Optimality conditions for nonlinear semidefinite programming via squared slack variables

In this work, we derive second-order optimality conditions for nonlinear semidefinite programming (NSDP) problems, by reformulating it as an ordinary nonlinear programming problem using squared slack variables. We first consider the correspondence between Karush-Kuhn-Tucker points and regularity conditions for the general NSDP and its reformulation via slack variables. Then, we obtain a pair of “no-gap” second-order optimality conditions that are essentially equivalent to the ones already considered in the literature. We conclude with the analysis of some computational prospects of the squared slack variables approach for NSDP.

[1]  Masao Fukushima,et al.  The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming , 2016, Journal of Optimization Theory and Applications.

[2]  Alexander Shapiro,et al.  First and second order analysis of nonlinear semidefinite programs , 1997, Math. Program..

[3]  Michael Stingl,et al.  PENNON: A code for convex nonlinear and semidefinite programming , 2003, Optim. Methods Softw..

[4]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[5]  R. Cominetti.  Metric regularity, tangent sets, and second-order optimality conditions , 1990 .

[6]  Renato D. C. Monteiro,et al.  Digital Object Identifier (DOI) 10.1007/s10107-004-0564-1 , 2004 .

[7]  Michael Stingl,et al.  PENLAB: A MATLAB solver for nonlinear semidefinite optimization , 2013 .

[8]  Gábor Pataki,et al.  On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues , 1998, Math. Oper. Res..

[9]  Alexander I. Barvinok,et al.  Problems of distance geometry and convex properties of quadratic maps , 1995, Discret. Comput. Geom..

[10]  Klaus Schittkowski,et al.  Test examples for nonlinear programming codes , 1980 .

[11]  Hidefumi Kawaski,et al.  An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems , 1988 .

[12]  Alexander Shapiro,et al.  Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets , 1999, SIAM J. Optim..

[13]  Hiroshi Yamashita,et al.  A survey of numerical methods for nonlinear semidefinite programming , 2015 .

[14]  J. Lasserre,et al.  Handbook on Semidefinite, Conic and Polynomial Optimization , 2012 .

[15]  Renato D. C. Monteiro,et al.  A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization , 2003, Math. Program..

[16]  J. Sturm.  Similarity and other spectral relations for symmetric cones , 2000 .

[17]  G. Pataki.  The Geometry of Semidefinite Programming , 2000 .

[18]  Klaus Schittkowski,et al.  More test examples for nonlinear programming codes , 1981 .

[19]  Florian Jarre.  Elementary Optimality Conditions for Nonlinear SDPs , 2012 .

[20]  Hidefumi Kawasaki.  An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems , 1988, Math. Program..

[21]  Anders Forsgren,et al.  Optimality conditions for nonconvex semidefinite programming , 2000, Math. Program..

[22]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[23]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[24]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .