55]Sumalee, A., Watling, D. P., Nakayama, S. Reliable network design problem:the case with uncertain demand and total travel time reliability [J]. Transportation Research Record,2006,1964: 81-90.
[56]Larsen, N., Mausser, H., Uryasev, S. Algorithms for optimization Value-at-Risk[C]. In. Pardolos P, Tsitsiringos. (Ed.) Financial Engineering, E-commerce and Supply Chain,2002, pp.19-46.
[57]Rockafellar, R.T., Uryasev, S. Optimization of Conditional Value-at-Risk. Journal of RiskfJ], 2000,2(3):21-41.
[58]Rockafellar, R.T., Uryasev, S. Conditional Value-at-Risk for general loss distributions [J]. Journal of Banking and Finance,2002,26(7):1443-1471.
[59]Chen, A., Kim, J., Zhou, Z., Chootinan, P. An alpha reliable network design problem [J]. Transportation Research Record,2007,2029:49-57.
[60]Chen, A., Ryu, S., Yang, C., Wong, S.C. Alpha reliable network design with multiple objectives and demand uncertainty[C]. Paper presented at the 89th Annual Meeting of the Transportation Research Board, Washington, DC, USA,.2010.
[61]Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Thinking coherently [J]. Risk,1997,10 (11): 68-71.
[62]Artzner, P., Delbaen, F., Eber, J.M., Heath, D., Coherent measures of risk [J]. Mathematical Finance,1999,9(3):203-228.
[63]Hellman, F. Towards the solution of large-scale and stochastic traffic network design problems [D]. Master's Thesis, Uppsala University, Uppsala, Sweden,2010.
[64]Bazaraa, M.S., Sherali, H.D., Shetty, C.M. Nonlinear Programming:Theory and Algorithms, Second Edition[M]. John Wiley & Sons, New York, NY,1993.
[65]Yin, Y., Lawphongpanich, S., Lou, Y. Estimating investment requirement for maintaining and improving highway systems[J]. Transportation Research A,2008,16(2):199-211
[66]Yin, Y. Robust optimal signal timing. Transportation Research Part B,2008,42 (10): 911-924.
[67]Lu, Y. Robust transportation network design under user equilibrium[D]. Master's Thesis MIT, Cambridge, MA,2007.
[68]Lou, Y., Yin, Y., Lawphongpainch, S. A robust approach to discrete network design with demand uncertainty [J]. Transportation Research Board,2009,2090:86-94.
[69]Zhang, L.H., Lawphongpanich, S., Yin, Y. An active set algorithm for discrete network design problems[C]. The 18th International Symposium on Transportation and Traffic Theory, Hong Kong, Springer. In:Lam, W.H.K., Wong, S.C., Lo, H.K.,2009. Transportation and Traffic Theory 2009:Golden Jubilee, pp.283-300.
[70]Ban, X., Lu, S., Ferris, M., Liu, H. Risk-averse second-best toll pricing[C]. Proceedings 18th International Symposium Transportation and Traffic System,2009,197-218.
[71]Lou, Y., Yin, Y., Lawphongpainch, S. Robust congesting pricing under boundedly rational user equilibrium [J]. Transportation Research Part B,2010,44(1):15-28.
[72]Chung, B.D., Yao, T, Xie, C., Thorsen, A. Robust optimization model for a dynamic design problem under demand uncertainty [J]. Networks and Spatial Economics,2011,11(2):371-389
[73]Yao, T., Mandala, R.S., Chung, B.D. Evacuation transportation planning under uncertainty:a robust optimization approach[J]. Networks and Spatial Economics,2009,9(2):171-189.
[74]Ben-Tal, A., Chung, B.D., Mandala, S.R., Yao, T. Robust optimization for emergency logistic planning:Risk mitigation in humanitarian relief supply chains [J]. Transportation Research Part B, 2011,45(8):1177-1189.
[75]Chung, B.D., Yao, T., Friesz, T.L., Liu H. Dynamic congestion pricing with demand uncertainty:A robust optimization approach [J]. Transportation Research Part B,2012,46(10): 1504-1518
[76]Houska, B., Diehl, M. Nonlinear robust optimization via sequential convex bilevel programming [J]. Mathematical Programming,2013,142(1-2),539-577.
[77]Ben-Tal, A., Goryashko, E., Guslitzer, A., Nemirovski, A. Adjustable robust solutions of uncertain linear programs [J]. Mathematical Programming,2004,99 (2):351-376.
[78]Zymler, S., Kuhn, D., Rustem, B. Distributionally robust joint chance constraints with second-order moment information [J]. Mathematical Programming,2013,137(1-2):167-198.
[79]Abdulaal, M.S., LeBlanc, L.J. Continuous equilibrium network design models[J]. Transportation Research Part B,1979,13(1):19-32.
[80]Davis, G. A. Exact local solution of the continuous network design problem via stochastic user equilibrium assignment [J]. Transportation Research Part B,1994,28(1):61-75
[81]Maher, M.J., Zhang, X., Van Vliet, D. A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows [J]. Transportation Research Part B,2001,35(1):23-40.
[82]Meng, Q., Yang, H., Bell, M.G.H.,2001. An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem [J]. Transportation Research Part B,2001,35(1):83-105.
