二阶锥规划和二阶锥互补问题的算法研究
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摘要
二阶锥规划是在一个仿射空间和有限个二阶锥的笛卡尔积的交集上极小化或者极大化一个线性函数问题.其约束是非线性的,但却是凸的,因此二阶锥规划属于凸规划.二阶锥规划是半定规划的特例,而线性规划、凸二次规划和二次约束的凸二次优化等可作为它的特例.由于其宽广的应用范围、特殊的锥结构和计算上的方便性,所以它有其独立的研究价值.由于它的广泛应用和原始-对偶内点法的迅速发展,二阶锥规划已经成为数学规划领域的一个重要研究方向.
二阶锥互补问题是一类均衡优化问题.近几年,人们借助欧几里得约当代数技术,在对称锥互补问题的研究方面取得了突破性进展并使之逐渐受到重视.二阶锥互补问题是二阶锥规划的推广,它包括线性二阶锥互补问题和非线性二阶锥互补问题.近几年来,人们对它的研究呈上升趋势,主要研究内容包括：解的存在性与特征,势函数和误差界,各种光滑化方法和优化方法,以及各种实际应用.关于非线性二阶锥规划及其互补问题的理论和算法,其研究方兴未艾,是当今人们关注的热点课题之一.
本文主要研究二阶锥规划及其互补问题的算法.论文共分八个部分：
第一部分,介绍二阶锥规划和二阶锥互补问题的模型,研究背景、意义和现状,并对现有算法加以总结,从而引出需要进一步解决的问题及本文所作的主要工作.
第二部分,简要介绍有关预备知识.主要介绍欧几里得约当代数、二阶锥规划的最优性条件、中心路径条件、互补条件和原始-对偶内点算法.
第三部分,定义了中心路径的一个宽邻域,并给出二阶锥规划问题的一个宽邻域原始-对偶路径跟踪内点算法,使得所有迭代点都跟踪这个宽领域,得到目前为止宽邻域路径跟踪内点算法最好的迭代复杂性界.
第四部分,给出一个与二阶锥关联的光滑函数,并研究该光滑函数的性质.基于此光滑函数,给出二阶锥规划问题的一个光滑牛顿类型算法.该算法把扰动最优性条件重新表述成一个线性方程组进行求解,得到比相应的内点法更强的结果.它可以始于任意初始点；每步迭代只需求解一个线性方程组,执行一次线搜索；在较弱的假设下,算法所产生的序列全局收敛,并且在无严格互补条件的情况下Q二次收敛于问题的最优解.
第五部分,受求解变分不等式问题的交替方向法的启发,给出二阶锥规划的一个带完全牛顿步的非内点全局收敛算法.该算法的主要思想是把原始-对偶最优性条件中的互补条件重新表述成一个投影方程.所给算法对初始点的可行性不作任何要求；每步迭代只需求解一个系数矩阵固定的线性方程组,执行两次简单的投影运算；无需执行任何线搜索；不要求约束系数矩阵的行向量组线性独立；算法所产生的序列全局收敛到问题的最优解,无需严格互补条件,这一结果强于相应的内点法和光滑算法.
第六部分,研究一类特殊的二阶锥规划问题带有P0函数的非线性互补问题.基于一个新的带有惩罚项的光滑函数,把问题近似成参数化的光滑方程组,并且给出一个新的非内部连续化方法.所给算法在每步迭代只需求解一个线性方程组,执行一次Armijo类型的线搜索.在不需要严格互补条件的情况下,证明了该算法是全局收敛和超线性收敛的.并且,在一个较弱的条件下该算法具有局部二阶收敛性.通过数值实验证实了算法的可行性和有效性.
第七部分,我们给出一类新的含有单参数的光滑二阶锥互补函数,在适当的条件下证明该类光滑函数是强制的.基于所给的单参数类二阶锥互补函数,给出求解单调二阶锥互补问题的一个光滑牛顿算法,在一定条件下证明了算法的全局收敛性和局部超线性收敛性.
最后,在第八部分,针对现有的二阶锥规划及其互补问题算法存在的问题,提出了有待进一步研究的课题.
Second-order cone programming is a kind of problem which minimizes or max-imizes a linear function over the intersection of an affine space with the Cartesian product of a finite number of second-order cones. The constraints of second-order cone programming are nonlinear, but they are convex, so it belongs to convex programming. Second-order cone programming is a special case of semidefinite programming, but it covers linear programming, convex quadratic programming, quadratically constraint convex quadratic optimization as well as other problems. The study of second-order cone programming has its independent value due to its wide range of applications, special cone constructer and computational convenience. Second-order cone program-ming has already become one of the important research direction in the mathematical programming fields.
Second-order cone complementarity problems are a class of equilibrium problems. In recent years, based on the technique of Euclidean Jordan Algebra, many researchers have achieved a breakthrough in the study of symmetric cone complementarity prob-lems, and made them attract a lot of attention. Second-order cone complementarity problems are generalizations of the second-order cone programming. They contains both linear and nonlinear second-order cone complementarity problems. In the recent years, the study of second-order cone complementarity problems is increasing. The study contains:the existence and properties of the solution, potential function and error bound, many smoothing method and optimization method, and many real appli-cations. However, the study of nonlinear second-order cone programming and nonlinear second-order cone complementarity problems is just in its initial state, and it is one of the current hot fields that researchers pay attention to.
The thesis mainly on the study of algorithms for second-order cone programming and second-order cone complementarity problems. It consists of the following eight parts:
(1) In the first chapter, we introduce the modeling, research background, signif-icance and the recent situations of second-order cone programming and second-order cone complementarity problems. The existing methods is discussed and summarized, which drops the problems need to be further solved and the main results of the paper.
(2) In the second part, we briefly introduce the preliminary knowledge. We mainly focus on Euclidean Jordan algebra, and optimality conditions, central path conditions, complementarity condition and interior-point method for second-order cone program-ming.
(3) We define a wide neighborhood of the central path, and present a large-update primal-dual path-following interior-point algorithm for second-order cone programming in the third chapter. Each iterate is always following the wide neighborhood. We show that the method has the best iteration complexity bound of the wide neighborhood algorithms for solving second-order cone programming.
(4) In the fourth chapter, a smoothing function corresponding to second-order cone is given, and its properties are studied. Based on the smoothing function, a smooth-ing Newton-type method is proposed for solving second-order cone programming. The method reformulates the perturbed optimality conditions into a system of linear equa-tions to solve. Its results are stronger than corresponding interior-point methods, and its initial point is arbitrary. At each iteration, the proposed algorithm need only to solve one system of linear equations and perform one line search. The sequence gen-erated by the method converges globally and Q-quadratically to the solution of the SOCP under a mild assumption, without strict complementarity condition.
(5) A globally convergent non-interior point algorithm with full Newton step for second-order cone programming is proposed in the fifth part. The main idea of the algorithm is that we cast the complementary equation of the primal-dual optimality conditions as a projection equation. This algorithm can start from an arbitrary point. Without performing any line search, we only need to solve a system of linear equa-tions with the same coefficient matrix and compute two simple projections at each iteration. It does not require the row vectors of coefficient matrix of constraints to be linearly independent. The proposed algorithm is globally convergent without strict complementarity conditions.
(6) In the sixth part, we study the special second-order cone programming-nonlinear complementarity problem with P0-function. Based on a new smoothing func-tion with penalty, the problem is approximated by a family of parameterized smooth equations and a new non-interior-point continuation method is presented. At each it-eration, the proposed algorithm only need to solve one system of linear equations and perform one Armijo-type line search. The algorithm is proved to be globally and locally superlinearly convergent without strict complementarity. Moreover, the quadratic con-vergence rate is achieved under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the proposed algorithm.
