Lipschitz and Hölder stability of optimization problems and generalized equations

This paper studies stability aspects of solutions of parametric mathematical programs and generalized equations, respectively, with disjunctive constraints. We present sufficient conditions that, under some constraint qualifications ensuring metric subregularity of the constraint mapping, continuity results of upper Lipschitz and upper Hölder type, respectively, hold. Furthermore, we apply the above results to parametric mathematical programs with equilibrium constraints and demonstrate, how some classical results for the nonlinear programming problem can be recovered and even improved by our theory.

[1]  Helmut Gfrerer,et al.  Second-Order Optimality Conditions for Scalar and Vector Optimization Problems in Banach Spaces , 2006, SIAM J. Control. Optim..

[2]  René Henrion,et al.  Calmness of constraint systems with applications , 2005, Math. Program..

[3]  Vladimir Shikhman.  Topological Aspects of Nonsmooth Optimization , 2011 .

[4]  Lei Guo,et al.  Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications , 2012, SIAM J. Optim..

[5]  Sonja Steffensen,et al.  Characterization of strong stability for C-stationary points in MPCC , 2012, Math. Program..

[6]  Alexey F. Izmailov,et al.  Sensitivity Analysis for Cone-Constrained Optimization Problems Under the Relaxed Constraint Qualifications , 2005, Math. Oper. Res..

[7]  M. Kojima.  Strongly Stable Stationary Solutions in Nonlinear Programs. , 1980 .

[8]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[9]  Helmut Gfrerer,et al.  First Order and Second Order Characterizations of Metric Subregularity and Calmness of Constraint Set Mappings , 2011, SIAM J. Optim..

[10]  Christian Kanzow,et al.  Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications , 2008 .

[11]  Christian Kanzow,et al.  Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications , 2008, Math. Program..

[12]  Helmut Gfrerer,et al.  JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics On Directional Metric Subregularity and Second-Order Optimality Conditions for a Class of Nonsmooth Mathematical Programs , 2012 .

[13]  Bernd Kummer,et al.  Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland's principle , 2009 .

[14]  René Henrion,et al.  On the Calmness of a Class of Multifunctions , 2002, SIAM J. Optim..

[15]  Stefan Scholtes,et al.  Convergence Properties of a Regularization Scheme for Mathematical Programs with Complementarity Constraints , 2000, SIAM J. Optim..

[16]  Helmut Gfrerer,et al.  JOHANNES KEPLER UNIVERSITY LINZ Institute of Computational Mathematics On Directional Metric Regularity, Subregularity and Optimality Conditions for Nonsmooth Mathematical Programs , 2012 .

[17]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[18]  Helmut Gfrerer,et al.  Optimality Conditions for Disjunctive Programs Based on Generalized Differentiation with Application to Mathematical Programs with Equilibrium Constraints , 2014, SIAM J. Optim..

[19]  Alexey F. Izmailov,et al.  Mathematical Programs with Vanishing Constraints: Optimality Conditions, Sensitivity, and a Relaxation Method , 2009, J. Optimization Theory and Applications.

[20]  Michal Červinka,et al.  On Stability of M-stationary Points in MPCCs , 2014 .

[21]  B. Mordukhovich.  Variational Analysis and Generalized Differentiation II: Applications , 2006 .

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

[23]  A. F. Izmailov.  Mathematical Programs with Complementarity Constraints: Regularity, Optimality Conditions, and Sensitivity , 2004 .

[24]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

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

[26]  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 .

[27]  D. Klatte.  Nonsmooth equations in optimization , 2002 .

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