[83]Meng, Q., Yang, H. Benefit distribution and equality in road network design [J]. Transportation Research Part B,2002,36(1):19-35.
[84]Gao, Z.Y., Sun, H.J., Shan, L.L. A continuous equilibrium network design model and algorithm for transit system[J]. Transportation Research Part B,2004,38(3):235-250.
[85]Chen, M., Bernstein, D.H. Solving the toll design problem with multiple user groups [J]. Transportation Research Part B,2004,38(1):61-79.
[86]Wang, D.Z.W., Lo, H.K. Global optimum of the linearized network design problem with equilibrium flows[J]. Transportation Research Part B,2010,44(4):482-492.
[87]Luathep, P., Sumalee, A., Lam, W.H.K., Li, Z.C., Lo, H.K. Global optimization method for mixed transportation network design problem:A mixed-integer linear programming approach[J]. Transportation Research Part B,2011,45(5):808-827.
[88]Wong, S.C., Yang, H, Reserve capacity of a signal-controlled road network [J]. Transportation Research Part B,1997,31(5):397-402.
[89]Gao, Z.Y., Song, Y.F. A reserve capacity model of optimal signal control with user-equilibrium route choice[J]. Transportation Research Part B,2002,36(4):313-323.
[90]Ceylan, H., Bell, M.G.H. Reserve capacity for a road network under optimized fixed time traffic signal control [J]. Journal of Intelligent Transportation Systems:Technology, Planning, and Operations,2004,8(2):87-99
[91]Chiou, S.W. Reserve capacity of signal-controlled road network. Applied Mathematics and Computation,2007,190(2):1602-1611.
[92]Chen, A., Kasikitwiwat, P. Modeling capacity flexibility of transportation networks [J]. Transportation Research Part A,2011,45(2):105-117.
[93]Lam, W.H.K., Tam, M.L., Bell, M.G.H. Optimal road tolls and parking charges for balancing the demand and supply of road transport facilities [J]. In:Taylor, M.A.P. (Ed.), Transportation and Traffic Theory in the 21st Century:Proceedings of the 15th International Symposium on Transportation and Traffic Theory. Elsevier Science, New York,2002,561-582.
[94]Szeto, W.Y., Lo, H. Strategies for road network design over time:robustness under uncertainty[J]. Transportmetrica,2005,1(1):47-63.
[95]Szeto, W.Y., Lo, H. Time-dependent transport network improvement and tolling strategies [J]. Transportation Research Part A,2008,42(2):376-391.
[96]Yang, H., Xiao, F., Huang, H.J. Private road competition and equilibrium with traffic equilibrium constraints [J]. Journal of Advanced Transportation,2009,43(1):21-45.
[97]Lo, H.K., Szeto, W.Y. Time-dependent transport network design under cost-recovery [J]. Transportation Research Part B,2009,43(1):142-158.
[98]Heydecker, B.G. Dynamic equilibrium network design[C]. In:Taylor, M.A.P. (Ed.), Transportation and Traffic Theory in the 21st Century:Proceedings of the 15th International Symposium on Transportation and Traffic Theory. Elsevier Science, New York,2002, pp.349-370.
[99]Szeto, W.Y., Lo, H. Transportation network improvement and tolling strategies:The issue of intergeneration equity[J]. Transportation Research Part A,2006,40(3):227-243
[100]Wu, D., Yin, Y., Lawphongpanich, S. Optimal selection of build-operate-transfer projects on transportation networks[J]. Transportation Research PartJB,2011,45(10):1699-1709.
[101]Friesz, T.L., Harker, P.T. Multicriteria spatial price equilibrium network design:theory and computational results[J]. Transportation Research Part B,1983,17(5):411-426
[102]Friesz, T.L., Anandalingam, G., Mehta, N.J., Nam, K., Shah, S.J., Tobin, R.L. The multiobjective equilibrium network design problem revisited:a simulated annealing approach. European Journal of Operational Research,1993,65(1):44-57.
[103]Gartner, N.H., Stamatiadis, C. Combining traffic assignment and adaptive control in a dynamic traffic management system[C]. In:Becker, U., Schnieder, E. (Eds.), Control in Transportation Systems 2000:Proceedings Volume from the 9th IFAC Symposium, vol.1. Elsevier Science, UK,2001, pp.281-286.
[104]Sohn, K. Multi-objective optimization of a road diet network design [J]. Transportation Research Part A,2011,45(6):499-511.
[105]Wardrop, J.G. Some theoretical aspects of road traffic research[J]. Proceedings of the Institution of Civil Engineers,1952,1(2):325-378.
[106]Sheffi, Y. Transportation Network:Equilibrium analysis with mathematical programming methods [M]. Prentice-Hall, Incorporated, Englewood Cliffs, NJ,1985.
[107]Friesz, T.L. Transportation network equilibrium, design and aggregation:key development and research opportunities [J]. Transportation Research Part A,1985,19(5-6):413-427.
[108]Suwansirikul, C, Friesz, T.L. Equilibrium decomposed optimization:a heuristic for the continuous equilibrium network design problem [J]. Transportation Science,1987,21(4):254-263
[109]Ferrari, P. Road pricing and network equilibrium [J]. Transportation Research Part B,1995, 29(5):357-372.
