不确定系统优化理论与应用研究
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摘要
不确定规划是不确定环境下的系统优化理论的重要内容。传统的不确定规划主要分为两大类：随机规划和模糊规划，并且在生产、经济及管理等诸多方面已有广泛的应用，特别是随机线性规划和模糊线性规划理论较为完善，应用更加广泛。随机或模糊机会约束规划是指在目标函数和约束条件中含有随机或模糊参数，机会的意思表示约束条件成立的概率或可能性。求解随机或模糊机会约束线性规划的传统方法是根据事先给定的置信水平，把相应的机会约束转换为各自确定的或清晰的数学规划问题进行计算。然而，这种做法只是随机或模糊机会约束线性规划的间接算法，有较大的局限性。这是因为，对较复杂的情形，得到的确定或清晰的数学规划通常是非线性规划，给计算带来复杂性，甚至有些问题难以转化为确定或清晰的数学规划问题。因此，如何根据随机或模糊机会约束线性规划的特点，提出和设计直接计算的方法是一个重要的问题。
不确定因素的存在确实会使决策系统含有随机或模糊参数，但不确定性决不完全单纯地是由随机或模糊单一因素来决定的，有可能是随机和模糊两种因素的混合或综合体现。提出同时带有随机和模糊参数的混合规划模型，不仅是不确定规划理论完善的需要，而且也是实际应用的需要。
随机最优控制也是随机不确定系统优化的一个重要内容。随机最优控制的目标函数，可分为折扣费用问题和平均期望费用问题等。具体的目标函数表达形式，往往根据实际应用问题的类型而变化，因而传统的随机最优控制问题出现了在多种目标函数下的理论研究形式，然而他们的研究手法和表现形式却非常相似，是否能在一个较为统一的框架下表现它们，则成了一些研究工作者的追求目标。随机最优控制的研究是促进倒向随机微分系统理论从出现到发展的重要因素之一，也正是倒向随机微分方程的出现，使得可以通过正向和倒向随机微分系统的耦合，用倒向随机微分方程的解定义随机最优控制的目标函数，统一多种传统的目标函数成为可能，在这一框架下研究随机最优控制问题则是传统随机最优控制的不平凡推广。
以上问题和想法促使作者进行以下研究：(1)克服不确定线性规划的计算需要转化成等价的确定性(或清晰)数学规划进行计算的不足，寻求直接计算的方法；(2)克服传统不确定规划模型中不确定因素的单一性，提出随机和模糊混合的不确定规划模型；(3)进一步强化倒向随机微分方程在随机不确定系统最优控制问题中的应用，实质性地推广传统的随机最优控制相关理论，扩大随机最优控制的应用领域，特别是在金融工作中的广泛应用。
到目前为止，尚没有作者对不确定规划理论和应用做过综述，也没有作者针对倒向随机微分方程在随机最优控制方面的应用做过综述。本论文首先综述了不确定规划理论及应用的研究成果和连续随机系统最优控制研究成果，以使人们对
不确定系统优化理论与应用研究摘要
不确定规划理论应用和随机最优控制研究有一个较为系统的了解。
本论文首次提出基于随机模拟方法的单纯形算法，为随机机会约束线性规划
提供了一种直接计算方法。借助随机线性规划单纯形算法的设计思想，重载随机
线性规划的相关概念，首次提出和设计了基于模糊模拟方法的单纯形算法，为模
糊机会约束线性规划也提供了一种直接计算方法。应用和数值计算举例，表明了
算法的有效性。直接计算方法的确立将进一步完善随机和模糊线性规划理论，也
将有助于推动随机和模糊线性规划在实际特别是大型实际问题中的应用。
本论文首次提出了同时带有随机和模糊参数的几类混合规划模型和算法。主
要包括:带有模糊参数的期望值模型、随机和模糊混合机会约束规划模型及其相
应的基于随机和模糊模拟方法的遗传算法。应用及数值问题举例表明了模型的实
际意义、合理性及其算法的有效性。这些工作不仅为在不确定性环境下更切合实
际地分析解决实际问题提供了适用的模型和算法，同时作为随机或模糊规划的一
种推广，也丰富了不确定规划的内容，为进一步进行完善不确定规划理论和应用
提供了引导和基础性的工作。
本论文以倒向随机微分方程为基础，通过正向和倒向随机微分系统的祸合，
用倒向随机微分方程的解定义随机最优控制的目标函数，研究了随机最优控制问
题，推广了传统随机最优控制问题的动态规划原理并研究了与随机最优控制相关
的H田卫ilton一Jacobi一Bellman方程等。主要包含:在正倒向随机微分系统系数弱于
LIPschitz一连续的情况下，推广了传统随机最优控制问题的动态规划原理及相关的
Hamilton一Jacobi一Bellman方程等。作为研究基础，给出了倒向随机微分方程新比
较定理，并推广通常比较定理到系数非Lipschitz一连续的情形.在Lipschitz一连续
的情况下，许多结果也是新的。
本论文最后在总结、综合和分析文中研究结果的基础上，沿着文中的研究思
整和途径，拐出了10个值得研究的方向和问题。
Uncertain programming is an important content of the optimization theory for uncertain systems. Traditional uncertain programming mainly contains stochastic programming and fuzzy programming, which have many applications in manufacture, economy and management etc. Especially, the theory of stochastic or fuzzy linear programming is more complete, so it has more applications than stochastic or fuzzy nonlinear programming. Stochastic or fuzzy chance constrained programming refer to the objective functions and the constraint conditions contain stochastic or fuzzy parameters, the meaning of chance is the probability or possibility that the constraint conditions are satisfied. The usual methods of dealing with stochastic or fuzzy chance constrained linear programming are converting the chance constraint conditions to respective definitive or clear mathematic programming problems to compute them, according to given belief