No Gap Second-order Optimality Conditions for a Matrix Cone Programming Induced by the Nuclear Norm

被引：0

作者：

Zhang, Ning ^{[1
]}

Zhang, Liwei ^{[1
]}

机构：

[1] Dalian Univ Technol, Sch Math Sci, Inst Operat Res & Control Theory, Dalian 116024, Peoples R China

来源：

ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH | 2016年 / 33卷 / 02期

关键词：

Nuclear norm; second-order tangent set; sigma term; no gap second-order optimality conditions;

D O I：

10.1142/S021759591650010X

中图分类号：

C93 [管理学]; O22 [运筹学];

学科分类号：

070105 ; 12 ; 1201 ; 1202 ; 120202 ;

摘要：

The first-order and the second-order directional derivatives of singular values are used to characterize the tangent cone, the normal cone and the second-order tangent set of the epigraph of the nuclear norm of matrices. Based on the variational geometry of the epigraph, the no gap second-order optimality conditions for the optimization problem, whose constraint is defined by the matrix cone induced by the nuclear norm, are established.

引用

页数：20

共 11 条

[1]

[Anonymous], 1996, PRINCETON MATH SER

[2]

Bonnans J.F., 2013, PERTURBATION ANAL OP

[3] Distributed optimization and statistical learning via the alternating direction method of multipliers [J].

Boyd S. ;

Parikh N. ;

Chu E. ;

Peleato B. ;

Eckstein J. .

Foundations and Trends in Machine Learning, 2010, 3 (01) :1-122

[4] An introduction to a class of matrix cone programming [J].

Ding, Chao ;

Sun, Defeng ;

Toh, Kim-Chuan .

MATHEMATICAL PROGRAMMING, 2014, 144 (1-2) :141-179

[5] MATHEMATICAL PROGRAMS WITH GEOMETRIC CONSTRAINTS IN BANACH SPACES: ENHANCED OPTIMALITY, EXACT PENALTY, AND SENSITIVITY [J].

Guo, Lei ;

Ye, Jane J. ;

Zhang, Jin .

SIAM JOURNAL ON OPTIMIZATION, 2013, 23 (04) :2295-2319

[6]

Johnson C. R., 1991, TOPICS MATRIX ANAL

[7] An implementable proximal point algorithmic framework for nuclear norm minimization [J].

Liu, Yong-Jin ;

Sun, Defeng ;

Toh, Kim-Chuan .

MATHEMATICAL PROGRAMMING, 2012, 133 (1-2) :399-436

[8]

Rockafellar RT., 1998, VARIATIONAL ANAL

[9] The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming [J].

Sun, Defeng ;

Sun, Jie ;

Zhang, Liwei .

MATHEMATICAL PROGRAMMING, 2008, 114 (02) :349-391

[10] Second-order directional derivatives of all eigenvalues of a symmetric matrix [J].

Torki, M .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2001, 46 (08) :1133-1150

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