[110]Ferrari, P. Road network toll pricing and social welfare. Transportation Research Part B,2002, 36(5):471-483.
[111]Yang, H., Lam, W.H.K. Optimal road tolls under conditions of queuing and congestion [J]. Transportation Research Part A,1996,30(5):319-332.
[112]Yang, H., Bell, M.G.H. Traffic restraint, road pricing and network equilibrium [J]. Transportation Research Part B,1997,31(4):303-314.
[113]Allosp, R.E. Some possibility for using traffic control to influence trip distribution and route choice[C]. Proceedings of the Sixteenth International Symposium on Transportation and Traflc Theory. Elsevier, Amsterdam,1974,345-374.
[114]Allsop, R.E., Charlesworth, J.A. Traffic in a signal-controlled road network:an example of different signal timings inducing different routes [J]. Traffic Engineering and Control,1977,18(5): 262-264.
[115]Cantarella, G.E., Improta, G., Sforza, A. Iterative procedure for equilibrium network traffic signal setting [J]. Transportation Research Part A,1991,25(5):241-249.
[116]Wong, S.C., Yang. C. An iterative group-based signal optimization scheme for traffic equilibrium networks [J]. Journal of Advanced Transportation,1999,33(2):201-217.
[117]Chiou, S.W. Optimization of area traffic control for equilibrium network flows. Transportation Science,1999,33(3):279-289.
[118]Yang, H., Yagar, S., Iida, Y., Asakura, Y. An algorithm for the inflow control problems on urban freeway networks with user optimal flows [J]. Transportation Research Part B,1994,28(2): 123-139.
[119]Yang, H., Yagar, S. Traffic assignment and traffic control in general freeway-arterial corridor systems [J]. Transportation Research Part B,1994,28(6):463-486
[120]Cho, H.J. Sensitivity analysis of equilibrium network flows and its application to development of solution methods for equilibrium network design problem [D]. PhD Dissertation. University of Pennsylvania, Philadelphia,1988.
[121]Friesz, T.L., Tobin, R.L., Cho, H.L., Metha, N.J. Sensitivity analysis based heuristic algorithms for mathematical programs with equilibrium constraints[J]. Mathematical Programming, 1990,48(2):265-284.
[122]Yang, H., Yagar, S. Traffic assignment and signal control in saturated road networks [J]. Transportation Research Part A,1995,29(2):125-139
[123]Yang, H. Sensitivity analysis for queuing equilibrium network flow and its application to traffic control [J]. Mathematical and Computer Modeling,1995,22(4-7):247-258.
[124]Yang, H. Sensitivity analysis for the elastic demand network equilibrium problem with application [J]. Transportation Research Part B,1997,31(1):55-70.
[125]Connors, R.D., Sumalee, A., Waiting, D.P. Sensitivity analysis of the variable demand probit stochastic user equilibrium with multiple user-classes [J]. Transportation Research Part B,2007, 41(6):593-615.
[126]Chiou, S.W. Bilevel programming for the continuous network design problem [J]. Transportation Research Part B,2005,39(4):361-383.
[127]Lawphongpanich, S., Yin, Y. Solving the Parteo-improving toll problem via manifold suboptimziation [J]. Transportation Research Part C,2010,18(2):234-246.
[128]Wu, D., Yin, Y., Lawphongpanich, S. Parteo-improve congestion pricing on multimodal transportation network [J]. European Journal of Operation Research,2011,210(3):660-669.
[129]Chiou, S.W. A subgradient optimization model for the continuous road network design problem [J]. Applied Mathematical Modeling,2009,33(3):1386-1396
[130]Yang, H., Zhang, X., Meng, Q. Modeling private highways with entry-exit based toll charges[J]. Transportation Research Part B,2004,38(3):463-486
[131]Gao, Z.Y., Sun, H.J., Zhang, H.Z. A globally convergent algorithm for transportation continuous network design problem [J]. Optimization and Enginnering,2007,8(3):241-257.
[132]Ban, X., Liu, H.X., Ferris, M., Ran, B. A general MPCC model and its solution algorithm for continuous network design problem [J]. Mathematical and Computer Modeling,2006,43(5-6): 493-505.
[133]Ban, X., Liu, H.X., Lu, J.G., Ferris, M. A decomposition scheme for continuous network design problem with asymmetric user equilibria[J]. Transportation Research Record,2006,1964: 185-192
[134]Asakura, Y., Sasaki, T. Formulation and feasibility test of optimal road network design model with endogenously determined travel demand[C]. Proceedings of the 5th World Conference on Transport Research, Yokohama, Japan, July,1990,351-365.
[135]Van Vuren, T., Van Vliet, D. Route Choice and Signal Control [M]. Newcastle upon Tyne, U.K.:Athenaeum Press,1992.
[136]Smith, M.J., Van Vuren, T. Traffic equilibrium with responsive traffic control [J]. Transportation Science,1993,27(2):118-132
[137]Koh, A. An adaptive differential evolution alogithm applied to highway network capacity optimization[C]. In:Avineri, E., Koppen, M., Dahal, K., Sunitiyoso, Y., Roy, R.,(Eds.), Application of Soft Computing:Update the state of Art,2009,211-220.