(7) In chapter seven, we introduce a class of one-parametric smoothing second-order cone complementarity functions. Under suitable assumptions, we prove that these functions are coercive. Based on this class of one-parametric smoothing functions, a smoothing Newton method is proposed to solve the monotone second-order cone complementarity problems. It is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Preliminary numerical results for randomly generated second-order cone programs are reported, which confirm the favorable theoretical properties of the method.
Finally, based on the problems of the existing methods, including the methods proposed in the thesis, the last chapter is concerned with some future research work.
二阶锥互补问题是一类均衡优化问题.近几年,人们借助欧几里得约当代数技术,在对称锥互补问题的研究方面取得了突破性进展并使之逐渐受到重视.二阶锥互补问题是二阶锥规划的推广,它包括线性二阶锥互补问题和非线性二阶锥互补问题.近几年来,人们对它的研究呈上升趋势,主要研究内容包括：解的存在性与特征,势函数和误差界,各种光滑化方法和优化方法,以及各种实际应用.关于非线性二阶锥规划及其互补问题的理论和算法,其研究方兴未艾,是当今人们关注的热点课题之一.
本文主要研究二阶锥规划及其互补问题的算法.论文共分八个部分：
第一部分,介绍二阶锥规划和二阶锥互补问题的模型,研究背景、意义和现状,并对现有算法加以总结,从而引出需要进一步解决的问题及本文所作的主要工作.
第二部分,简要介绍有关预备知识.主要介绍欧几里得约当代数、二阶锥规划的最优性条件、中心路径条件、互补条件和原始-对偶内点算法.
第三部分,定义了中心路径的一个宽邻域,并给出二阶锥规划问题的一个宽邻域原始-对偶路径跟踪内点算法,使得所有迭代点都跟踪这个宽领域,得到目前为止宽邻域路径跟踪内点算法最好的迭代复杂性界.
第四部分,给出一个与二阶锥关联的光滑函数,并研究该光滑函数的性质.基于此光滑函数,给出二阶锥规划问题的一个光滑牛顿类型算法.该算法把扰动最优性条件重新表述成一个线性方程组进行求解,得到比相应的内点法更强的结果.它可以始于任意初始点；每步迭代只需求解一个线性方程组,执行一次线搜索；在较弱的假设下,算法所产生的序列全局收敛,并且在无严格互补条件的情况下Q二次收敛于问题的最优解.
第五部分,受求解变分不等式问题的交替方向法的启发,给出二阶锥规划的一个带完全牛顿步的非内点全局收敛算法.该算法的主要思想是把原始-对偶最优性条件中的互补条件重新表述成一个投影方程.所给算法对初始点的可行性不作任何要求；每步迭代只需求解一个系数矩阵固定的线性方程组,执行两次简单的投影运算；无需执行任何线搜索；不要求约束系数矩阵的行向量组线性独立；算法所产生的序列全局收敛到问题的最优解,无需严格互补条件,这一结果强于相应的内点法和光滑算法.
第六部分,研究一类特殊的二阶锥规划问题带有P0函数的非线性互补问题.基于一个新的带有惩罚项的光滑函数,把问题近似成参数化的光滑方程组,并且给出一个新的非内部连续化方法.所给算法在每步迭代只需求解一个线性方程组,执行一次Armijo类型的线搜索.在不需要严格互补条件的情况下,证明了该算法是全局收敛和超线性收敛的.并且,在一个较弱的条件下该算法具有局部二阶收敛性.通过数值实验证实了算法的可行性和有效性.
第七部分,我们给出一类新的含有单参数的光滑二阶锥互补函数,在适当的条件下证明该类光滑函数是强制的.基于所给的单参数类二阶锥互补函数,给出求解单调二阶锥互补问题的一个光滑牛顿算法,在一定条件下证明了算法的全局收敛性和局部超线性收敛性.
最后,在第八部分,针对现有的二阶锥规划及其互补问题算法存在的问题,提出了有待进一步研究的课题.
Second-order cone programming is a kind of problem which minimizes or max-imizes a linear function over the intersection of an affine space with the Cartesian product of a finite number of second-order cones. The constraints of second-order cone programming are nonlinear, but they are convex, so it belongs to convex programming. Second-order cone programming is a special case of semidefinite programming, but it covers linear programming, convex quadratic programming, quadratically constraint convex quadratic optimization as well as other problems. The study of second-order cone programming has its independent value due to its wide range of applications, special cone constructer and computational convenience. Second-order cone program-ming has already become one of the important research direction in the mathematical programming fields.
Second-order cone complementarity problems are a class of equilibrium problems. In recent years, based on the technique of Euclidean Jordan Algebra, many researchers have achieved a breakthrough in the study of symmetric cone complementarity prob-lems, and made them attract a lot of attention. Second-order cone complementarity problems are generalizations of the second-order cone programming. They contains both linear and nonlinear second-order cone complementarity problems. In the recent years, the study of second-order cone complementarity problems is increasing. The study contains:the existence and properties of the solution, potential function and error bound, many smoothing method and optimization method, and many real appli-cations. However, the study of nonlinear second-order cone programming and nonlinear second-order cone complementarity problems is just in its initial state, and it is one of the current hot fields that researchers pay attention to.
The thesis mainly on the study of algorithms for second-order cone programming and second-order cone complementarity problems. It consists of the following eight parts:
(1) In the first chapter, we introduce the modeling, research background, signif-icance and the recent situations of second-order cone programming and second-order cone complementarity problems. The existing methods is discussed and summarized, which drops the problems need to be further solved and the main results of the paper.
(2) In the second part, we briefly introduce the preliminary knowledge. We mainly focus on Euclidean Jordan algebra, and optimality conditions, central path conditions, complementarity condition and interior-point method for second-order cone program-ming.
(3) We define a wide neighborhood of the central path, and present a large-update primal-dual path-following interior-point algorithm for second-order cone programming in the third chapter. Each iterate is always following the wide neighborhood. We show that the method has the best iteration complexity bound of the wide neighborhood algorithms for solving second-order cone programming.
(4) In the fourth chapter, a smoothing function corresponding to second-order cone is given, and its properties are studied. Based on the smoothing function, a smooth-ing Newton-type method is proposed for solving second-order cone programming. The method reformulates the perturbed optimality conditions into a system of linear equa-tions to solve. Its results are stronger than corresponding interior-point methods, and its initial point is arbitrary. At each iteration, the proposed algorithm need only to solve one system of linear equations and perform one line search. The sequence gen-erated by the method converges globally and Q-quadratically to the solution of the SOCP under a mild assumption, without strict complementarity condition.
(5) A globally convergent non-interior point algorithm with full Newton step for second-order cone programming is proposed in the fifth part. The main idea of the algorithm is that we cast the complementary equation of the primal-dual optimality conditions as a projection equation. This algorithm can start from an arbitrary point. Without performing any line search, we only need to solve a system of linear equa-tions with the same coefficient matrix and compute two simple projections at each iteration. It does not require the row vectors of coefficient matrix of constraints to be linearly independent. The proposed algorithm is globally convergent without strict complementarity conditions.
(6) In the sixth part, we study the special second-order cone programming-nonlinear complementarity problem with P0-function. Based on a new smoothing func-tion with penalty, the problem is approximated by a family of parameterized smooth equations and a new non-interior-point continuation method is presented. At each it-eration, the proposed algorithm only need to solve one system of linear equations and perform one Armijo-type line search. The algorithm is proved to be globally and locally superlinearly convergent without strict complementarity. Moreover, the quadratic con-vergence rate is achieved under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the proposed algorithm.
