On Lipschitzian Stability of Optimal Solutions of Parametrized Semi-Infinite Programs

We study continuity properties of optimal solutions of parametrized semi-infinite programming problems. The involved constraints are formulated in a form of cone constraints and then a slightly modified general result of Shapiro and Bonnans 1992 on Lipschitzian stability of optimal solutions is applied. It is shown that under certain second-order sufficient conditions, optimal solutions of the semi-infinite programs are Lipschitzian stable provided a regularity assumption related to a linearization of the considered programs is satisfied.

[1]  Diethard Klatte.  Stability of Stationary Solutions in Semi-Infinite Optimization via the Reduction Approach , 1992 .

[2]  Stephen M. Robinson,et al.  Regularity and Stability for Convex Multivalued Functions , 1976, Math. Oper. Res..

[3]  R. P. Hettich,et al.  Semi-infinite programming: Conditions of optimality and applications , 1978 .

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

[5]  J. Frédéric Bonnans,et al.  Second-order Sufficiency and Quadratic Growth for Nonisolated Minima , 1995, Math. Oper. Res..

[6]  J. F. Bonnans,et al.  Sensitivity analysis of parametrized programs under cone constraints , 1992 .

[7]  A. Shapiro.  Sensitivity analysis of nonlinear programs and differentiability properties of metric projections , 1988 .

[8]  H. Th. Jongen,et al.  On sufficient conditions for local optimality in semi-infinite programming , 1987 .

[9]  Jacques Gauvin,et al.  Directional Behaviour of Optimal Solutions in Nonlinear Mathematical Programming , 1988, Math. Oper. Res..

[10]  B. N. Pshenichnyi.  Necessary Conditions for an Extremum , 1971 .

[11]  H. Maurer,et al.  Differential stability in infinite-dimensional nonlinear programming , 1980 .

[12]  A. Shapiro.  Perturbation analysis of optimization problems in banach spaces , 1992 .

[13]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[14]  J. F. Bonnans,et al.  Expansion of exact and approximate solutions in nonlinear programming , 1992 .

[15]  J. Zowe,et al.  Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems , 1979 .

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

[17]  A. Auslender,et al.  First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions , 1990 .

[18]  Alexander Shapiro,et al.  Second-Order Derivatives of Extremal-Value Functions and Optimality Conditions for Semi-Infinite Programs , 1985, Math. Oper. Res..