ISSN:
1436-4646
Keywords:
Constraint qualifications
;
characterization of optimum
;
duality
;
convex programming
;
constrained best approximation
;
partially finite programs
;
infinite programs
;
closure of sums of cones
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract In this paper we study constraint qualifications and duality results for infinite convex programs (P)μ = inf{f(x): g(x) ∈ − S, x ∈ C}, whereg = (g 1,g 2) andS = S 1 ×S 2,S i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg i are, respectively, convex andS i -convex,i = 1, 2. In particular, we consider the special case whenS 2 is in afinite dimensional space,g 2 is affine andS 2 is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g 2 is affine andS = S 2 is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong 2 and the finite dimensionality assumption onS 2, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01581074
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