(7) In chapter seven, we introduce a class of one-parametric smoothing second-order cone complementarity functions. Under suitable assumptions, we prove that these functions are coercive. Based on this class of one-parametric smoothing functions, a smoothing Newton method is proposed to solve the monotone second-order cone complementarity problems. It is proved that the proposed algorithm is globally and locally superlinearly convergent under suitable assumptions. Preliminary numerical results for randomly generated second-order cone programs are reported, which confirm the favorable theoretical properties of the method.
Finally, based on the problems of the existing methods, including the methods proposed in the thesis, the last chapter is concerned with some future research work.
引文
[1]I. Adler and F. Alizadeh. Primal-dual interior point algorithms for convex quadratically constrained and semidefinite optimization problems. Technical Re-port RRR-111-95, Rutcor, Rutgers University,1995.
[2]W. Ai and S. Zhang. An O((?)L) iteration primal-dual path-following method, based on wide neighborhoods and large updates, for monotone LCP. SIAM Journal on Optimization,2005,16 (2),400-417.
[3]F. Alizadeh and S.H. Schmieta. Optimization with semidefinite, quadratic and linear constraints. Technical Report rrr 23-97, Rutcor, Rutgers University, November 1997.
[4]F. Alizadeh, J.A. Haeberly and M.L. Overton. Primal-Dual Interior-Point Meth-ods for Semidefinite Programming:Convergence Rates, Stabilty and Numerical Results. SIAM Journal on Optimization,1998,8 (3),746-768.
[5]F. Alizadeh and D. Goldfarb. Second-order cone programming. Mathematical Programming,2003,95,3-51.
[6]K.D. Andersen, E. Christiansen, A.R. Conn and M.L. Overton. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM Journal on Scientific Computing,2000,22 (1),243-262.
[7]R. Andreani, A. Friedlander, M.P. Mello and S.A. Santos. Box-constrained min-imization reformulations of complementarity problems in second-order cones. Journal of Global Optimization,2008,40 (4),505-527.
[8]A.B. Berkelaar, J.F. Sturm and S. Zhang. Polynomial primal-dual cone affine scaling for semidefinite programming. Applied Numerical Mathematics,1999, 29,317-333.
[9]J. Burke and S. Xu. A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem. Mathematical Programming, Ser. A,2000,87,113-130.
[10]B. Chen and P.T. Harker. A non-interior-point continuation method for linear complementarity problem. SIAM Journal on Matrix Analysis and Applications, 1993,14 (4),1168-1190.
[11]B. Chen and P.T. Harker. Smooth approximations to nonlinear complementar-ity problems. Artificial Neural Networks-ICANN'96, Berlin, SIAM Journal on Optimization,1997,7 (2),403-420.
[12]B. Chen and N. Xiu. A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions. SIAM Journal on Optimization,1999,9 (3),605-623.
[13]B. Chen and X. J. Chen. A global linear and local quadratic continuation smooth-ing method for variational inequalities with box constraints. Computational Op-timization and Applications,2000,17,131-158.
[14]J.-S. Chen and P. Tseng. An unconstrained smooth minimization reformula-tion of the second-order cone complementarity problem. Mathematical Program-ming, Series B,2005,104,293-327.
[15]J.-S. Chen. Two classes of merit functions for the second-order cone comple-mentarity problem. Mathematical Methods of Operations Research,2006,64, 495-519.
[16]J.-S. Chen and S. Pan. A descent method for a reformulation of the second-order cone complementarity problem. Journal of Computational and Applied Mathematics,2008,213,547-558.
[17]J.-S. Chen, D. Sun and J. Sun. The SC1 property of the squared norm of the SOC Fischer-Burmeister function. Operations Research Letters,2008,36,385-392.
[18]J.-S. Chen and S. Pan. A one-parametric class of merit functions for the second-order cone complementarity problem. Computational Optimization and Appli-cations, DOI:10.1007/s10589-008-9182-9.
[19]X. Chen, L. Qi and D. Sun. Global and superlinear convergence of the smooth-ing Newton method and its application to general box constrained variational inequalities. Mathematics of Computation,1998,67 (222),519-540.
[20]X.D. Chen, D. Sun and J. Sun. Complementarity functions and numerical ex-periments on some smoothing Newton methods for second-order-cone comple-mentarity problems. Computational Optimization and Applications,2003,25, 39-56.
[21]X. Chen and P. Tseng. Non-Interior continuation methods for solving semi-definite complementarity problems. Mathematical Programming, Series A,2003, 95 (3),431-474.
[22]C.B. Chua. The primal-dual second-order cone approximations algorithm for symmetric cone programming. Foundations of Computational Mathematics, 2007,271-302.
[23]F.H. Clarke. Optimization and non-smooth analysis. John Wiley and Sons, New York,1993.
[24]R. Debnath, M. Muramatsu and H. Takahashi. An efficient support vector ma-chine learning method with second-order cone programming for large-scale prob-lems. Applied Intelligence,2005,23,219-239.
[25]S. Effati and M.M. Moghadam. Conversion of some classes of fractional pro-gramming to second-order cone programming and solving it by potential reduc-tion interior point method. Applied Mathematics and Computation,2006,181, 563-578.
[26]E. Erdogan and G. Iyengar. An active set method for single-cone second-order cone programs. SIAM Journal on Optimization,2006,17 (2),459-484.
[27]F. Facchinei and C. Kanzow. A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Mathematical Pro-gramming,1997,76,493-512.
[28]F. Facchinei and C. Kanzow. Beyond monotonicity in regularization methods for nonlinear complementarity problems. SIAM Journal on Control and Opti-mization,1999,37 (4),1150-1161.
[29]J. Faraut and A. Koranyi. Analysis on symmetric cones. Oxford University Press, London and New York,1994, ISBN 0-198-53477-9.
[30]L. Faybusovich. Euclidean Jordan algebras and interior-point algorithms. Posi-tivity,1997,1,331-357.
[31]L. Faybusovich. Linear systems in Jordan algebras and primal-dual interior-point algorithms. Journal of Computational and Applied Mathematics,1997, 86,149-175.
[32]L. Faybusovich. A Jordan-algebraic approach to potential-reduction algorithms. Technical report, Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA,1998.
[33]L. Faybusovich and Y. Lu. Jordan-algebraic aspects of nonconvex optimization over symmetric cones. Applied Mathematics and Optimization,2006,53,67-77.
[34]M. Fiedler and V. Ptgk. Some generalizations of positive definiteness and mono-tonicity. Numerische Mathematik,1966,9,163-172.
[35]M. Fukushima, Z.Q. Luo and P. Tseng. Smoothing functions for second-order-cone complimentarity problems. SIAM Journal on Optimization,2001,12 (2), 436-460.
[36]D. Goldfarb, S. Liu and S. Wang. A logarithmic barrier function algorithm for convex quadratically constrained convex quadratic programming. SIAM Journal on Optimization,1991,1 (2),252-267.
[37]D. Goldfarb and K. Scheinberg. Product-form Cholesky factorization in inte-rior point methods for second-order cone programming. Mathematical Program-ming, Ser. A,2005,103,153-179.
D. Goldfarb and W. Yin. Second-order cone programming methods for total variation-based image restoration. SIAM Journal on Scientific Computing,2005, 27 (2),622-645.
[39]M.S. Gowda and M.A. Tawhid. Existence and limiting behavior of trajectories associated with P0-Equations. Computational Optimization and Applications, 1999,12,229-251.
