对称锥优化问题的扰动分析
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摘要
本论文主要建立对称锥的变分分析并给出对称锥优化问题扰动分析的理论结果,主要内容可概括如下:
1.第2章基于欧氏Jordan代数的性质,研究了对称锥的变分性质。首先推导了对称锥上投影算子B-次微分的计算公式以及对称锥的切锥、二阶切集的表达式。然后定义了对称锥上的线性-二次函数,建立了线性-二次函数与对称锥二阶切集的支撑函数的关系。
2.第3章在前一章研究的基础上,阐述了对称锥优化问题的扰动理论结果。首先证明了对称锥具有外二阶正则性,从而给出对称锥优化问题无间隙的二阶最优性条件。其次引入对称锥优化问题两种形式的强二阶充分性条件,其中之一通过一线性-二次函数定义,另一个用二阶切集的支撑函数定义。之后,对于上述两种强二阶充分条件重合的非凸对称锥优化问题,得到了下述条件的等价性:约束非退化条件下的强二阶充分条件,KKT条件对应的广义方程的解的强正则性,KKT条件对应的非光滑映射(简称KKT映射)的Clarke广义微分的非奇异性,优化问题局部最优解的强稳定性,以及其他4条结论相互等价。特别地,对于凸对称锥优化问题,不需要任何假设,上述9条性质均等价,且等价于KKT映射的B-次微分的非奇异性。最后,对于线性对称锥优化问题,首先刻画了对偶严格约束规范与二阶充分性条件的等价关系;然后证明了上述10条等价性质与原始对偶约束非退化条件的等价性。
3.第4章具体推导了二阶锥、半正定实对称矩阵锥、半正定复Hermite矩阵锥以及半正定四元数Hermite矩阵锥二阶切集的表达式,并证明了这四种情况下二阶切集的支撑函数与对应的线性-二次函数是相等的,从而得到了当K同构于有限多个上述四种类型的锥的卡氏积时,第3章提出的两种形式的强二阶充分性条件等价,因此对于上述形式的K,第3章中关于锥优化问题扰动理论的等价性结论是成立的。
This dissertation focuses on the study of variational analysis of symmetric cones and perturbation analysis of optimization over symmetric cones. The main results, obtained in this dissertation, may be summarized as follows:
1. Chapter 2, based on the theory of Euclidean Jordan algebras, develops the elements in the variational analysis of symmetric cones related to the optimization problem. The formulas of tangent cones, second order tangent sets of symmetric cones and the B-subdifferential of the projection operator onto a symmetric cone are derived. A linear-quadratic function is introduced and the relationship with the supporting function of the second order tangent set is established.
2. Chapter 3, based on the variational analysis developed in Chapter 2, studies the perturbation analysis of the nonlinear symmetric conic optimization problem. Firstly, the outer second order regularity of symmetric cones is proved and the no gap optimality conditions are obtained. Secondly, two types of strong second order sufficient optimality conditions for the optimization over symmetric cones are proposed, one of which is described through the linear-quadratic form and the other through the supporting function of the second order tangent set. For the nonconvex conic optimization problems whose two strong second order sufficient conditions coincide, we show that for a locally optimal solution, the following conditions are equivalent: the strong second order sufficient condition and primal constraint non-degeneracy, the strong regularity of the generalized equation corresponding to the KKT system, the nonsingularity of the Clarke's generalized subdifferential of the nonsmooth mapping (KKT mapping for short) corresponding to the KKT conditions at the KKT point, strong stability of the optimal solution and other 4 conditions. Moreover, for a convex conic programming problem, the equivalence among all the listed conditions still holds when the cone is symmetric. Besides, each of these conditions is equivalent to the nonsingularity of B-subdifferential of the KKT mapping. Especially, for a linear conic programming problem, the relationship between the second order sufficient condition and the dual strict constraint qualification is characterized. Moreover, in this case the above 10 equivalent conditions for convex programming are proven to be equivalent to both the primal and dual constraint non-degeneracies.