level. However, these methods are indirect algorithm, which have some limitations. In the case of complex situation, the obtained definitive or clear mathematic programming problems are usually nonlinear programming which bring complexity to computing, and even some stochastic or fuzzy problems are difficult to convert to definitive or clear problems. Therefore, It is important problem how to give and design direct computing methods for stochastic or fuzzy chance constrained linear programming according to which characteristic.
The existence of uncertain factors really cause that decision systems contain stochastic or fuzzy parameters, but the uncertainty is not just stochastic or fuzzy, which might be hybrid of two factors of stochastic and fuzzy. It is need for not only theory but also applications to establish the hybrid programming models that contains stochastic and fuzzy parameters.
Stochastic optimal control is an important content of the optimization theory for uncertain systems too. The objective function for the stochastic optimal control can be classified by the discounted cost problem and average expectation cost problem etc. The expression of specific objective function often depends its actual application problems, thus there are many types of theory study under the several objective functions in the usual stochastic optimal control, but the study methods are very similar. So it is the aim of many authors to give a uniform objective function for studying stochastic optimal control problems. For the appearance of the backward stochastic differential equations (BSDE), the studies of the stochastic optimal control problems are one of the main factors, and along with studies of BSDE. a uniform objective function for the stochastic optimal control can be defined using the solution of BSDE by the coupled forward-backward stochastic differential equations. It is not trivial generalization for the usual theory of the stochastic optimal control to study the stochastic optimal control problems.
The above problems motivated the author to: (1) conquer the lack of the indirect
computing methods for the uncertain linear programming to seek the direct computing method; (2) conquer the Singularity of stochastic or fuzzy factor in the usual uncertain programming models to give the hybrid programming models which contains stochastic and fuzzy parameters; (3) further strengthen the applications of BSDE in the stochastic optimal control to extend the related theories of the usual stochastic optimal control, and to enlarge the applied field.
Upon to date, there is no existing review on uncertain programming theory and its applications, and there is no existing review on the applications of BSDE in the stochastic optimal control problems. In the dissertation, recent studies on uncertain programming theories and their applications and the optimal control for continuous stochastic systems are first systematically overviewed.