[40]M.S. Gowda, R. Sznajder and J. Tao. Some P-properties for linear transforma-tions on Euclidean Jordan algebras. Linear Algebra and Its Applications,2004, 393,203-232.
[41]D.R. Han, H.K. Lo and I. Galligani. New alternating direction method for a class of nonlinear variational inequality problems. Journal of Optimization Theory and Applications,2002,112 (3),549-560.
[42]Q. Han. Projection and contraction methods for semidefinite programming. Ap-plied Mathematics and Computation,1998,95,275-289.
[43]P.T. Harker and J.S. Pang. Finite-dimensional variational inequality and non-linear complementarity problems:A survey of theory, algorithms and applica-tions. Mathematical Programming,1990,48 (1),161-220.
[44]S. Hayashi, N. Yamashita and M. Fukushima. Robust Nash equilibria and second-order cone complementarity problems. Journal of Nonlinear and Con-vex Analysis,2005,6 (2),283-296.
[45]S. Hayashi, N. Yamashita and M. Fukushima. A combined smoothing and reg-ularized method for monotone second-order cone complementarity problems. SIAM Journal on Optimization,2005,15 (2),593-615.
[46]B.S. He. A class of projection and contraction methods for monotone variational inequalities. Applied Mathematics and Optimization,1997,35,69-76.
[47]B.S. He, L.Z. Liao and R. Glowinski. Improvements of some projection methods for monotone nonlinear variational inequalities. Journal of Optimization Theory and Applications,2002,112 (1),111-128.
[48]M. Heiler and C. Schnorr. Learning sparse representations by non-negative ma-trix factorization and sequential cone programming. Journal of Machine Learn-ing Research,2006,7,1385-1407.
[49]Z. Huang and J. Han. Non-interior continuation method for solving the mono-tone semidefinite complementarity problem. Applied Mathematics and Opti-mization,2003,47 (3),195-211.
[50]Z.H. Huang, J. Han and Z. Chen. A predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear com-plementarity problem with a P0 function. Journal of Optimization Theory and Applications,2003,117 (1),39-68.
[51]Z.H. Huang and T. Ni. Smoothing algorithms for complementarity prob-lems over symmetric cones. Computational Optimization and Applications, DOI:10.1007/s10589-008-9180-y.
[52]F. Jarre. On the convergence of the method of analytic centers when applied to convex quadratic programs. Mathematical Programming,1991,49,341-358.
[53]H. Jiang. Smoothed Fischer-Burmeister equation methods for the complemen-tarity problem. Technical Report, Department of Mathematics, The University of Melbourne, Parville, Victoria, Australia,1997.
[54]Y. Kanno and M. Ohsaki. Minimum principle of complementary energy of cable networks by using second-order cone programming. International Journal of Solids and Structures,2003,40,4437-4460.
[55]Y. Kanno, J.A.C. Martins and A.P. Da Costa. Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem. International Journal for Numerical Methods in Engineering,2006,65,62-83.
[56]Y. Kanno and M. Ohsaki. Contact analysis of cable networks by using second-order cone programming. SIAM Journal on Scientific Computing,2006,27 (6), 2032-2052.
[57]C. Kanzow. Some noninterior continuation methods for linear complementarity problems. SIAM Journal on Matrix Analysis and Applications,1996,17 (4), 851-868.
[58]C. Kanzow and H. Kleinmichel. A new class of semismooth newton-type meth-ods for nonlinear complementarity problems. Computational Optimization and Applications,1998,11,227-251.
[59]N. Karmarkar. A new polynomial-time algorithm for linear programming. Com-binatorica,1984,4 (4),373-395.
[60]H. Kato and M. Fukushima. An SQP-type algorithm for nonlinear second-order cone programs. Optimization Letters,2007,1,129-144.
[61]S. Kim, M. Kojima and M. Yamashita. Second order cone programming relax-ation of positive semidefinite constraint. Research Reports on Mathematical and Computing Sciences, Series B:Operations Research,2002, B-381.
[62]K. Kobayashi, S. Kim and M. Kojima. Sparse second order cone programming formulations for convex optimization problems. Journal of the Operations Re-search Society of Japan,2008,51 (3),241-264.
[63]M. Kojima and M. Muramatsu. An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones. Mathematical Pro-gramming, Series A,2007,110,315-336.
[64]L.C. Kong, J. Sun and N. Xiu. A regularized smoothing newton method for sym-metric cone complementarity problems. SIAM Journal on Optimization 2008, 19 (3),1028-1047.
[65]Y.J. Kuo and H.D. Mittelmann. Interior point methods for second-order cone programming and OR applications. Computational Optimization and Applica-tions,2004,28,255-285.
[66]X. Liu and J. Sun. A robust primal-dual interior-point algorithm for nonlinear programs. SIAM Journal on Optimization,2004,14 (4),1163-1186.
[67]X. Liu, Z. Huang. A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones. Mathematical Methods of Operations Research,2009,70,385-404.
[68]X. Liu and T. Ni. Smoothing Newton algorithm for linear programming over symmetric cones. Transactions of Tianjin University,2009,15 (3),216-221.
[69]Y. J. Liu, L.W. Zhang and Y.H. Wang. Analysis of a smoothing method for sym-metric conic linear programming. Journal of Applied Mathematics and Comput-ing,2006,22 (1-2),133-148.
[70]Y.J. Liu and L.W. Zhang. Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal-ysis,2007,67,1359-1373.
[71]M.S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret. Applications of second-Order cone programming. Linear Algebra and its Applications,1998,284,193-228.
[72]Y. Lu and Y.X. Yuan. An interior-point trust-region algorithm for general sym-metric cone programming. SIAM Journal on Optimization,2007,18 (1),65-86.
[73]C. Ma and X. Chen. The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function. Journal of Computational and Applied Mathematics,2008,216,1-13.
[74]B.K. Mccrimmon. Jordan algebras and their applications. Bulletin of the Amer-ican Mathematical Society,84 (4),1978.
[75]S. Mehrotra and J. Sun. A method of analytic centers for quadratically cons-trianed convex quadratic programming. SIAM Journal on Numerical Analysis, 1991,28 (2),529-544.
[76]R. Mifflin. Semismooth and semiconvex functions in constraint optimization. SIAM Journal on Control and Optimization,1977,15 (6),959-972.
[77]R.D.C. Monteiro and Y. Zhang. A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite program-ming. Mathematical Programming,1998,81,281-299.
[78]R.D.C. Monteiro and T. Tsuchiya. Polynomial convergence of primal-dual algo-rithms for the second-order cone program based on the MZ-family of directions. Mathematical Programming, Ser. A,2000,88,61-83.
[79]M. Muramatsu. On a commutative class of search directions for linear program-ming over symmetric cones. Journal of Optimization Theory and Applications, 2002,112 (3),595-625.
[80]A. Nemirovski and K. Scheinberg. Extension of Karmarkar's algorithm onto convex quadratically constrained quadratic programming. Mathematical Pro-gramming,1996,72,273-289.
[81]A. Nemirovski and L. Tuncel. "Cone-free" primal-dual path-following and potential-reduction polynomial time interior-point methods. Mathematical Pro-gramming, Series A,2005,102,261-294.
[82]Y.E. Nesterov and A.S. Nemirovsky. Interior point polynomial algorithms in convex programming. Society for Industrial andApplied Mathematics (SIAM), Philadelphia,1994.
[83]Y.E. Nesterov and M.J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research,1997,22 (1), 1-42.
