New Farkas-type Constraint Qualifications in Convex Infinite Programming

This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly in…nite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily realvalued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas’lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint quali…cations and/or a certain closedness assumption. 1. Introduction This paper deals with optimization problems of the form (P) Minimize f(x) subject to ft(x) 0; t 2 T; x 2 C; where T is an arbitrary (possibly in…nite) index set, C is a non-empty closed convex subset of a locally convex Hausdor¤ topological vector space X, and f; ft : X ! R [ f+1g, t 2 T; are proper lower semicontinuous (l.s.c., in brief) convex functions. Throughout the paper we assume that the (convex) constraint system (1.1) := fft(x) 0; t 2 T ; x 2 Cg; is consistent, with solution set represented by A (A 6= ;). Date : 15/12/2005. N. DINH: Department of Mathematics, International University, Vietnam National University-HCM city, Linh Trung ward, Thu Duc district, Ho Chi Minh city, Vietnam. M.A. GOBERNA and M.A. LÓPEZ: Department of Statistics and Operations Research, University of Alicante, 03080 Alicante, Spain. T.Q. SON: Nha Trang College of Education, Nha Trang, Vietnam. This research was partially supported by MEC of Spain and FEDER of EU, Grant MTM2005-08572-C03-01, and by Project B.2005.23.68 of the MOET, Vietnam. 1 2 N. DINH, M.A. GOBERNA, M.A. LÓPEZ, AND T.Q. SON The system is called linear when ft(x) = at(x) bt; at 2 X (topological dual of X); bt 2 R; t 2 T; and C = X: Moreover, it is called in…nite (ordinary or …nite) if the dimension of X and the number of constraints (jT j) are in…nite (…nite, respectively). If exactly one of these numbers is …nite, then is called semi-in…nite (typically, T is in…nite and X = Rn). An optimization problem is called in…nite (…nite, semi-in…nite) when its constraint system is in…nite (…nite, semi-in…nite, respectively). The objective of the paper is to provide optimality conditions, duality theorems, and stability theorems for (P). To do that we introduce new Farkas-type constraint quali…cations and new versions of Farkas lemma. The classical Farkas lemma characterizes those linear inequalities which are consequences of a consistent ordinary linear inequality system (i.e., they are satis…ed by every solution of the system). Farkas-type results for convex systems (characterizing families of inequalities which are consequences of a consistent convex system ) are fundamental in convex optimization and in other …elds as game theory, set containment problems, etc. Since the literature on Farkas lemma, and its extensions, is very wide (see, e.g., the survey in [15]), we just mention here some works giving Farkas-type results for the kind of systems considered in the paper: [3], [11], [16], and [21] for semi-in…nite systems, [8], [14], [19], and [22] for in…nite systems, and [9], [17], and [18] for cone convex systems. The paper is organized as follows. Section 2 contains the necessary notations and recalls some basic results on convexity and convex systems. Section 3 extends to in…nite convex systems two constraint quali…cations (c.q., in brief) which play a crucial role in linear semi-in…nite programming, one of them (the so-called Farkas-Minkowski property, FM in brief) being of global nature whereas the other one is a local property (and so it is called locally Farkas-Minkowski, LFM in short). Section 4 provides new asymptotic and non-asymptotic versions of Farkas’ lemma characterizing those reverse-convex inequalities f (x) which are consequences of . The non-asymptotic Farkas’lemma requires the FM c.q. together with a certain closedness condition involving ft; t 2 T; and f (which holds whenever f is linear or it is continuous at some feasible point), and it provides a characterization of the containment of convex sets in reverse-convex sets. Under these two assumptions we obtain, in Section 5, a Karush-Kuhn-Tucker (KKT) optimality condition for (P), we show that the LFM c.q. holds everywhere if the constraint system is FM, and, what is more important, that the LFM c.q. is, in a certain sense, the weakest condition guarateeing that (P) satis…es the KKT condition at the optimal solutions. Finally, in Section 6, a strong duality theorem and an optimality condition for (P), in terms of saddle points of the associated Lagrange function, are established. The strong duality theorem allows us to show that the optimal value of (P) is stable (in di¤erent senses) relatively to small arbitrary perturbations of the right-hand side function (the null function). NEW FARKAS-TYPE CONSTRAINT QUALIFICATIONS 3 2. Preliminaries For a set D X, the closure of D will be denoted by clD and the convex cone generated by D [ f0g by coneD. The closure with respect to the weak -topology of a subset E of either the dual space X or the product space X R will be represented also by clE. We represent by R ) + the positive cone in R(T ), the so-called space of generalized …nite sequences = ( t)t2T such that t 2 R; for each t 2 T; and with only …nitely many t di¤erent from zero. The supporting set of 2 R(T ) is supp := ft 2 T j t 6= 0g: Observe that R(T ) is the topological dual of RT ; endowed with the product topology, and