In the dissertation, the simplex method basing on stochastic simulations is first presented, which provides a direct approach for computing stochastic chance constrained line
不确定因素的存在确实会使决策系统含有随机或模糊参数，但不确定性决不完全单纯地是由随机或模糊单一因素来决定的，有可能是随机和模糊两种因素的混合或综合体现。提出同时带有随机和模糊参数的混合规划模型，不仅是不确定规划理论完善的需要，而且也是实际应用的需要。
随机最优控制也是随机不确定系统优化的一个重要内容。随机最优控制的目标函数，可分为折扣费用问题和平均期望费用问题等。具体的目标函数表达形式，往往根据实际应用问题的类型而变化，因而传统的随机最优控制问题出现了在多种目标函数下的理论研究形式，然而他们的研究手法和表现形式却非常相似，是否能在一个较为统一的框架下表现它们，则成了一些研究工作者的追求目标。随机最优控制的研究是促进倒向随机微分系统理论从出现到发展的重要因素之一，也正是倒向随机微分方程的出现，使得可以通过正向和倒向随机微分系统的耦合，用倒向随机微分方程的解定义随机最优控制的目标函数，统一多种传统的目标函数成为可能，在这一框架下研究随机最优控制问题则是传统随机最优控制的不平凡推广。
以上问题和想法促使作者进行以下研究：(1)克服不确定线性规划的计算需要转化成等价的确定性(或清晰)数学规划进行计算的不足，寻求直接计算的方法；(2)克服传统不确定规划模型中不确定因素的单一性，提出随机和模糊混合的不确定规划模型；(3)进一步强化倒向随机微分方程在随机不确定系统最优控制问题中的应用，实质性地推广传统的随机最优控制相关理论，扩大随机最优控制的应用领域，特别是在金融工作中的广泛应用。
到目前为止，尚没有作者对不确定规划理论和应用做过综述，也没有作者针对倒向随机微分方程在随机最优控制方面的应用做过综述。本论文首先综述了不确定规划理论及应用的研究成果和连续随机系统最优控制研究成果，以使人们对
不确定系统优化理论与应用研究摘要
不确定规划理论应用和随机最优控制研究有一个较为系统的了解。
本论文首次提出基于随机模拟方法的单纯形算法，为随机机会约束线性规划
提供了一种直接计算方法。借助随机线性规划单纯形算法的设计思想，重载随机
线性规划的相关概念，首次提出和设计了基于模糊模拟方法的单纯形算法，为模
糊机会约束线性规划也提供了一种直接计算方法。应用和数值计算举例，表明了
算法的有效性。直接计算方法的确立将进一步完善随机和模糊线性规划理论，也
将有助于推动随机和模糊线性规划在实际特别是大型实际问题中的应用。
本论文首次提出了同时带有随机和模糊参数的几类混合规划模型和算法。主
要包括:带有模糊参数的期望值模型、随机和模糊混合机会约束规划模型及其相
应的基于随机和模糊模拟方法的遗传算法。应用及数值问题举例表明了模型的实
际意义、合理性及其算法的有效性。这些工作不仅为在不确定性环境下更切合实
际地分析解决实际问题提供了适用的模型和算法，同时作为随机或模糊规划的一
种推广，也丰富了不确定规划的内容，为进一步进行完善不确定规划理论和应用
提供了引导和基础性的工作。
本论文以倒向随机微分方程为基础，通过正向和倒向随机微分系统的祸合，
用倒向随机微分方程的解定义随机最优控制的目标函数，研究了随机最优控制问
题，推广了传统随机最优控制问题的动态规划原理并研究了与随机最优控制相关
的H田卫ilton一Jacobi一Bellman方程等。主要包含:在正倒向随机微分系统系数弱于
LIPschitz一连续的情况下，推广了传统随机最优控制问题的动态规划原理及相关的
Hamilton一Jacobi一Bellman方程等。作为研究基础，给出了倒向随机微分方程新比
较定理，并推广通常比较定理到系数非Lipschitz一连续的情形.在Lipschitz一连续
的情况下，许多结果也是新的。
本论文最后在总结、综合和分析文中研究结果的基础上，沿着文中的研究思
整和途径，拐出了10个值得研究的方向和问题。
Uncertain programming is an important content of the optimization theory for uncertain systems. Traditional uncertain programming mainly contains stochastic programming and fuzzy programming, which have many applications in manufacture, economy and management etc. Especially, the theory of stochastic or fuzzy linear programming is more complete, so it has more applications than stochastic or fuzzy nonlinear programming. Stochastic or fuzzy chance constrained programming refer to the objective functions and the constraint conditions contain stochastic or fuzzy parameters, the meaning of chance is the probability or possibility that the constraint conditions are satisfied. The usual methods of dealing with stochastic or fuzzy chance constrained linear programming are converting the chance constraint conditions to respective definitive or clear mathematic programming problems to compute them, according to given belief level. However, these methods are indirect algorithm, which have some limitations. In the case of complex situation, the obtained definitive or clear mathematic programming problems are usually nonlinear programming which bring complexity to computing, and even some stochastic or fuzzy problems are difficult to convert to definitive or clear problems. Therefore, It is important problem how to give and design direct computing methods for stochastic or fuzzy chance constrained linear programming according to which characteristic.