[138]Sun, Y., Turnquist, M.A. Investment in transportation network design under demand uncertainty:simulated annealing approach [J]. Transportation Research Record,2007,2039:67-74.
[139]Cree, N.D., Maher, M.J., Paechter, B. The continuous equilibrium optimal network design problem:a genetic approach [C]. Transportation Networks:Recent Methodological Advances, Selected Proceedings of the 4th EURO Transportation Meeting, Newcastle, UK,1998,163-174
[140]Yin, Y. Genetic-algorithm-based approach for bilevel programming models[J]. Journal of Transportation Engineering,2000,126(2):115-120.
[141]Sumalee, A. Multi-concentric optimal charging cordon design [J]. Transportmetrica,2007, 3(1):41-71.
[142]Mousko, K. A tabu-serch based approach for network design [C]. Ph.D. Dissertation. The University of Texas at Austin, Austin, TX,1992.
[143]Xu, T.Z, Wei, H., Hu, G.H Study on continuous network design problem using simulated annealing and genetic algorithm [J]. Expert system and Application,2009,36(2):1322-1328
[144]Friesz, T.L., Hsunjung, H.L., Tobin, R.L.., Metha, N.J., Anandalingam, G. A simlated annealing appraoch to the network design problem with variational inequility constaints[J]. Transporatation Science,1992,26(1):18-26.
[145]Li, C.M., Yang, H., Zhu, D.L., Meng, Q. A global opitmization for continuous network design problems [J]. Transportation Reserach Part B,2012,46(9):1144-1158.
[146]Marcotte, P.A. Network optimization with continuous control parameters [J]. Transportation Science,1983,17(2):181-197.
[147]Lawphongpanish, S., Hearn, D.W. An MPEC approach to second-best toll pricing [J]. Mathematical Programming,2004,101(1):33-55.
[148]Joseph, Y.J., Chow, P.E., Regan, A.C., Arkhipov, D.I.. A faster converging global heuristic for continuous network design using radial basis functions [C]. The 89th Transportation Research Board annual meeting,2010.
[149]LeBlanc, L.J. An algorithm for the discrete network design problem [J]. Transportation Science,1975,9(3):183-199.
[150]Poorzahedy, H., Turnquist, M.A. Approximate algorithms for the discrete network design problem [J]. Transportation Research Part B,1982,16(1):45-55.
[151]Chen, M, Alfa, A.S. A network design algorithm using a stochastic incremental traffic assignment problem [J]. Transportation Science,1991,25(3):215-224.
[152]Drezner, Z., Salhi, S. Using hybid metaheurisitics for the one-way and two-way network design problem [J]. Naval Research Logistic,2002,49(5):449-463.
[153]Drezner, Z., Wesolowsky, G.O. Selecting an optimum configuration of one-way and two-way routes [J]. Transportation Science,1997,31(4):386-394.
[154]Drener, Z., Wesolowsky, G.O. Network design:selecting and design of links and facility location [J]. Transportation Research Part A,2003,37(3):241-256.
[155]Farvareshhe, H., Sepehri, M.M. A branch and bound algorithm for bi-level discrete network design [J]. Networks and Spatial Economics,2013,13(1):67-106.
[156]Solanki, R.S., Gorti, J.K., Southworth, F. Using decomposition in large-scale highway network design with a quasi-optimization heuristic [J]. Transportation Research Part B,1998,32(2): 127-140.
[157]Farvareshhe, H., Sepehri, M.M. A single-level mixed integer linear formulation for a bi-level discrete problem [J]. Transportation Research Part E,2011,47(5):623-640.
[158]Gao, Z.Y., Wu, J.J., Sun, H.J. Solution algorithm for the bi-level discrete network design problem [J]. Transportation Research Part B,2005,39(6):479-495.
[159]Wang, S., Meng, Q., Yang, H. Global optimization methods for the discrete network design problem. Transportation Research Part B,2013,50:42-60.
[160]Poorzahedy, H., Rouhanni, O.M. Hybrid meta-heuristic algorithm for solving network design problem [J]. European Journal of Operations Research,2007,182(2):578-596.
[161]Poorzahedy, H., Abulghasemi. Application of ant system to network design problem [J]. Transportation,2005,32(3):251-273.
[162]Kim, B.J., Kim, W. An equilibrium network design model with a social cost function for multimodal networks [J]. Annual Regional Science,2006,40(3):473-491.
[163]Xiong, Y., Schneider, J.B., Transportation network design using a cumulative genetic algorithm and neutral network [J]. Transportation Research Record,1995,1364:37-44.
[164]Lee, C.K., Yang, K.I. Network design of one-way street with simulated annealing [J]. Papers in Regional Science,1994,73(2):119-134
[165]Ekstrom, J., Sumalee, A., Lo, H.K. Optimizing toll location and levels using a mixed integer linear approximation approach [J]. Transportation Research Part B,2012,46(7):834-854.
[166]Zhang, H.Z., Gao, Z.Y. Bilevel programming model and solution methods for mixed network design problem[J]. Journal of Systems Science and Complexity,2009,22(3):446-459
[167]Cantarella, G.E., Vitetta, G. The multi-criteria road network design problem in an urban area. Transportation,2006,33(6):567-588.
