Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs

Abstract

Abstract. The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical MangasarianFromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.

Existence theorem and optimality conditions for a class of convex semi-infinite problems with noncompact index sets

“…In few papers dedicated to study of optimality for SIP problems without the assumption of the compactness of the index set (see, for example, [3,5,13], and references therein), it is supposed that the Farkas-Minkowski Constraint Qualification (CQ) is satisfied. In this paper, we do not have such an assumption for our problem (7) (see Theorem below).…”

“…In this case the optimality conditions of Theorem 2 are the same as the optimality conditions formulated in [3,5,13] under the assumption that problem (7) satisfies the Farkas-Minkowski CQ.…”

Convex SIP Problems with Finitely Representable Compact Index Sets: Immobile Indices and the Properties of the Auxiliary NLP Problem

“…In the recent papers [3,21], et al, the advanced techniques of variational analysis are applied to broad classes of infinite programming problems, including nonsmooth and nonconvex problems with arbitrary index sets. Under certain differentiability assumptions and the closedness CQ, new verifiable necessary optimality conditions are derived.…”

“…Therefore one can formulate the optimality conditions for SIP problem in terms of such conditions for the auxiliary NLP problem. The discovered properties of the NLP problem permit one to use the most efficient optimality conditions that should give new optimality conditions for SIP that differ from the known ones (see, e.g., [1,8,11,12,21,23,25]). Notice here that the assumptions we make for the convex SIP problem, are less restrictive than those that are usually made in the literature.…”

“…Without the additional conditions, the known optimality conditions may not hold true. Different CQs have been proposed in the literature for different classes of SIP problems (see [1,12,21] …”