The existence of uncertain factors really cause that decision systems contain stochastic or fuzzy parameters, but the uncertainty is not just stochastic or fuzzy, which might be hybrid of two factors of stochastic and fuzzy. It is need for not only theory but also applications to establish the hybrid programming models that contains stochastic and fuzzy parameters.
Stochastic optimal control is an important content of the optimization theory for uncertain systems too. The objective function for the stochastic optimal control can be classified by the discounted cost problem and average expectation cost problem etc. The expression of specific objective function often depends its actual application problems, thus there are many types of theory study under the several objective functions in the usual stochastic optimal control, but the study methods are very similar. So it is the aim of many authors to give a uniform objective function for studying stochastic optimal control problems. For the appearance of the backward stochastic differential equations (BSDE), the studies of the stochastic optimal control problems are one of the main factors, and along with studies of BSDE. a uniform objective function for the stochastic optimal control can be defined using the solution of BSDE by the coupled forward-backward stochastic differential equations. It is not trivial generalization for the usual theory of the stochastic optimal control to study the stochastic optimal control problems.
The above problems motivated the author to: (1) conquer the lack of the indirect
computing methods for the uncertain linear programming to seek the direct computing method; (2) conquer the Singularity of stochastic or fuzzy factor in the usual uncertain programming models to give the hybrid programming models which contains stochastic and fuzzy parameters; (3) further strengthen the applications of BSDE in the stochastic optimal control to extend the related theories of the usual stochastic optimal control, and to enlarge the applied field.
Upon to date, there is no existing review on uncertain programming theory and its applications, and there is no existing review on the applications of BSDE in the stochastic optimal control problems. In the dissertation, recent studies on uncertain programming theories and their applications and the optimal control for continuous stochastic systems are first systematically overviewed.
In the dissertation, the simplex method basing on stochastic simulations is first presented, which provides a direct approach for computing stochastic chance constrained line
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任建芳．赵瑞清，鲍兰平(2000)．随机最优证券投资组合模型．系统工程理论与实践，2000；(9)：14-18
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Peng, S. and Wu, Z.(1999) . Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J Control Optim., 1999; 37(3) : 825-859
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Situ Rong(1997). On solutions of backward stochastic differential equations and applications. Stochastic Processes and Applications. 1997; 66:209-236
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Tang, J. and Wang D.(1997). An interactive approach based on GA. for a type of quadratic programming problems with fuzzy objective and resources. Computers and Operations Research, 1997; 24(5): 413-422
Tang, J., Wang D. Fung, R.Y.K.(1998). Model and method based on GA for non-linear programming problems with fuzzy objective and resources. Int. J. of System Science, 1998; 29(8): 907-913
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Yazenin, A.V.(1996). On the problem of possibilistic optimization. Fuzzy Set and Systems, 1996; 81:133-140
Zadeh, L.A.(1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Set and Systems, 1978; 1:3-28
Zariphopoulou, Th.(1994). Optimal investment/consumption models with constraints. SIAM J. Contr. Optim., 1994; 32:59-85
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陈增敬(1996)．股票债券市场中的一个随机规划问题．山东大学学报(自然科学版)，1996；31(1)：42-47
陈增敬，胡发胜(1997)．股票债券市场中一般的随机规划问题．经济数学，1997；14(1)：20-24
丁晓东(1993)．关于泛函型多维随机微分方程弱解的存在性．纺织基础科学学报，1993；3：146-149
丁晓东，吴让泉，邵世煌(2001)．一类含有模糊和随机参数的期望值规划模型及应用．控制与决策，2001；16(Suppl)：745-748
丁晓东，吴让泉，邵世煌(2002)．含有模糊和随机参数的混合机会约束规划模型．控制与决策，2002；17(4)
方述成，汪定伟(1997)．模糊数学与模糊优化．北京：科学出版社，1997
费为银(2002)．基于随机控制的动态资产研究．东华大学博士论文．2002
龚光鲁(1987)．随机微分方程引论．北京：北京大学出版社，1987
何声武，汪嘉冈，严加安(1995)．半鞅与随机积分．北京：科学出版社，1995
胡金焱，嵇少林(2001)．一类优化问题得到向随机微分方程方法．山东大学学报，2001；36(9)：282-286
黄小原(2001)．金融工程：信息与控制领域的挑战．信息与控制．2001；30(2)：150-154
黄志远(1988)．随机分析学基础．武汉：武汉大学出版社，1988
李小军(2000)．Lagrange方法和期权定价．应用概率统计，2000；16(4)：373-378
刘宝碇，赵瑞清(1998)．随机规划与模糊规划．北京：清华大学出版社，1998
刘宝碇，赵瑞清(1999)．不确定规划进一步的研究问题．指挥技术学院学报，1999；10(6)：1-5
刘海龙，苗实，潘德惠(2000)．风险敏感性最优控制问题研究．控制与决策，2000：15(1)：107-109
刘海龙，吴冲锋(2001)．非完全市场最优消费和投资策略研究．系统工程，2001；19(1)：39-42
刘坤会(1986)．Richard模型最佳控制的存在性及其费用的函数结构．应用数学学报，1986：9(4)：385-408
刘坤会(1990)．一类脉冲控制最佳费用的解析表达式．应用数学学报，1990；13(4)：385-390
秦学志，吴冲锋(2001)．基于鞅和线性规划对偶原理的或有要求权定价方法．控制与决策，2001；16(Suppl．)：846-848
任建芳．赵瑞清，鲍兰平(2000)．随机最优证券投资组合模型．系统工程理论与实践，2000；(9)：14-18
孙世良，赵新有(2001)．随机控制理论的现状和展望．太原理工大学学报，2001；32(2)：209-212
唐家福，汪定伟(1998)．模糊非线性规划对称模型基于遗传算法的模糊最优解．控制理论与应用，1998；15(4)：525-530
唐家福．汪定伟(2000)．模糊优化理论与方法的研究综述．控制理论与应剧，2000；17(2)：159-164
吴臻，徐爱平(2000)．状态约束下完全耦合的正倒向随机控制问题的最大值原理．山东大学学报(自然科学版)，2000；35(4)：373-380
严加安，彭实戈，方诗赞和吴黎明(1997)．随机分析选讲．科学出版社，1997
张建中，许绍吉(1990)．线性规划．北京：科学出版社，1990
赵莉萍，王建华，龙刚(2000)．混合遗传算法在随机规划问题中的应用．华东理工大学学报．2000；26f51：547-551