Metric Subregularity and Constraint Qualifications for Convex Generalized Equations in Banach Spaces

Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the normal cones, and thereby we provide some characterizations for convex generalized equations to have the metric subregularity. As applications, we establish formulas of the modulus of calmness and provide several characterizations of the calmness. Extending the classical concept of extreme boundary, we introduce a notion of recession cores of closed convex sets. Using this concept, we establish global metric subregularity (i.e., error bound) results for generalized equations.

[1]  René Henrion,et al.  Subdifferential Conditions for Calmness of Convex Constraints , 2002, SIAM J. Optim..

[2]  Hui Hu,et al.  Characterizations of the Strong Basic Constraint Qualifications , 2005, Math. Oper. Res..

[3]  A. Kruger,et al.  A covering theorem for set-valued mappings , 1988 .

[4]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

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

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

[7]  Xi Yin Zheng,et al.  Characterizations of Error Bounds for Convex Multifunctions on Banach Spaces , 2004, Math. Oper. Res..

[8]  Wu Li,et al.  Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities , 2000, SIAM J. Optim..

[9]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[10]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[11]  C. Ursescu.  Multifunctions with convex closed graph , 1975 .

[12]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[13]  Diethard Klatte,et al.  Constrained Minima and Lipschitzian Penalties in Metric Spaces , 2002, SIAM J. Optim..

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

[15]  J. Burke,et al.  Weak sharp minima revisited Part I: basic theory , 2002 .

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

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

[18]  J. Aubin.  Set-valued analysis , 1990 .

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

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

[21]  Wei Hong Yang,et al.  Error Bounds for Abstract Linear Inequality Systems , 2002, SIAM J. Optim..

[22]  A. Lewis,et al.  Error Bounds for Convex Inequality Systems , 1998 .

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

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

[25]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[26]  Xi Yin Zheng,et al.  Metric Regularity and Constraint Qualifications for Convex Inequalities on Banach Spaces , 2003, SIAM J. Optim..

[27]  Wu Li,et al.  Abadie's Constraint Qualification, Metric Regularity, and Error Bounds for Differentiable Convex Inequalities , 1997, SIAM J. Optim..

[28]  Wu Li,et al.  Global Error Bounds for Convex Multifunctions and Applications , 1998, Math. Oper. Res..

[29]  G. Jameson.  Ordered Linear Spaces , 1970 .

[30]  M. Ferris,et al.  On the Clarke subdifferential of the distance function of a closed set , 1992 .

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

[32]  Asen L. Dontchev,et al.  Regularity and Conditioning of Solution Mappings in Variational Analysis , 2004 .

[33]  S. M. Robinson.  Normed convex processes , 1972 .

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

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

[36]  Constantin Zalinescu,et al.  A Nonlinear Extension of Hoffman's Error Bounds for Linear Inequalities , 2003, Math. Oper. Res..