Conditions For Metric Regularity Research Articles

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.

Lipschitzian Properties and Stability of a Class of First-Order Stochastic Dominance Constraints

Considering first-order stochastic dominance constraints for random variables arising as optimal values of stochastic programs with linear recourse, verifiable sufficient conditions for metric regularity are presented. A growth condition developed in [R. Henrion and W. Römisch, Math. Program., 84 (1999), pp. 55--88] has a crucial role in the analysis of the present paper. Implications regarding stability and sensitivity of optimal values and optimal solutions of stochastic optimization problems involving the dominance constraints considered conclude the paper.

Point-based sufficient conditions for metric regularity of implicit multifunctions

We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation.

Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Introducing probabilistic constraints leads in general to nonconvex, nonsmooth or even disconti- nuous optimization models. In this paper, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to derive bounds for the deviation of solution sets when the probability measure is replaced by empirical estimates.