3. In Chapter 4, formulas for second order tangent sets of second order cone, the cone of positively semidefinite symmetric real matrices, the cone of positively semidefinite complex Hermitian matrices, and the cone of positively semidefinite quaternion Hermitian matrices, are developed and in each case the linear-quadratic form is proved to coincide with the supporting function of the second order tangent set. Therefore, when K is represented as a cone isomorphic to a Cartesian product of a finite number of these four types of cones, the two strong second order sufficient conditions are equivalent and all equivalent conditions about perturbation analysis in Chapter 3 are valid when K is of this kind of form.
1.第2章基于欧氏Jordan代数的性质,研究了对称锥的变分性质。首先推导了对称锥上投影算子B-次微分的计算公式以及对称锥的切锥、二阶切集的表达式。然后定义了对称锥上的线性-二次函数,建立了线性-二次函数与对称锥二阶切集的支撑函数的关系。
2.第3章在前一章研究的基础上,阐述了对称锥优化问题的扰动理论结果。首先证明了对称锥具有外二阶正则性,从而给出对称锥优化问题无间隙的二阶最优性条件。其次引入对称锥优化问题两种形式的强二阶充分性条件,其中之一通过一线性-二次函数定义,另一个用二阶切集的支撑函数定义。之后,对于上述两种强二阶充分条件重合的非凸对称锥优化问题,得到了下述条件的等价性:约束非退化条件下的强二阶充分条件,KKT条件对应的广义方程的解的强正则性,KKT条件对应的非光滑映射(简称KKT映射)的Clarke广义微分的非奇异性,优化问题局部最优解的强稳定性,以及其他4条结论相互等价。特别地,对于凸对称锥优化问题,不需要任何假设,上述9条性质均等价,且等价于KKT映射的B-次微分的非奇异性。最后,对于线性对称锥优化问题,首先刻画了对偶严格约束规范与二阶充分性条件的等价关系;然后证明了上述10条等价性质与原始对偶约束非退化条件的等价性。
3.第4章具体推导了二阶锥、半正定实对称矩阵锥、半正定复Hermite矩阵锥以及半正定四元数Hermite矩阵锥二阶切集的表达式,并证明了这四种情况下二阶切集的支撑函数与对应的线性-二次函数是相等的,从而得到了当K同构于有限多个上述四种类型的锥的卡氏积时,第3章提出的两种形式的强二阶充分性条件等价,因此对于上述形式的K,第3章中关于锥优化问题扰动理论的等价性结论是成立的。
This dissertation focuses on the study of variational analysis of symmetric cones and perturbation analysis of optimization over symmetric cones. The main results, obtained in this dissertation, may be summarized as follows:
1. Chapter 2, based on the theory of Euclidean Jordan algebras, develops the elements in the variational analysis of symmetric cones related to the optimization problem. The formulas of tangent cones, second order tangent sets of symmetric cones and the B-subdifferential of the projection operator onto a symmetric cone are derived. A linear-quadratic function is introduced and the relationship with the supporting function of the second order tangent set is established.
2. Chapter 3, based on the variational analysis developed in Chapter 2, studies the perturbation analysis of the nonlinear symmetric conic optimization problem. Firstly, the outer second order regularity of symmetric cones is proved and the no gap optimality conditions are obtained. Secondly, two types of strong second order sufficient optimality conditions for the optimization over symmetric cones are proposed, one of which is described through the linear-quadratic form and the other through the supporting function of the second order tangent set. For the nonconvex conic optimization problems whose two strong second order sufficient conditions coincide, we show that for a locally optimal solution, the following conditions are equivalent: the strong second order sufficient condition and primal constraint non-degeneracy, the strong regularity of the generalized equation corresponding to the KKT system, the nonsingularity of the Clarke's generalized subdifferential of the nonsmooth mapping (KKT mapping for short) corresponding to the KKT conditions at the KKT point, strong stability of the optimal solution and other 4 conditions. Moreover, for a convex conic programming problem, the equivalence among all the listed conditions still holds when the cone is symmetric. Besides, each of these conditions is equivalent to the nonsingularity of B-subdifferential of the KKT mapping. Especially, for a linear conic programming problem, the relationship between the second order sufficient condition and the dual strict constraint qualification is characterized. Moreover, in this case the above 10 equivalent conditions for convex programming are proven to be equivalent to both the primal and dual constraint non-degeneracies.
