A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorith...A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.展开更多
As a basic mathematical structure,the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many o...As a basic mathematical structure,the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems.In this paper,a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone.It is shown that the proposed algorithm is globally convergent and well-defined.Moreover,it can start from any point and only needs to solve one system of linear equations at most at each iteration.Under suitable assumptions,global linear and local quadratic convergence is established with Euclidean Jordan algebras.Numerical results indicate that the algorithm is efficient.The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100,,1 000 respectively and the problems of each size were generated randomly for 10 times.The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations.It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the secondorder cones quickly.Moreover,a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones,and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.展开更多
The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships bet...The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships between the two classes of set-valued maps are investigated, and some properties of them are shown. A Gordan type alternative theorem under the assumption of generalized cone-subconvexlikeness of set-valued maps is proved by applying convex separation theorems involving the relative interiors in infinite dimensional spaces. Finally a necessary optimality condition theorem is shown for a general kind of set-valued vector optimization in a sense of weak E-minimizer.展开更多
Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algor...Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton （AHO） directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O（rln（εo/ε）） is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.展开更多
The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity proble...The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.展开更多
单一线性锥规划方法求解最优潮流(optimal power flow,OPF)问题时,锥变量仅局限于某个单一锥集合,使得锥规划模型的构造缺乏灵活性,建模难度较大。为此,基于混合线性锥规划(mixed cone linear programming,MCLP)方法,提出了求解OPF问题...单一线性锥规划方法求解最优潮流(optimal power flow,OPF)问题时,锥变量仅局限于某个单一锥集合,使得锥规划模型的构造缺乏灵活性,建模难度较大。为此,基于混合线性锥规划(mixed cone linear programming,MCLP)方法,提出了求解OPF问题的3种MCLP模型—MCLP-OPF。该模型采用不同的锥变量来构建原始OPF问题的锥松弛模型,锥变量可同时取自半正定锥、二阶锥和非负多面体锥。引入MCLP-OPF问题的可行域"厚度",并根据该"厚度"大小选择直接内点法或齐次自对偶(homogeneous self-dual,HSD)内点法求解。从C-703节点等6个测试系统的仿真结果可以看到,相较于半定规划法,MCLP-OPF提高了锥规划方法的建模效率、求解效率和存储效率,更适于求解大规模电力系统问题。展开更多
基金the National Science Foundation(60574075, 60674108)
文摘A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.
基金Supported by National Natural Science Foundation of China (No.10871144)the Seed Foundation of Tianjin University (No.60302023)
文摘As a basic mathematical structure,the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems.In this paper,a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone.It is shown that the proposed algorithm is globally convergent and well-defined.Moreover,it can start from any point and only needs to solve one system of linear equations at most at each iteration.Under suitable assumptions,global linear and local quadratic convergence is established with Euclidean Jordan algebras.Numerical results indicate that the algorithm is efficient.The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100,,1 000 respectively and the problems of each size were generated randomly for 10 times.The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations.It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the secondorder cones quickly.Moreover,a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones,and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.
文摘The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships between the two classes of set-valued maps are investigated, and some properties of them are shown. A Gordan type alternative theorem under the assumption of generalized cone-subconvexlikeness of set-valued maps is proved by applying convex separation theorems involving the relative interiors in infinite dimensional spaces. Finally a necessary optimality condition theorem is shown for a general kind of set-valued vector optimization in a sense of weak E-minimizer.
基金supported by the National Natural Science Foundation of China (Nos. 71061002 and 11071158)the Natural Science Foundation of Guangxi Province of China (Nos. 0832052 and 2010GXNSFB013047)
文摘Based on the ideas of infeasible interior-point methods and predictor-corrector algorithms, two interior-point predictor-corrector algorithms for the second-order cone programming （SOCP） are presented. The two algorithms use the Newton direction and the Euler direction as the predictor directions, respectively. The corrector directions belong to the category of the Alizadeh-Haeberly-Overton （AHO） directions. These algorithms are suitable to the cases of feasible and infeasible interior iterative points. A simpler neighborhood of the central path for the SOCP is proposed, which is the pivotal difference from other interior-point predictor-corrector algorithms. Under some assumptions, the algorithms possess the global, linear, and quadratic convergence. The complexity bound O（rln（εo/ε）） is obtained, where r denotes the number of the second-order cones in the SOCP problem. The numerical results show that the proposed algorithms are effective.
基金Supported by the Funds of Ministry of Education of China for PhD (20020141013)the NNSF of China (10471015).
文摘The paper uses Euclidean Jordan algebras as a basic tool to extend smoothing functions, which include the Chen-Mangasarian class and the Fischer-Burmeister smoothing functions, to symmetric cone complementarity problems. Computable formulas for these functions and their Jacobians are derived. In addition, it is shown that these functions are Lipschitz continuous with respect to parameter # and continuously differentiable on J × J for any μ 〉 0.
文摘单一线性锥规划方法求解最优潮流(optimal power flow,OPF)问题时,锥变量仅局限于某个单一锥集合,使得锥规划模型的构造缺乏灵活性,建模难度较大。为此,基于混合线性锥规划(mixed cone linear programming,MCLP)方法,提出了求解OPF问题的3种MCLP模型—MCLP-OPF。该模型采用不同的锥变量来构建原始OPF问题的锥松弛模型,锥变量可同时取自半正定锥、二阶锥和非负多面体锥。引入MCLP-OPF问题的可行域"厚度",并根据该"厚度"大小选择直接内点法或齐次自对偶(homogeneous self-dual,HSD)内点法求解。从C-703节点等6个测试系统的仿真结果可以看到,相较于半定规划法,MCLP-OPF提高了锥规划方法的建模效率、求解效率和存储效率,更适于求解大规模电力系统问题。