[84]Y.E. Nesterov and M.J. Todd. Primal-dual interior-point methods for self-scaled cones. SIAM Journal on Optimization,1998,8 (2),324-364.
[85]M.A. Noor. A modified extragradient method for general monotone variational inequalitiesA modified extragradient method for general monotone variational inequalities. Computers and Mathematics with Applications,1999,38,19-24.
6] S. Pan and Y.X. Jiang. Smoothing Newton method for minimizing the sum of p-norms. Journal of Optimization Theory and Applications,2008,137 (2), 255-275.
[87]S. Pan, J.-S. Chen. A damped Gauss-Newton method for the second-order cone complementarity problem. Applied Mathematics and Optimization,2009,59 (3),293-318.
[88]S. Pan and J.-S. Chen. A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions. Computational Opti-mization and Applications,2010,45 (1),59-88.
[89]X. Peng and I. King. Robust BMPM training based on second-order cone pro-gramming and its application in medical diagnosis. Neural Networks,2008,21, 450-457.
[90]J. Peng, C. Roos and T. Terlaky. Primal-dual interior-point methods for second-order conic optimization based on self-regular proximities. SIAM Journal on Optimization,2002,13 (1),179-203.
[91]F.A. Potra and S.J. Wright. Interior-point methods. Journal of Computational and Applied Mathematics,2000,124,281-302.
[92]H.D. Qi and L.Z. Liao. A smoothing Newton method for general nonlinear com-plementarity problems. Computational Optimization and Applications,2000, 17,231-253.
[93]L. Qi. Convergence analysis of some algorithms for solving nonsmooth equations. Mathematics of Operations Research,1993,18 (1),227-244.
[94]L. Qi and J. Sun. A nonsmooth version of Newton's method. Mathematical Programming,1993,58,353-367.
[95]L. Qi and D. Sun. Improving the convergence of non-interior point algorithms for nonlinear complementarity problems. Mathematics of Computation,2000, 69 (229),283-304.
[96]L. Qi, D. Sun and G. Zhou. A new look at smoothing Newton methods for non-linear complementarity problems and box constrained variational inequalities. Mathematical Programming, Ser. A,2000,87,1-35.
[97]L. Qi and D. Sun. Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. Journal of Optimization Theory and Applications,2002,113 (1),121-147.
[98]B.K. Rangarajan. Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM Journal on Optimization,2006,16 (4),1211-1229.
[99]T. Sasakawa and T. Tsuchiya. Optimal magnetic shield design with second-order cone programming. SIAM Journal on Scientific Computing,2003,24 (6), 1930-1950.
[100]S.H. Schmieta. Application of Jordan algebras to the design and analysis of interior-point algorithms for linear, quadratically constrained quadratic, and semi-definite programming. PhD thesis, Rutgers Center for Operations Re-search, Rutgers University,1999.
[101]S.H. Schmieta and F. Alizadeh. Extension of commutative class of primal-dual interior point algorithms to symmetric cones. Technical Report RRR 13-99, Rutcor, Rutgers University,1999.
[102]S.H. Schmieta and F. Alizadeh. Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Mathematics of Operations Research,2001,26 (3),543-564.
[103]S.H. Schmieta and F. Alizadeh. Extension of primal-dual interior point algo-rithms to symmetric cones. Mathematical Programming, Series A,2003,96, 409-438.
[104]P.K. Shivaswamy, C. Bhattacharyya and A.J. Smola. Second order cone pro-gramming approaches for handling missing and uncertain data. Journal of Ma-chine Learning Research,2006,7,1283-1314.
[105]C.K. Sim and G. Zhao. A note on treating a second order cone program as a special case of semidefinite program. Mathematical Programming, Series A, 2005,102,609-613.
[106]J.F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over sym-metric cones. Optimization Methods and Software,11-12 (1999) 625-653, Spe-cial issue on Interior Point Methods (CD supplement with software).
[107]J.F. Sturm. Similarity and other spectral relations for symmetric cones. Linear Algebra and its Applications,2000,312,135-154.
[108]T. Sueyoshi and K. Sekitani. Computational strategy for Russell measure in DEA:Second-order cone programming. European Journal of Operational Re-search,2007,180,459-471.
[109]D. Sun and L. Qi. Solving variational inequality problems via smoothing-nonsmooth reformulations. Journal of Computational and Applied Mathematics, 2001,129,37-62.
[110]D. Sun and J. Sun. Semismooth matrix-valued functions. Mathematics of Op-erations Research,2002,27 (1),150-169.
[111]D. Sun and J. Sun. Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Mathematical Programming, Ser. A,2005,103, 575-581.
[112]D. Sun and J. Sun. Lowner's operator and spectral functions in Euclidean Jordan algebras. Mathematics of Operations Research,2008,33 (2),421-445.
[113]K. Taji and M. Miyamoto. A globally convergent smoothing Newton method for nonsmooth equations and its application to complementarity problems. Com-putational Optimization and Applications,2002,22,81-101.
[114]M. Todd. Semidefinite optimization. Acta Numerica,2001,10,515-560.
[115]K.C. Toh, R.H. Tutuncu and M.J. Todd. On the implementation and usage of SDPT3-a MATLAB software package for semidefinite-quadratic-linear pro-gramming, version 4.0,2006.
[116]P. Tseng. Error bounds and superlinear convergence analysis of some Newton-type methods in optimization. In:G. Di Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Related Topics, Kluwer Academic Publishers, Boston,2000, 445-462.
[117]T. Tsuchiya. A polynomial primal-dual path-following algorithm for second-order cone programming. Research Memorandum No.649, The Institute of Sta-tistical Mathematics, Tokyo, Japan, October (Revised:December 1997).
[118]T. Tsuchiya. A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optimization Methods and Software,1999,11,141-182.
[119]C. Witzgall. Optimal location of a central facility, mathematical models and concepts. Technical Report 8388, National Bureau of Standards,Washington, DC,1964.
[120]S.J. Wright. Primal-dual interior point methods. Society for Industrial and Ap-plied Mathematics (SIAM). Philadelaphia, PA,1997.
[121]Y. Xia. A Newton's method for perturbed second-order cone programs. Com-putational Optimization and Applications,2007,37 (3),371-408.
[122]Y. Xia. Second order cone programming relaxation for quadratic assignment problems. Optimization Methods and Software,2008,23 (3),441-449.
[123]Y. Xia and F. Alizadeh. The Q method for second order cone programming. Computers and Operations Research,2008,35,1510-1538.
[124]N.H. Xiu and J.Z. Zhang. Some recent advances in projection-type methods for variational inequalities. Journal of Computational and Applied Mathematics, 2003,152,559-585.
[125]S. Yan and Y. Ma. Robust supergain beamforming for circular array via second-order cone programming. Applied Acoustics,2005,66,1018-1032.
[126]S. Yan and Y. Ma. Optimal design and verification of temporal and spatial filters using second-order cone programming Approach. Science in China:Series F Information Sciences,2006,49 (2),235-253.
[127]Z. Yu. Solving semidefinite programming problems via alternating direction methods. Journal of Computational and Applied Mathematics,2006,193,437-445.
[128]J.Z. Zhang and N.H. Xiu. Local convergence behavior of some projection-type methods for affine variational inequalities. Journal of Optimization Theory and Applications,2001,108 (1),205-216.
[129]L. Zhang, J. Han and Z. Huang. Superlinear/quadratic one-step smoothing New-ton method for P0-NCP. Acta Math Sinica,2005,26 (2),117-128.
[130]Y.B. Zhao. Extended projection methods for monotone variational inequalities. Journal of Optimization Theory and Applications,1999,100 (1),219-231.
