ACTA MATHEMATICA VIETNAMICA 125 Volume 34, Number 1, 2009, pp. 125–155 QUALIFICATION AND OPTIMALITY CONDITIONS FOR DC PROGRAMS WITH INFINITE CONSTRAINTS N. DINH, B. S. MORDUKHOVICH AND T. T. A. NGHIA Dedicated to Nguyen Van Hien on the occasion of his sixty-fifth birthday Abstract. The paper is devoted to the study of a new class of optimiza- tion problems with objectives given as differences of convex (DC) functions and constraints described by infinitely many convex inequalities. We consider such problems in the general framework of locally convex topological vector spaces, although the major results obtained in the paper are new even in finite dimensions when the problems under consideration reduce to DC semi-infinite programs. The main attention is paid to deriving qualified necessary optimal- ity conditions as well as necessary and sufficient optimality conditions for DC infinite and semi-infinite programs and to establishing relations between vari- ous qualification conditions. The results obtained are applied to and specified for particular classes of DC programs involving polyhedral convex functions in DC objectives, programs with cone constraints as well as those with positive semi-definite constraints described via the L¨owner partial order. 1. Introduction This paper deals with a new class of DC infinite programs given in the form:  minimize ϑ(x) - θ(x) subject to ϑ t (x) ≤ 0, t ∈ T, and x ∈ Θ, (1.1) where ϑ : X → R := (-∞, ∞] and ϑ t : X → R as t ∈ T are proper, convex, lower semicontinuous (l.s.c.) functions with values in the extended real line R, where θ : X → R is also a proper, l.s.c., convex function while real-valued, and where Θ ⊂ X is a closed and convex subset of a locally convex Hausdorff topological vector space. The above assumptions are standing throughout the whole paper. An important feature of problem (1.1) is that the index set T is arbitrary, i.e., Received June 2, 2008; in revised form November 12, 2008. 2000 Mathematics Subject Classification. 90C30, 49J52, 49J53. Key words and phrases. Convex and variational analysis, Topological vector spaces, Gener- alized differentiation, Differences of convex functions, Semi-infinite and infinite programming, Cone constraints, Semi-definite constraints, Constraint qualifications, Optimality conditions. Research of this author was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846.