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.
Given a system of linear equations and inequalities in n variables, a famous result due to A. J. Hoffman (1952) says that the distance of any point in R" to the solution set of this system is bounded above by the product of a positive constant and the absolute residual. We shall discuss explicit representations of this constant in dependence upon the pair of norms used for the estimation. A method for computing a special form of Hoffman constants is proposed. Finally, we use these results in the analysis of Lipsehitz continuity for solutions of parametric quadratic programs.
Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several rst order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.
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