NDLI: From the Farkas Lemma to the Hahn--Banach Theorem

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Dinh, N. Goberna, M. A. López, M. A. Mo, T. H.
Copyright Year	2014
Abstract	This paper provides new versions of the Farkas lemma characterizing those inequalities of the form $f(x)\geq 0$ which are consequences of a composite convex inequality $(S\circ g)(x)\leq 0$ on a closed convex subset of a given locally convex topological vector space $X$, where $f$ is a proper lower semicontinuous convex function defined on $X,$ $S$ is an extended sublinear function, and $g$ is a vector-valued $S$-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of $f$, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn--Banach--Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn--Banach theorem and the Mazur--Orlicz theorem for extended sublinear functions.
Starting Page	678
Ending Page	701
Page Count	24
File Format	PDF
ISSN	10526234
DOI	10.1137/120901805
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	2
Volume Number	24
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-04-08
Access Restriction	Subscribed
Subject Keyword	Hahn--Banach theorem Hahn--Banach--Lagrange theorem Mazur--Orlicz theorem Convex sets in topological linear spaces; Choquet theory Farkas lemma Functional inequalities, including subadditivity, convexity, etc. Theorems of Hahn-Banach type; extension and lifting of functionals and operators
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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