Characterizations of local upper Lipschitz property of perturbed solutions to nonlinear second-order cone programs

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping.

[1]  Benjamin Pfaff,et al.  Perturbation Analysis Of Optimization Problems , 2016 .

[2]  Jean-Pierre Aubin,et al.  Lipschitz Behavior of Solutions to Convex Minimization Problems , 1984, Math. Oper. Res..

[3]  R. Rockafellar,et al.  Characterizations of Lipschitzian Stability in Nonlinear Programming , 2020 .

[4]  R. Tyrrell Rockafellar,et al.  Sensitivity analysis for nonsmooth generalized equations , 1992, Math. Program..

[5]  Diethard Klatte,et al.  Nonsmooth Equations in Optimization: "Regularity, Calculus, Methods And Applications" , 2006 .

[6]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[7]  Shaohua Pan,et al.  Locally upper Lipschitz of the perturbed KKT system of Ky Fan $k$-norm matrix conic optimization problems , 2015, 1509.00681.

[8]  Defeng Sun,et al.  On the Coderivative of the Projection Operator onto the Second-order Cone , 2008 .

[9]  Defeng Sun,et al.  Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems , 2017, SIAM J. Optim..

[10]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[11]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[12]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[13]  J. F. Bonnans,et al.  Local analysis of Newton-type methods for variational inequalities and nonlinear programming , 1994 .

[14]  Diethard Klatte.  Upper Lipschitz behavior of solutions to perturbed. C1,1 programs , 2000, Math. Program..

[15]  S. M. Robinson.  Generalized equations and their solutions, Part I: Basic theory , 1979 .

[16]  Defeng Sun,et al.  Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Quadratic and Semi-Definite Programming , 2015, 1508.02134.

[17]  Z.-Q. Luo,et al.  Error bounds and convergence analysis of feasible descent methods: a general approach , 1993, Ann. Oper. Res..

[18]  Defeng Sun,et al.  Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems , 2003, Comput. Optim. Appl..

[19]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[20]  Jirí V. Outrata,et al.  Erratum: On The Aubin Property of Critical Points to Perturbed Second-Order Cone Programs , 2017, SIAM J. Optim..

[21]  S. M. Robinson.  First Order Conditions for General Nonlinear Optimization , 1976 .

[22]  Asen L. Dontchev,et al.  Characterizations of Lipschitz Stability in Optimization , 1995 .

[23]  Marie Frei,et al.  Recent Developments In Well Posed Variational Problems , 2016 .

[24]  Jirí V. Outrata,et al.  On the Aubin Property of Critical Points to Perturbed Second-Order Cone Programs , 2011, SIAM J. Optim..

[25]  D. Klatte.  Book review: Implicit Functions and Solution Mappings:A View from Variational Analysis. Second Edition. By A. L. Dontchev and R. T. Rockafellar. Springer, New York, 2014 , 2015 .

[26]  B. Mordukhovich.  Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis , 1994 .

[27]  J. Frédéric Bonnans,et al.  Perturbation analysis of second-order cone programming problems , 2005, Math. Program..

[28]  Boris S. Mordukhovich,et al.  Graphical Derivatives and Stability Analysis for Parameterized Equilibria with Conic Constraints , 2014, 1412.0550.

[29]  Boris S. Mordukhovich,et al.  Second-Order Variational Analysis in Conic Programming with Applications to Optimality and Stability , 2015, SIAM J. Optim..

[30]  Adam B. Levy,et al.  Implicit multifunction theorems for the sensitivity analysis of variational conditions , 1996, Math. Program..