[168]Sun, Y., Song, R., He, S., Chen, Q. Mixed transportation network design based immune clone anneal algorithm [J]. Journal of Transportation System Engineering and Information Technology, 2009,9(3):103-108
[169]Lin, D.Y., Karoonsoontawong, A., Waller S.T.. A Dantzig-Wolfe decomposition based heuristic scheme for bi-level dynamic network problem [J]. Networks and Spatial Economics,2011, 11(1):101-126.
[170]Ziliaskopoulos, A.K. A linear programming model for the single destination system optimum dynamic traffic assignment problem [J]. Transportation Science,2000,34(1):37-49.
[171]Lighthill, M.H., Whitham, G.B. On kinematics wave:II A theory of traffic flow on long crowed roads [J]. Proceedings of the Royal Society, London, Series A,,1955,229(1178):317-345.
[172]Waller, S.T., Mousokos, K.C., Kamaryiannis, D., Ziliaskopoloulos, A.K., A linear model for the continuous network design problem [J]. Computer-Aided Civil Infrastructure Engineering,2006, 21(5):334-345.
[173]Ukkusuri, S.V. Linear programs for the user optimum dynamic traffic assignment problem [D]. Master Thesis, University of Illinois at Urbana-Champaign,2002.
[174]Karoonsoontawong, A., Waller, S.T. Dynamic network design problem:linear bi-level programming and metaheuristic approaches [J]. Transportation Research Record,2006,1964: 104-117.
[175]Lin, D.Y., Unnikrishnan, A., Waller S.T. A genteic algorithm for bi-level linear programming dynamic network design problem [J]. Transportation Letters:the international journal of transportation research,2009,1(4):281-294.
[176]Lin, D.Y. A dual variable approximation-based descent method for a bi-level continuous dynamic network design problem [J]. Computer-Aided Civil Infrastructure Engineering,2011,26(8): 581-594.
[177]Mulevy, J.M., Vanderbei, R.J., Zenios, S.A. Robust optimization of large-scale systems [J]. Operations Research,1995,43(2):264-281.
[178]Soyster, L.A. Convex programming with set inclusive constraints and applications to inexact linear programming [J]. Operations Research,1973,21(5):1154-1157.
[179]Bertsimas, D., Sim, M. Robust discrete optimization and network flow [J]. Mathematical Programming,2003,98(1/3):49-71
[180]Bertsimas, D., Sim, M. The price of robustness [J]. Operations Research,,2004,52(1):35-53
[181]Atamturk, A. Strong formulation of robust mixed 0-1 programming [J]. Mathematical Programming,2006,108(2):235-250
[182]Ben-Tal, A., Nemirovski, A.,1998. Robust convex optimization [J]. Mathematics of Operations Research,23(4):769-805
[183]Ben-Tal, A., Nemirovski, A.,1999. Robust solutions of uncertain linear programs [J]. Operation Research Letter,25(1):1-13
[184]Ben-Tal, A., Nemirovski, A.,2000. Robust solutions of linear programming problems contaminated with uncertain data [J]. Mathematical Programming,88(3):411-424
[185]EL Ghaoui, L., Oustry, F., Lebert, H. Robust solutions to uncertain semiefinite programming [J]. SIAM Journal on Optimization,1998,9(1):35-52.
[186]Ben-Tal, A., EL Ghaoui, L., Nemirovski, A. Robust semidefinite programming[C]. In: Wolkowicz, H., Saignal, R., Vandeberghe, L., eds., Handbook on semidefinte programming, Kluwer Academic Publishers,2000,303-328
[187]Ben-Tal, A., Nemirovski, A. Robust optimization-methodology and application [J]. Mathematical Programming,2002,92(3):453-480.
[188]Diehl, M., Bock, H.G., Kostina, E. An approximation technique for robust nonlinear optimization [J]. Mathematical Programming,2006,107(1-2):213-230
[189]Zhang, Y. General robust-optimization formulation for nonlinear programming [J]. Journal of Optimization Theory and Applications,2007,132(1):111-124
[190]Takeda, A., Taguchi, S., Tutiincu, R.H. Adjustable Robust Optimization Models for a nonlinear two-period system [J]. Journal of Optimization Theory and Application,2008,136(2):275-295.
[191]Demyanov, A.V., Demyanov,V.F., Malozemov, V.N. Minmaxmin problem revisited [J]. Optimization Methods and Software,2002,17(5):783-804.
[192]Ben-Tal, A., EL Ghaoui, L., Nemirovski, A. Robust Optimization [M]. Princeton Series in Applied Mathematics, Princeton University Press,2009.
[193]Bertsimas, D., Brown, D.B., Caramanis, C. Theory and application of robust optimization [J]. SIAM Review,2011,53(3):464-501.
[194]Popescu, I. A semidefinite programming approach to optimal-moment bounds for convex classes of distributions [J]. Mathematics of Operations Research,2005,30(3):632-657.
[195]Delage, E., Ye, Y. Distributionally robust optimization under uncertain moment application to data-driven problem [J]. Operations Research,2010,58(3):595-612.