[131]Y.B. Zhao and D. Li. A new path-following algorithm for nonlinear P* comple-mentarity problems. Computational Optimization and Applications,2005,34, 183-214.
[132]L.F. Zuluaga and J.F. Pena. A conic programming approach to generalized Tchebycheff inequalities. Mathematics of Operation Research,2005,30 (2),369-388.
[2]W. Ai and S. Zhang. An O((?)L) iteration primal-dual path-following method, based on wide neighborhoods and large updates, for monotone LCP. SIAM Journal on Optimization,2005,16 (2),400-417.
[3]F. Alizadeh and S.H. Schmieta. Optimization with semidefinite, quadratic and linear constraints. Technical Report rrr 23-97, Rutcor, Rutgers University, November 1997.
[4]F. Alizadeh, J.A. Haeberly and M.L. Overton. Primal-Dual Interior-Point Meth-ods for Semidefinite Programming:Convergence Rates, Stabilty and Numerical Results. SIAM Journal on Optimization,1998,8 (3),746-768.
[5]F. Alizadeh and D. Goldfarb. Second-order cone programming. Mathematical Programming,2003,95,3-51.
[6]K.D. Andersen, E. Christiansen, A.R. Conn and M.L. Overton. An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms. SIAM Journal on Scientific Computing,2000,22 (1),243-262.
[7]R. Andreani, A. Friedlander, M.P. Mello and S.A. Santos. Box-constrained min-imization reformulations of complementarity problems in second-order cones. Journal of Global Optimization,2008,40 (4),505-527.
[8]A.B. Berkelaar, J.F. Sturm and S. Zhang. Polynomial primal-dual cone affine scaling for semidefinite programming. Applied Numerical Mathematics,1999, 29,317-333.
[9]J. Burke and S. Xu. A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem. Mathematical Programming, Ser. A,2000,87,113-130.
[10]B. Chen and P.T. Harker. A non-interior-point continuation method for linear complementarity problem. SIAM Journal on Matrix Analysis and Applications, 1993,14 (4),1168-1190.
[11]B. Chen and P.T. Harker. Smooth approximations to nonlinear complementar-ity problems. Artificial Neural Networks-ICANN'96, Berlin, SIAM Journal on Optimization,1997,7 (2),403-420.
[12]B. Chen and N. Xiu. A global linear and local quadratic noninterior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions. SIAM Journal on Optimization,1999,9 (3),605-623.
[13]B. Chen and X. J. Chen. A global linear and local quadratic continuation smooth-ing method for variational inequalities with box constraints. Computational Op-timization and Applications,2000,17,131-158.
[14]J.-S. Chen and P. Tseng. An unconstrained smooth minimization reformula-tion of the second-order cone complementarity problem. Mathematical Program-ming, Series B,2005,104,293-327.
[15]J.-S. Chen. Two classes of merit functions for the second-order cone comple-mentarity problem. Mathematical Methods of Operations Research,2006,64, 495-519.
[16]J.-S. Chen and S. Pan. A descent method for a reformulation of the second-order cone complementarity problem. Journal of Computational and Applied Mathematics,2008,213,547-558.
[17]J.-S. Chen, D. Sun and J. Sun. The SC1 property of the squared norm of the SOC Fischer-Burmeister function. Operations Research Letters,2008,36,385-392.
[18]J.-S. Chen and S. Pan. A one-parametric class of merit functions for the second-order cone complementarity problem. Computational Optimization and Appli-cations, DOI:10.1007/s10589-008-9182-9.
[19]X. Chen, L. Qi and D. Sun. Global and superlinear convergence of the smooth-ing Newton method and its application to general box constrained variational inequalities. Mathematics of Computation,1998,67 (222),519-540.
[20]X.D. Chen, D. Sun and J. Sun. Complementarity functions and numerical ex-periments on some smoothing Newton methods for second-order-cone comple-mentarity problems. Computational Optimization and Applications,2003,25, 39-56.
[21]X. Chen and P. Tseng. Non-Interior continuation methods for solving semi-definite complementarity problems. Mathematical Programming, Series A,2003, 95 (3),431-474.
[22]C.B. Chua. The primal-dual second-order cone approximations algorithm for symmetric cone programming. Foundations of Computational Mathematics, 2007,271-302.
[23]F.H. Clarke. Optimization and non-smooth analysis. John Wiley and Sons, New York,1993.
[24]R. Debnath, M. Muramatsu and H. Takahashi. An efficient support vector ma-chine learning method with second-order cone programming for large-scale prob-lems. Applied Intelligence,2005,23,219-239.
[25]S. Effati and M.M. Moghadam. Conversion of some classes of fractional pro-gramming to second-order cone programming and solving it by potential reduc-tion interior point method. Applied Mathematics and Computation,2006,181, 563-578.
[26]E. Erdogan and G. Iyengar. An active set method for single-cone second-order cone programs. SIAM Journal on Optimization,2006,17 (2),459-484.
[27]F. Facchinei and C. Kanzow. A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Mathematical Pro-gramming,1997,76,493-512.
[28]F. Facchinei and C. Kanzow. Beyond monotonicity in regularization methods for nonlinear complementarity problems. SIAM Journal on Control and Opti-mization,1999,37 (4),1150-1161.
[29]J. Faraut and A. Koranyi. Analysis on symmetric cones. Oxford University Press, London and New York,1994, ISBN 0-198-53477-9.
[30]L. Faybusovich. Euclidean Jordan algebras and interior-point algorithms. Posi-tivity,1997,1,331-357.
[31]L. Faybusovich. Linear systems in Jordan algebras and primal-dual interior-point algorithms. Journal of Computational and Applied Mathematics,1997, 86,149-175.
[32]L. Faybusovich. A Jordan-algebraic approach to potential-reduction algorithms. Technical report, Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA,1998.
[33]L. Faybusovich and Y. Lu. Jordan-algebraic aspects of nonconvex optimization over symmetric cones. Applied Mathematics and Optimization,2006,53,67-77.
[34]M. Fiedler and V. Ptgk. Some generalizations of positive definiteness and mono-tonicity. Numerische Mathematik,1966,9,163-172.
[35]M. Fukushima, Z.Q. Luo and P. Tseng. Smoothing functions for second-order-cone complimentarity problems. SIAM Journal on Optimization,2001,12 (2), 436-460.
[36]D. Goldfarb, S. Liu and S. Wang. A logarithmic barrier function algorithm for convex quadratically constrained convex quadratic programming. SIAM Journal on Optimization,1991,1 (2),252-267.
[37]D. Goldfarb and K. Scheinberg. Product-form Cholesky factorization in inte-rior point methods for second-order cone programming. Mathematical Program-ming, Ser. A,2005,103,153-179.
D. Goldfarb and W. Yin. Second-order cone programming methods for total variation-based image restoration. SIAM Journal on Scientific Computing,2005, 27 (2),622-645.
[39]M.S. Gowda and M.A. Tawhid. Existence and limiting behavior of trajectories associated with P0-Equations. Computational Optimization and Applications, 1999,12,229-251.
[40]M.S. Gowda, R. Sznajder and J. Tao. Some P-properties for linear transforma-tions on Euclidean Jordan algebras. Linear Algebra and Its Applications,2004, 393,203-232.
[41]D.R. Han, H.K. Lo and I. Galligani. New alternating direction method for a class of nonlinear variational inequality problems. Journal of Optimization Theory and Applications,2002,112 (3),549-560.
[42]Q. Han. Projection and contraction methods for semidefinite programming. Ap-plied Mathematics and Computation,1998,95,275-289.
