On Lipschitz Semicontinuity Properties of Variational Systems with Application to Parametric Optimization

In this paper, two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In the consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fréchet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed.

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

[2]  Stephen M. Robinson,et al.  Variational conditions with smooth constraints: structure and analysis , 2003, Math. Program..

[3]  Anthony V. Fiacco,et al.  Introduction to Sensitivity and Stability Analysis in Nonlinear Programming , 2012 .

[4]  R. Rockafellar,et al.  Sensitivity analysis of solutions to generalized equations , 1994 .

[5]  Jane J. Ye,et al.  A note on optimality conditions for bilevel programming problems , 1997 .

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

[7]  Boris S. Mordukhovich,et al.  Enhanced metric regularity and Lipschitzian properties of variational systems , 2011, J. Glob. Optim..

[8]  A. Ioffe.  Metric regularity and subdifferential calculus , 2000 .

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

[10]  Jen-Chih Yao,et al.  Calmness of efficient solution maps in parametric vector optimization , 2011, J. Glob. Optim..

[11]  Marcin Studniarski,et al.  Weak Sharp Minima: Characterizations and Sufficient Conditions , 1999, SIAM J. Control. Optim..

[12]  Boris S. Mordukhovich,et al.  Failure of Metric Regularity for Major Classes of Variational Systems , 2008 .

[13]  Jane J. Ye,et al.  Optimality conditions for bilevel programming problems , 1995 .

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

[15]  Alexander D. Ioffe,et al.  On Metric and Calmness Qualification Conditions in Subdifferential Calculus , 2008 .

[16]  Xi Yin Zheng,et al.  Metric Subregularity and Constraint Qualifications for Convex Generalized Equations in Banach Spaces , 2007, SIAM J. Optim..

[17]  A. Ioffe.  Necessary and Sufficient Conditions for a Local Minimum. 1: A Reduction Theorem and First Order Conditions , 1979 .

[18]  Boris S. Mordukhovich,et al.  Metric regularity and Lipschitzian stability of parametric variational systems , 2010 .

[19]  Boris S. Mordukhovich,et al.  Variational analysis of extended generalized equations via coderivative calculus in Asplund spaces , 2009 .

[20]  Frank H. Clarke,et al.  A New Approach to Lagrange Multipliers , 1976, Math. Oper. Res..

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

[22]  A. Kruger,et al.  Error Bounds: Necessary and Sufficient Conditions , 2010 .

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

[24]  Xi Yin Zheng,et al.  Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces , 2010, SIAM J. Optim..

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

[26]  Adam B. Levy.  Supercalm Multifunctions For Convergence Analysis , 2006 .

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

[28]  A. Uderzo.  On Some Regularity Properties in Variational Analysis , 2009 .

[29]  N. D. Yen.  Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application , 1997 .

[30]  A. Uderzo.  On a Quantitative Semicontinuity Property of Variational Systems with Applications to Perturbed Quasidifferentiable Optimization , 2014 .

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

[32]  Huynh van Ngai,et al.  Implicit multifunction theorems in complete metric spaces , 2013, Math. Program..

[33]  James V. Burke,et al.  Calmness and exact penalization , 1991 .

[34]  D. Azé,et al.  Variational pairs and applications to stability in nonsmooth analysis , 2002 .

[35]  I. Ekeland,et al.  Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces , 1976 .

[36]  Karolin Papst,et al.  Techniques Of Variational Analysis , 2016 .

[37]  Nguyen Dong Yen,et al.  Normal coderivative for multifunctions and implicit function theorems , 2008 .

[38]  Marius Durea,et al.  Openness stability and implicit multifunction theorems: Applications to variational systems , 2011, 1102.0415.

[39]  Boris S. Mordukhovich,et al.  Coderivative calculus and metric regularity for constraint and variational systems , 2009 .

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

[41]  R. Rockafellar,et al.  Variational conditions and the proto-differentiation of partial subgradient mappings , 1996 .