[196]Ben-Tal, A., Bertsimas, D., Brown, D.B. A soft robust model for optimization under ambiguity [J]. Operations Research,2010,58 (4):1220-1234.
[197]Chen, W., Sim, M., Sun, J., Teo, C.P. From CVaR to uncertainty set:implications in the joint chance-constrained optimization [J]. Operations Research,2010,58(2):470-48.
[198]EL Ghaoui, L., Oks, M., Oustry, F. Worst-case Value-at-Risk and robust portfolio optimization:a conic programming approach [J]. Operations Research,2003,51(4):543-556.
[199]Bertsimas, D., Doan, X., Natarajan, K., Teo, C.P. Models for minmax stochastic linear optimization problems with risk aversion [J]. Mathematics of Operations Research,2010,35(3): 580-602.
[200]Delage, E.H. Distributionally robust optimization in context of data-driven problems [J]. Ph.D.thesis, Standford University, Palo Alto, CA,2009.
[201]Gabrel,V., Murant, C, Thiele, A. Recent advances in robust optimization and robustness:an overview[R]. Technical report, LAMSADE, Universite Paris-Dauphine, Paris, France,2012.
[202]Ukkusuri, S.V., Ramadurai, G, Patil, G.,2010. A robust transportation signal control problem accounting for traffic dynamics. Computers & Operations Research,37(5):869-879
[203]Mudchanatongsuk, S., Ordonez, F., Liu, J. Robust solution for network design under transportation cost and demand uncertainty [J]. Journal of the Operations Research Society,2008, 59(5):652-662.
[204]Chiou, S.W. Optimization of robust area traffic control with equilibrium flow under demand uncertainty [J]. Computers & Operations Research,2014,41:399-411.
[205]Lemarechal, C., Ouorou, A., Petrou, G. Robust network design in telecommunications under polytope demand uncertainty [J]. European Journal of Operations Research,2010,206(3):634-641.
[206]Falk, J.E., Soland, R.M. An algorithm for separable nonconvex programming problems. Management Science,1969,15 (9):550-569.
[207]Scheel, H., Scholte, S. Mathematical programs with complementarity constraints:stationarity, optimality, and sensitivity [J]. Mathematics of Operations Research,2000,22(1):1-22
[208]Brooke, A., Kendirick, D., Meeraus A. GAMS:A User's Guide [M]. The Scientific Press, 1992, South San Franciso, California,1992.
[209]Durd, A.S. CONOPT-A Large-Scale GRG Code [J]. Journal on Computing,1994,6(2): 207-216.
[210]Hearn, D.W., Ramana, M.V., Solving congestion toll pricing models [C]. In:Marcotte, P., Nguyen, S. (Eds.), Equilibrium and Advanced Transportation Modeling. Kluwer Academic Publishers, Nowell, MA,1998,109-124.
[211]Daganzo, C.F., Sheffi, Y. On stochastic models of traffic assignment [J]. Transportation Science,1977,11(3):253-274.
[212]Abdel-Aty, M., Kitamura, R., Jovanis, P. Exploring route choice behavior using geographical information system-based alternative paths and hypothetical travel time information input [J]. Transportation Research Record,1995,1493:74-80
[213]Small, K.A., Noland, R., Chu, X., Lewis, D.,1999. Valuation of travel-time savings and predictability in congested conditions for highway user-cost estimation. NCHRP Report No.431, Transportation Research Board, National Research Council, USA
[214]Brownstone, D., Gabrei, S.A., Golob, T.F., Kazimi, C., Amelsfort, D.V. Drivers' willingness-to-pay to reduce travel time:evidence from San Diego 1-15 congestion pricing project [J]. Transportation Research Part A,2003,37(4):373-387.
[215]Liu, H., Recker, W., Chen, A. Uncovering the contribution of travel time reliability to dynamic route choice using real-time loop data[J]. Transportation Research Part A,2004,38(6): 435-453
[216]de Plama, A., Picard, A. Route choice decision under travel time reliability [J]. Transportation Research Part A,2005,39 (4):295-32.
[217]Fosgerau, M., Karlstrom, A. The value of reliability [J]. Transportation Research Part B,2011, 44(l):38-49.
[218]Fosgerau, M., Engelson, L. The value of travel time variance [J]. Transportation Research Part B,2007,45(1):1-8.
[219]Mirchandani, P., Soroush, H. Generalized traffic equilibrium with probabilistic travel times and perceptions [J]. Transportation Science,1987,21(3):133-152.
[220]Yin, Y, Idea, H. Assessing performance reliability of road network under non-recurrent congestion [J]. Transportation Research Record,2001,1771:148-155.
[221]Chen, A., Ji, Z., Recker, W. Travel time reliability with risk sensitive travelers [J]. Transportation Research Record,2002,1783:27-33.
[222]Yin, F., Lam, W.H.K., Idea, H. New technology and the modeling of risk-taking behavior in congested road networks [J]. Transportation Research Part C,2004,12(3/4):171-192
[223]Watling, D. User equilibrium traffic network assignment with stochastic travel times and late arrival penalty [J]. European Journal of Operational Research,2006,175(3):1539-1556.
[224]Di, S., Pan, C., Ran, B. Stochastic multiclass traffic assignment with consideration of risk-taking behaviors [J]. Transportation Research Record,2008,2085:111-123.
