Coderivative Analysis of Variational Systems

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite-dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications.

[1]  B. Morduhovic.  Calculus of second-order subdifferentials in infinite dimensions , 2002 .

[2]  Boris S. Mordukhovich,et al.  Coderivatives of set-valued mappings: Calculus and applications , 1997 .

[3]  Adam B. Levy,et al.  Stability of Locally Optimal Solutions , 1999, SIAM J. Optim..

[4]  Boris S. Mordukhovich,et al.  On Second-Order Subdifferentials and Their Applications , 2001, SIAM J. Optim..

[5]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

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

[7]  Lionel Thibault,et al.  Subdifferentials of compactly lipschitzian vector-valued functions , 1980 .

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

[9]  R. Rockafellar,et al.  The radius of metric regularity , 2002 .

[10]  René Henrion,et al.  Metric regularity and quantitative stability in stochastic programs with probabilistic constraints , 1999, Math. Program..

[11]  L. Thibault,et al.  Coderivatives of multivalued mappings, locally compact cones and metric regularity , 1999 .

[12]  R. Phelps.  Convex Functions, Monotone Operators and Differentiability , 1989 .

[13]  Boris S. Mordukhovich,et al.  Compactly epi-Lipschitzian convex sets and functions in normed spaces , 2000 .

[14]  R. Tyrrell Rockafellar,et al.  Tilt Stability of a Local Minimum , 1998, SIAM J. Optim..

[15]  B. Mordukhovich.  Lipschitzian stability of constraint systems and generalized equations , 1994 .

[16]  Boris S. Mordukhovich,et al.  Restrictive metric regularity and generalized differential calculus in Banach spaces , 2004, Int. J. Math. Math. Sci..

[17]  Boris S. Mordukhovich,et al.  Sequential normal compactness versus topological normal compactness in variational analysis , 2003 .

[18]  D. Varberg.  Convex Functions , 1973 .

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

[20]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[21]  Boris S. Mordukhovich,et al.  Calculus of sequential normal compactness in variational analysis , 2003 .

[22]  Jean-Paul Penot,et al.  Compactness Properties, Openness Criteria and Coderivatives , 1998 .

[23]  B. Mordukhovich,et al.  Stablity of Set-Valued Mappings In Infinite Dimensions: Point Criteria and Applications , 1997 .

[24]  R. Tyrrell Rockafellar,et al.  Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets , 1996, SIAM J. Optim..

[25]  B. S. Mordukhovich,et al.  Mixed Coderivatives of Set–Valued Mappings in Variational Analysis , 1998 .

[26]  Boris S. Mordukhovich,et al.  Nonconvex differential calculus for infinite-dimensional multifunctions , 1996 .

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