[43]P.T. Harker and J.S. Pang. Finite-dimensional variational inequality and non-linear complementarity problems:A survey of theory, algorithms and applica-tions. Mathematical Programming,1990,48 (1),161-220.
[44]S. Hayashi, N. Yamashita and M. Fukushima. Robust Nash equilibria and second-order cone complementarity problems. Journal of Nonlinear and Con-vex Analysis,2005,6 (2),283-296.
[45]S. Hayashi, N. Yamashita and M. Fukushima. A combined smoothing and reg-ularized method for monotone second-order cone complementarity problems. SIAM Journal on Optimization,2005,15 (2),593-615.
[46]B.S. He. A class of projection and contraction methods for monotone variational inequalities. Applied Mathematics and Optimization,1997,35,69-76.
[47]B.S. He, L.Z. Liao and R. Glowinski. Improvements of some projection methods for monotone nonlinear variational inequalities. Journal of Optimization Theory and Applications,2002,112 (1),111-128.
[48]M. Heiler and C. Schnorr. Learning sparse representations by non-negative ma-trix factorization and sequential cone programming. Journal of Machine Learn-ing Research,2006,7,1385-1407.
[49]Z. Huang and J. Han. Non-interior continuation method for solving the mono-tone semidefinite complementarity problem. Applied Mathematics and Opti-mization,2003,47 (3),195-211.
[50]Z.H. Huang, J. Han and Z. Chen. A predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the nonlinear com-plementarity problem with a P0 function. Journal of Optimization Theory and Applications,2003,117 (1),39-68.
[51]Z.H. Huang and T. Ni. Smoothing algorithms for complementarity prob-lems over symmetric cones. Computational Optimization and Applications, DOI:10.1007/s10589-008-9180-y.
[52]F. Jarre. On the convergence of the method of analytic centers when applied to convex quadratic programs. Mathematical Programming,1991,49,341-358.
[53]H. Jiang. Smoothed Fischer-Burmeister equation methods for the complemen-tarity problem. Technical Report, Department of Mathematics, The University of Melbourne, Parville, Victoria, Australia,1997.
[54]Y. Kanno and M. Ohsaki. Minimum principle of complementary energy of cable networks by using second-order cone programming. International Journal of Solids and Structures,2003,40,4437-4460.
[55]Y. Kanno, J.A.C. Martins and A.P. Da Costa. Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem. International Journal for Numerical Methods in Engineering,2006,65,62-83.
[56]Y. Kanno and M. Ohsaki. Contact analysis of cable networks by using second-order cone programming. SIAM Journal on Scientific Computing,2006,27 (6), 2032-2052.
[57]C. Kanzow. Some noninterior continuation methods for linear complementarity problems. SIAM Journal on Matrix Analysis and Applications,1996,17 (4), 851-868.
[58]C. Kanzow and H. Kleinmichel. A new class of semismooth newton-type meth-ods for nonlinear complementarity problems. Computational Optimization and Applications,1998,11,227-251.
[59]N. Karmarkar. A new polynomial-time algorithm for linear programming. Com-binatorica,1984,4 (4),373-395.
[60]H. Kato and M. Fukushima. An SQP-type algorithm for nonlinear second-order cone programs. Optimization Letters,2007,1,129-144.
[61]S. Kim, M. Kojima and M. Yamashita. Second order cone programming relax-ation of positive semidefinite constraint. Research Reports on Mathematical and Computing Sciences, Series B:Operations Research,2002, B-381.
[62]K. Kobayashi, S. Kim and M. Kojima. Sparse second order cone programming formulations for convex optimization problems. Journal of the Operations Re-search Society of Japan,2008,51 (3),241-264.
[63]M. Kojima and M. Muramatsu. An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones. Mathematical Pro-gramming, Series A,2007,110,315-336.
[64]L.C. Kong, J. Sun and N. Xiu. A regularized smoothing newton method for sym-metric cone complementarity problems. SIAM Journal on Optimization 2008, 19 (3),1028-1047.
[65]Y.J. Kuo and H.D. Mittelmann. Interior point methods for second-order cone programming and OR applications. Computational Optimization and Applica-tions,2004,28,255-285.
[66]X. Liu and J. Sun. A robust primal-dual interior-point algorithm for nonlinear programs. SIAM Journal on Optimization,2004,14 (4),1163-1186.
[67]X. Liu, Z. Huang. A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones. Mathematical Methods of Operations Research,2009,70,385-404.
[68]X. Liu and T. Ni. Smoothing Newton algorithm for linear programming over symmetric cones. Transactions of Tianjin University,2009,15 (3),216-221.
[69]Y. J. Liu, L.W. Zhang and Y.H. Wang. Analysis of a smoothing method for sym-metric conic linear programming. Journal of Applied Mathematics and Comput-ing,2006,22 (1-2),133-148.
[70]Y.J. Liu and L.W. Zhang. Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal-ysis,2007,67,1359-1373.
[71]M.S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret. Applications of second-Order cone programming. Linear Algebra and its Applications,1998,284,193-228.
[72]Y. Lu and Y.X. Yuan. An interior-point trust-region algorithm for general sym-metric cone programming. SIAM Journal on Optimization,2007,18 (1),65-86.
[73]C. Ma and X. Chen. The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function. Journal of Computational and Applied Mathematics,2008,216,1-13.
[74]B.K. Mccrimmon. Jordan algebras and their applications. Bulletin of the Amer-ican Mathematical Society,84 (4),1978.
[75]S. Mehrotra and J. Sun. A method of analytic centers for quadratically cons-trianed convex quadratic programming. SIAM Journal on Numerical Analysis, 1991,28 (2),529-544.
[76]R. Mifflin. Semismooth and semiconvex functions in constraint optimization. SIAM Journal on Control and Optimization,1977,15 (6),959-972.
[77]R.D.C. Monteiro and Y. Zhang. A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite program-ming. Mathematical Programming,1998,81,281-299.
[78]R.D.C. Monteiro and T. Tsuchiya. Polynomial convergence of primal-dual algo-rithms for the second-order cone program based on the MZ-family of directions. Mathematical Programming, Ser. A,2000,88,61-83.
[79]M. Muramatsu. On a commutative class of search directions for linear program-ming over symmetric cones. Journal of Optimization Theory and Applications, 2002,112 (3),595-625.
[80]A. Nemirovski and K. Scheinberg. Extension of Karmarkar's algorithm onto convex quadratically constrained quadratic programming. Mathematical Pro-gramming,1996,72,273-289.
[81]A. Nemirovski and L. Tuncel. "Cone-free" primal-dual path-following and potential-reduction polynomial time interior-point methods. Mathematical Pro-gramming, Series A,2005,102,261-294.
[82]Y.E. Nesterov and A.S. Nemirovsky. Interior point polynomial algorithms in convex programming. Society for Industrial andApplied Mathematics (SIAM), Philadelphia,1994.
[83]Y.E. Nesterov and M.J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research,1997,22 (1), 1-42.
[84]Y.E. Nesterov and M.J. Todd. Primal-dual interior-point methods for self-scaled cones. SIAM Journal on Optimization,1998,8 (2),324-364.
[85]M.A. Noor. A modified extragradient method for general monotone variational inequalitiesA modified extragradient method for general monotone variational inequalities. Computers and Mathematics with Applications,1999,38,19-24.
6] S. Pan and Y.X. Jiang. Smoothing Newton method for minimizing the sum of p-norms. Journal of Optimization Theory and Applications,2008,137 (2), 255-275.
[87]S. Pan, J.-S. Chen. A damped Gauss-Newton method for the second-order cone complementarity problem. Applied Mathematics and Optimization,2009,59 (3),293-318.