[225]Bell, M.G.H. A game theory approach to measuring the performance reliability of transportation networks [J]. Transportation Research Part B,2000,34(6):533-554.
[226]Bell, M.G.H., Cassir, C., Risk-averse user equilibrium traffic assignment:an application of game theory [J]. Transportation Research Part B,2002,36(8):671-681.
[227]Szeto, W.Y., O'Brien, L., O'Mathony, M., Risk-averse traffic assignment with elastic demands:NCP formulation and solution method for assessing performance reliability [J]. Networks and Spatial Economics,2006,6(3/4):313-332.
[228]Sumalee, A., Connors, R.D., Luthep, P. Network equilibrium under cumulative prospect theory and endogenous stochastic demand and supply [C]. In:Lam WHK, Wong SC, Lo HK (Eds.). Proceedings of 18th International Symposium of Transportation and Traffic Theory,2009, pp.19-38.
[229]Xu, H., Zhou, J., Xu, W. A direction-making rule for modeling travelers'route choice behavior based cumulative prospect theory [J]. Transportation Research Part C,2011,19 (2): 218-228.
[230]Zhang, C., Chen, X., Sumalee, A. Robust Wardrop's user equilibrium assignment under stochastic demand and supply:expected residual minimization approach [J]. Transportation Research Part B,2011,45(3):534-552.
[231]Uchida, T., Iida, Y. Risk assignment:a new traffic assignment model considering the risk travel time[C]. In:Daganzo CF(Ed.). Proceedings of 12th International Symposium on Transportation and Traffic Theory, Amsterdam,1993,89-105.
[232]Shao, H., Lam, W.H.K., Tam, M.L. Demand driven traffic assignment problem based on travel time reliability [J]. Transportation Research Record,2006,1985:220-230.
[233]Shao, H., Lam, W.H.K., Tam, M.L. A reliability-based stochastic traffic assignment model for network with multiple user classes under uncertain in demand [J]. Network and Spatial Economics, 2006,6(3/4):313-332.
[234]Shao, H., Lam, W.H.K., Tam, M.L., Yuan, X.M. Modeling rain effects on risk-taking behaviors of multi-user classes in road networks with uncertainty [J]. Journal of Advanced Transportation,2008,42(3):265-290.
[235]Siu, B.W.Y., Lo, H.K. Doubly uncertain transport network:degradable link capacity and perception variations in traffic conditions [J]. Transportation Research Record,2006,1964:59-69
[236]Siu, B.W.Y., Lo, H.K. Doubly uncertain transport network:Degradable capacity and stochastic demand [J]. European Journal of Operational Research,2008,191(1):166-181
[237]Lam, W.H.K., Shao, H., Sumalee, A. Modeling impacts of adverse weather conditions on a road network with uncertainties in demand and supply [J]. Transportation Research Part B,2008,42 (10):890-910.
[238]Zhou, Z., Chen, A. Comparative analysis of the three user equilibrium models under stochastic demand [J]. Journal of Advanced Transportation,2008,42(3):239-263.
[239]Chen, A., Zhou, Z. The alpha-reliable mean-excess traffic equilibrium model with stochastic travel times [J]. Transportation Research Part B,2010,44(4):493-513.
[240]Chen, A., Zhou, Z., Lam, W.H.K. Modeling stochastic perception error in the mean-excess traffic equilibrium model [J]. Transportation Research Part B,2011,45(10):1619-1640.
[241]Nie, Y. Multi-class percentile user equilibrium with flow-dependent stochasticy. Transportation Research Part B,2011,45(10):1641-1659.
[242]Jorion, P. Value-at-Risk:the new benchmark for controlling market risk. Irwin Professional Publishing, Chicago,1996.
[243]Hall, P.R. Travel outcome and performance:the effect of uncertainty on accessibility [J]. Transportation Research Part B,1983,17(4):275-290.
[244]Yamai, Y., Yoshiba, Y. On the validity of value-at-risk:comparative analysis with expected shortfall [J]. Monetary and Economic Studies,2002,20(1):57-85
[245]Wu, X., Nie, Y. Modeling heterogeneous risk-taking behavior in route choice:a stochastic dominance approach [J]. Transportation Research Part A,2011,45 (9):896-915
[246]Xu, X., Chen, A., Zhou, Z., Cheng, L. A multi-class mean-excess traffic equilibrium model with elastic demand [J]. Journal of Advanced Transportation,2012, DOI:10.1002/atr.205.
[247]Zhu, S.S., Fukushima, M. Worst-Case Conditional Value-at-Risk with application to robust portfolio management [J]. Operations Research,2009,57(5):1155-1168
[248]Ordonez, F., Stier-Moses, N.E. Wardrop equilibria with risk-averse users [J]. Transportation Science,2010,44(1):63-86
[249]Facchinei, F., Pang, J.S. Finite-dimensional variational inequalities and complementarity problem[M], Springer-Verlag, New York,2003.