3. In Chapter 4, formulas for second order tangent sets of second order cone, the cone of positively semidefinite symmetric real matrices, the cone of positively semidefinite complex Hermitian matrices, and the cone of positively semidefinite quaternion Hermitian matrices, are developed and in each case the linear-quadratic form is proved to coincide with the supporting function of the second order tangent set. Therefore, when K is represented as a cone isomorphic to a Cartesian product of a finite number of these four types of cones, the two strong second order sufficient conditions are equivalent and all equivalent conditions about perturbation analysis in Chapter 3 are valid when K is of this kind of form.
引文
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[42]Wang Y,Zhang L W.Nonsingularity in second-order cone programming via the smoothing metric projector.Submitted to Science in China Series A:Mathematics,2008.
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[44]Sun D F.The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications.Mathematics of Operations Research,2006,31:761-776.
[45]Chan Z X,Sun D F.Constraint nondegeneracy,strong regularity and nonsingularity in semidefinite programming.SIAM Journal on Optimization,2008,19:370-396.
[46]Sun D F,Wang Y,Zhang L W.Optimization over symmetric cones:variational and sensitivity analysis.Reported,2008.
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[2]Koecher M.Positivit(a|¨)tsbereiche in R~n.American Journal of Mathematics,1958,79:575-596.
[3]Koecher M.On real Jordan algebras.Bulletin of The American Mathematical Society,1962,68:374-377.
[4]Koecher M.Jordan Algebras and Their Applications.Lecture notes,University of Minnesota,Minneapolis,1962.
[5]Rothaus O S.Domians of positivity.Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg,1960,24:189-235.
[6]Vinberg E B.The theory of convex homogeneous cones.Transactions of the Moscow Mathematical Society,1963,12:303-358.
[7]Faybusovich L.Euclidean Jordan algebras and interior-point algorithms.Positivity,1997,1(4):331-357.
[8]Faybusovich L.Linear systems in Jordan algebras and primal-dual interior point algorithms.Journal of Computational and Applied Mathematics,1997,86:149-175.
[9]Schmieta S H,Alizadch F.Associative and Jordan algebras,and polynomial time interior-point algorithms for symmetric cones.Mathematics of Operations Research,2001,26(3):543-564.
[10]Schmieta S H,Alizadeh F.Extension of primal-dual interior point algorithms to symmetric cones.Mathematical Programming,Series A,2003,96(3):409-438.
[11]Vandenberghe L,Boyd S.Semidefinite programming.SIAM Review,1996,38:49-95.
[12]Vandenberghe L,Boyd S.Applications of semidefinite programming,Applied Numerical Mathematics,1999,29:283-299.
[13]Henry W,Romesh S,Lieven V.Handbook of Semidefinite Programming:Theory,Algorithms,and Applications.Printed in the United States of America,ISBN,2000.
[14]Klerk E D.Aspects Of Semidefinite Programming.Netherlands:Kluwer Academic Publishers,2002.
[15]刘勇进.对称锥优化的某些光滑函数及其应用.(博士学位论文).大连:大连理工大学,2004.
[16]Roos C,Bai Y Q,Chaerani D.Conic optimization models for robust truss topology design with exampies.OR Transactions,2004,8(1):1-40.
[17]Nemirovski A.Advances in Convex Optimization:Conic Programming.Volume Ⅰ(Plenary Lectures)of International Congress of Mathematicians,Madrid 2006,European Mathematical Society,2006.
[18]修乃华,韩继业.对称锥互补问题.数学进展,2007,1:3-14.
[19]孔令臣.对称锥互补问题的互补函数和价值函数研究.(博士学位论文).北京:北京交通大学,2007.
[20]Huang Z H,Han J.Non-interior continuation method for solving the monotone semidefinite comple-mentarity problem.Applied Mathematics Optimization,2003;47:195-211.
[21]Huang Z H,Qi L Q,Sun D F.Sub-quadratic convergence of a smoothing Newton algorithm for the P_0-and monotone LCP.Mathematical Programming,2004,99:423-441.
[22]Huang Z H.The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP.IMAJ.Numerical Analysis,2005,25:670-684.