[88]S. Pan and J.-S. Chen. A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions. Computational Opti-mization and Applications,2010,45 (1),59-88.
[89]X. Peng and I. King. Robust BMPM training based on second-order cone pro-gramming and its application in medical diagnosis. Neural Networks,2008,21, 450-457.
[90]J. Peng, C. Roos and T. Terlaky. Primal-dual interior-point methods for second-order conic optimization based on self-regular proximities. SIAM Journal on Optimization,2002,13 (1),179-203.
[91]F.A. Potra and S.J. Wright. Interior-point methods. Journal of Computational and Applied Mathematics,2000,124,281-302.
[92]H.D. Qi and L.Z. Liao. A smoothing Newton method for general nonlinear com-plementarity problems. Computational Optimization and Applications,2000, 17,231-253.
[93]L. Qi. Convergence analysis of some algorithms for solving nonsmooth equations. Mathematics of Operations Research,1993,18 (1),227-244.
[94]L. Qi and J. Sun. A nonsmooth version of Newton's method. Mathematical Programming,1993,58,353-367.
[95]L. Qi and D. Sun. Improving the convergence of non-interior point algorithms for nonlinear complementarity problems. Mathematics of Computation,2000, 69 (229),283-304.
[96]L. Qi, D. Sun and G. Zhou. A new look at smoothing Newton methods for non-linear complementarity problems and box constrained variational inequalities. Mathematical Programming, Ser. A,2000,87,1-35.
[97]L. Qi and D. Sun. Smoothing functions and smoothing Newton method for complementarity and variational inequality problems. Journal of Optimization Theory and Applications,2002,113 (1),121-147.
[98]B.K. Rangarajan. Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM Journal on Optimization,2006,16 (4),1211-1229.
[99]T. Sasakawa and T. Tsuchiya. Optimal magnetic shield design with second-order cone programming. SIAM Journal on Scientific Computing,2003,24 (6), 1930-1950.
[100]S.H. Schmieta. Application of Jordan algebras to the design and analysis of interior-point algorithms for linear, quadratically constrained quadratic, and semi-definite programming. PhD thesis, Rutgers Center for Operations Re-search, Rutgers University,1999.
[101]S.H. Schmieta and F. Alizadeh. Extension of commutative class of primal-dual interior point algorithms to symmetric cones. Technical Report RRR 13-99, Rutcor, Rutgers University,1999.
[102]S.H. Schmieta and F. Alizadeh. Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones. Mathematics of Operations Research,2001,26 (3),543-564.
[103]S.H. Schmieta and F. Alizadeh. Extension of primal-dual interior point algo-rithms to symmetric cones. Mathematical Programming, Series A,2003,96, 409-438.
[104]P.K. Shivaswamy, C. Bhattacharyya and A.J. Smola. Second order cone pro-gramming approaches for handling missing and uncertain data. Journal of Ma-chine Learning Research,2006,7,1283-1314.
[105]C.K. Sim and G. Zhao. A note on treating a second order cone program as a special case of semidefinite program. Mathematical Programming, Series A, 2005,102,609-613.
[106]J.F. Sturm. Using SeDuMi 1.02, a MATLAB toolbox for optimization over sym-metric cones. Optimization Methods and Software,11-12 (1999) 625-653, Spe-cial issue on Interior Point Methods (CD supplement with software).
[107]J.F. Sturm. Similarity and other spectral relations for symmetric cones. Linear Algebra and its Applications,2000,312,135-154.
[108]T. Sueyoshi and K. Sekitani. Computational strategy for Russell measure in DEA:Second-order cone programming. European Journal of Operational Re-search,2007,180,459-471.
[109]D. Sun and L. Qi. Solving variational inequality problems via smoothing-nonsmooth reformulations. Journal of Computational and Applied Mathematics, 2001,129,37-62.
[110]D. Sun and J. Sun. Semismooth matrix-valued functions. Mathematics of Op-erations Research,2002,27 (1),150-169.
[111]D. Sun and J. Sun. Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Mathematical Programming, Ser. A,2005,103, 575-581.
[112]D. Sun and J. Sun. Lowner's operator and spectral functions in Euclidean Jordan algebras. Mathematics of Operations Research,2008,33 (2),421-445.
[113]K. Taji and M. Miyamoto. A globally convergent smoothing Newton method for nonsmooth equations and its application to complementarity problems. Com-putational Optimization and Applications,2002,22,81-101.
[114]M. Todd. Semidefinite optimization. Acta Numerica,2001,10,515-560.
[115]K.C. Toh, R.H. Tutuncu and M.J. Todd. On the implementation and usage of SDPT3-a MATLAB software package for semidefinite-quadratic-linear pro-gramming, version 4.0,2006.
[116]P. Tseng. Error bounds and superlinear convergence analysis of some Newton-type methods in optimization. In:G. Di Pillo, F. Giannessi (Eds.), Nonlinear Optimization and Related Topics, Kluwer Academic Publishers, Boston,2000, 445-462.
[117]T. Tsuchiya. A polynomial primal-dual path-following algorithm for second-order cone programming. Research Memorandum No.649, The Institute of Sta-tistical Mathematics, Tokyo, Japan, October (Revised:December 1997).
[118]T. Tsuchiya. A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optimization Methods and Software,1999,11,141-182.
[119]C. Witzgall. Optimal location of a central facility, mathematical models and concepts. Technical Report 8388, National Bureau of Standards,Washington, DC,1964.
[120]S.J. Wright. Primal-dual interior point methods. Society for Industrial and Ap-plied Mathematics (SIAM). Philadelaphia, PA,1997.
[121]Y. Xia. A Newton's method for perturbed second-order cone programs. Com-putational Optimization and Applications,2007,37 (3),371-408.
[122]Y. Xia. Second order cone programming relaxation for quadratic assignment problems. Optimization Methods and Software,2008,23 (3),441-449.
[123]Y. Xia and F. Alizadeh. The Q method for second order cone programming. Computers and Operations Research,2008,35,1510-1538.
[124]N.H. Xiu and J.Z. Zhang. Some recent advances in projection-type methods for variational inequalities. Journal of Computational and Applied Mathematics, 2003,152,559-585.
[125]S. Yan and Y. Ma. Robust supergain beamforming for circular array via second-order cone programming. Applied Acoustics,2005,66,1018-1032.
[126]S. Yan and Y. Ma. Optimal design and verification of temporal and spatial filters using second-order cone programming Approach. Science in China:Series F Information Sciences,2006,49 (2),235-253.
[127]Z. Yu. Solving semidefinite programming problems via alternating direction methods. Journal of Computational and Applied Mathematics,2006,193,437-445.
[128]J.Z. Zhang and N.H. Xiu. Local convergence behavior of some projection-type methods for affine variational inequalities. Journal of Optimization Theory and Applications,2001,108 (1),205-216.
[129]L. Zhang, J. Han and Z. Huang. Superlinear/quadratic one-step smoothing New-ton method for P0-NCP. Acta Math Sinica,2005,26 (2),117-128.
[130]Y.B. Zhao. Extended projection methods for monotone variational inequalities. Journal of Optimization Theory and Applications,1999,100 (1),219-231.
[131]Y.B. Zhao and D. Li. A new path-following algorithm for nonlinear P* comple-mentarity problems. Computational Optimization and Applications,2005,34, 183-214.
[132]L.F. Zuluaga and J.F. Pena. A conic programming approach to generalized Tchebycheff inequalities. Mathematics of Operation Research,2005,30 (2),369-388.