[250]Bernstein, D., Gabriel, S.A. The traffic equilibrium problem with nonadditive path costs [J]. Transportation Science,1997,31(4):337-348
[251]Xu, M., Chen, A., Qu, Y.C., Gao, Z.Y. A semismooth Newton method for traffic equilibrium problem with a general nonadditive route cost [J]. Applied Mathematical Modeling,2011,35(6): 3048-3062
[252]Chen, A., Zhou, Z., Xu, X.D.,2012. A self-adaptive gradient projection algorithm for the nonadditive traffic equilibrium problem [J]. Computers & Operations Research,2012,39(2): 127-138
[253]Panicucci, B., Pappalardo, M., Passacantando, M.,2007. A path-based double projection method for solving the asymmetric traffic network equilibrium problem [J]. Optimization Letters, 1(2):171-185
[254]Lo, H.K., Chen, A. Traffic equilibrium problem with route-specific costs:formulation and algorithms [J]. Transportation Research Part B,2000,34 (6):493-513
[255]Nguyen, S., Dupuis, C. An efficient of method for computing traffic equilibria in network with asymmetric transportation costs [J]. Transportation Science,1984,18(2):185-202
[256]Nemirovski, A., Shapiro, A. Convex approximations of chance constrained programs [J]. SIAM Journal on Optimization,,2006,17(4):969-996
[257]Janson, B.N. Network design effects of dynamic traffic assignment. Journal of Transportation Engineering [J],1995,121(1):18-26.
[258]Waller, S.T. Optimization and control of stochastic dynamic Transportation system: formulations, solution methodologies, and Computational experience [D], Northwestern University, 2000.
[259]Jeon, K., Ukkusuri, S.V., Waller, S.T. Heuristic approach for discrete network design problem accounting for dynamic traffic assignment conditions:formulation, solution methodologies, implementations and computation experiences [C]. The 84th Annual Meeting of the Transportation Research Board, Washington, D.C.,1995.
[260]Waller, S.T., Ziliaskopoulos, A.K. A chance-constrained based stochastic dynamic traffic assignment model:analysis, formulation and solution algorithms [J]. Transportation Research Part C, 2006,14(6):418-427
[261]Yazici, M.A., Ozbay, K. Evacuation network modeling via dynamic traffic assignment with probabilistic demand and capacity constraints [J]. Transportation Research Record,2010, 2196:11-20
[262]Li, Y., Waller, S.T., Ziliaskopoulos, T. A decomposition scheme for system optimal dynamic traffic assignment models [J]. Networks and Spatial Economics,2003,3(4):441-455
[263]Zhu, F., Ukkusuri, S.V. A cell based dynamic system optimum model with non-holding back flows [J]. Transportation Research Part C,2013,36,367-380
[264]Ben-Tal, A., Boyd, S., Nemirovski, A. Extending scope of robust optimization:robust counterparts of uncertain problems [J]. Mathematical Programming,2006,107(1):63-89
[265]Chung, B.D., Yao, T., Zhang, B. Dynamic traffic assignment under uncertainty:A distributional robust joint chance-constrained Approach [J]. Networks and Spatial Economics,2012, 12(1):167-181
[266]Alizadeh, F., Goldfrb, D. Second-order Cone Programming. [J] Mathematical Programming, 2003,95(1):3-51
[267]Calafiore, G., EL Ghaoui, L. On distributionally robust chance-constrained linear programs with applications [J]. Journal of Optimization Theory and Application,2006,130(1):1-22
[268]Prekopa, A. On probabilistic constrained programming [C]. In:Proceedings of the Princeton Symposium on Mathematical Programming,113-138. Princeton University Press, Princeton,1970.
[269]Shapiro, A., Kleywegt, A.J. Minimax analysis of stochastic problems [J]. Optimization Methods and Software,2002,17(3):523-542.
[270]Chen, W., Sim, M. Goal-driven optimization. Operations Research,2009,57(2):342-357
[271]LOfberg, J.YALMIP:a toolbox for modeling and optimization in MATLAB [C]. In: Proceedings of the CACSD conference, Taipei,2004.
[272]Sturn, J.S., Using Sedumi 1.02, a Matlab toolbox for optimization over symmetric cones [J]. Optimization Methods and Software,1999,11(1-4):625-653.
[273]Luo, Z.Q., Pang, J.S., Ralph, D. Mathematical programs with Equilibrium Constraints [M], Cambridge University Press, Cambridge, England,1996.
[274]Huang, D.S., Zhu, S.S., Fabozzi, F.J., Fukushima, M. Portfolio selection undet distributional uncertainty:a relative robust CvaR approach [J]. European Journal of Operations Research,2010, 203(1):185-194.
[275]Ralph, D., Wright, S.J. Some properties of regularization and penalization schemes for MPECs [J]. Optimization Method for Software,2004,19(5):527-556
[276]Hoheisel, T., Kanzow, C., Schwartz, A. Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints [J]. Mathematical Programming,2013,137(1-2),257-288.
[277]Scholte, S.,2001. Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM Journal on Optimization,11(4):918-936
[278]Chen, X., Zhang Y.H., Uncertain Linear programs:Extended affinely adjustable robust counterparts. Operations Research,2009,57(6):1469-1482.
[279]Isii, K. The extrema of probability determined by generalized moments (i) bounded random variables. Annals of the Institue of Statistical Mathemtics,1960,12(2):119-134.