[23]Huang Z H,Sun J.A non-interior continuation algorithm for the P_0 or P_* LCP with strong global local convergence properties.Applied Mathematics and Optimization,2005,26:237-262.
[24]Huang Z H.Global Lipschitzian error bounds for semidefinite complementarity problems with emphasis on NCPs.Applied Mathematics and Computation,2005,162:1237-1258.
[25]Rockafellar R T.Conjugate Duality and Optimization.SIAM Philadelphia,PA,Regional Conference Series in Applied Mathematics,1974.
[26]Malanowski K.Second order conditions and constraints qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces.Applied Mathematics and Optimization,1992,25(1):51-79.
[27]Shapiro A,Bonnans J F.Sensitivity analysis of parameterized programs under cone constraints.SIAM Journal on Control and Optimization,1992,30(6):1409-1422.
[28]Rockafellar R T,Wets R J B.Variational Analysis.New York:Springer,1998.
[29]Bonnans J F,Shapiro A.Perturbation Analysis of Optimization Problems.New York:Springer,2000.
[30]Klatte D,Kummer B.Nonsmooth Equations in Optimization:Regularity,Calculus,Methods and Applications.Boston:Kluwer Academic Publishers,2002.
[31]Facchinei F,Pang J S.Finite-Dimensional Variational Inequalities and Complementarity Problems:Volume Ⅰ.New York:Springer,2003.
[32]Fiacco A V.Introduction to Sensitivity and Stability Analysis in Nonlinear Programming.New York:Academic Press,1983.
[33]Robinson S M.Strongly regular generalized equations.Mathematics of Operations Research,1980,5:43-62.
[34]Kojima M.Strongly stable stationary solutions in nonlinear programs.S.M.Robinson,ed.Analysis and Computation of Fixed Points.New York:Academic Press,1980,93-138.
[35]Jongen H Th,Mobert T,R(u|¨)ckmann J et al.On inertia and Schur complement in optimization.Linear Algebra and Its Applications,1987,95:97-109.
[36]Bonnans J F,Sulem A.Pseudopower expansion of solutions of generalized equations and constrained optimization problems.Mathematical Programming,1995,70:123-148.
[37]Dontchev A L,Rockafellar R T.Characterizations of strong regularity for variational inequalities over polyhedral convex sets.SIAM Journal on Optimization,1996,6:1087-1105.
[38]Shapiro A.perturbation analysis of optimization problems in Banach spaces.Numerical Functional Analysis and Optimization,1992,13:97-116.
[39]Klatte D,Kummer B.Strong stability in nonlinear programming revisited.Journal of the Australian Mathematical Society,Series B,1999,40:336-352.
[40]Bonnans J F,Ramfrez C H.Perturbation analysis of second order cone programming problems.Mathematical Programming,Series B,2005,104:205-227.
[41]Wang Y,Zhang L W.Properties of equation reformulation of the Karush-Kuhn-Tucker condition for nonlinear second order cone optimization problems.Mathematical Methods of Operations Research,2008,DOI:10.1007/s00186-008-0241-x.
[42]Wang Y,Zhang L W.Nonsingularity in second-order cone programming via the smoothing metric projector.Submitted to Science in China Series A:Mathematics,2008.
[43]王韵,张立卫.Hilbert空间里的一类双层规划的最优性条件.运筹学学报,2008,12(3):90-102.
[44]Sun D F.The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications.Mathematics of Operations Research,2006,31:761-776.
[45]Chan Z X,Sun D F.Constraint nondegeneracy,strong regularity and nonsingularity in semidefinite programming.SIAM Journal on Optimization,2008,19:370-396.
[46]Sun D F,Wang Y,Zhang L W.Optimization over symmetric cones:variational and sensitivity analysis.Reported,2008.
[47]Zarantonello E H.Projections on convex sets in Hilbert space and spectral theory Ⅰ and Ⅱ.In E.H.Zarantonello,editor,Contributions to Nonlinear Functional Analysis,New York,Academic Press,1971:237-424.
[48]Meng F,Sun D,Zhao G.Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization.Mathematical Programming,2005,104:561-581.
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