Year: 2025
Type: article
Abstract: This paper deals with the characterization, in terms of closedness of certain sets regarding other sets, of Farkas lemmas determining when the upperlevel set of a convex function $f$ f contains a set of the form $C\cap \mathbb{A}^{-1}\left ( D\right ) $ C ∩ A − 1 ( D ) , where $C$ C and $D$ D are convex sets (not necessarily cones) in locally convex spaces $X$ X (with topological dual $X^{\prime }$ X ′ ) and $Y$ Y , respectively, while $\mathbb{A}$ A is a continuous linear operator from $X$ X to $Y$ Y . More in detail, each of the mentioned characterizations of Farkas type lemmas consists in the closedness of certain subset of either one of the “primal” spaces $X\times Y\times \mathbb{R}$ X × Y × R and $Y\times \mathbb{R}$ Y × R , or of the “dual” space $X^{\prime }\times \mathbb{R}$ X ′ × R , regarding some singleton set of the corresponding space. Moreover, the paper also provides an existence theorem for the feasible set $C\cap \mathbb{A}^{-1}\left ( D\right ) $ C ∩ A − 1 ( D ) in terms of the closedness of certain subset of the dual space $X^{\prime }\times \mathbb{R}$ X ′ × R regarding the singleton set formed by the null element. These results are illustrated with significant applications to constrained convex minimization problems and to functional approximation by polynomials.
Authors: N. Dinh, M. A. Goberna, M. Volle
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4413411734