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Zero duality gaps in infinite-dimensional programming. (English) Zbl 0687.90077

We study the following infinite-dimensional programming problem

$\left(P\right)\phantom{\rule{1.em}{0ex}}\mu :=inf{f}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}subject\phantom{\rule{1.em}{0ex}}to\phantom{\rule{1.em}{0ex}}x\in C,\phantom{\rule{1.em}{0ex}}{f}_{i}\left(x\right)\le ,\phantom{\rule{1.em}{0ex}}i\in I,$

where I is an index set with possibly infinite cardinality and C is an infinite-dimensional set. Zero duality gap results are presented under suitable regularity hypotheses for convex- like (nonconvex) and convex infinitely constrained programs (P). Various properties of the value function of the convex-like program and its connections to the regularity hypotheses are studied. Relationships between the zero duality gap property, semicontinuity, and $ϵ$- subdifferentiability of the value function are examined. In particular, a characterization for the value function without convexity is given, using the $ϵ$-subdifferential of the value function.

Reviewer: V.Jeyakumar
##### MSC:
 90C30 Nonlinear programming 90C34 Semi-infinite programming 49N15 Duality theory (optimization) 90C48 Programming in abstract spaces
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