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Primal and Dual Characterizations for Farkas type Lemmas in Terms of Closedness Criteria

Nguyen Dinh, Miguel A. Goberna, Michel Volle

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 given convex function contains the intersection, say F, of a convex set of a locally convex space X with the inverse image by a continuous linear operator from X to another locally convex space Y of certain convex subset of 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 XxYxR and YxR, or of the "dual" space X'xR, regarding some singleton set of the corresponding space. Moreover, the paper also provides an existence theorem for the feasible set F in terms of the closedness of certain subset of the dual space X'xR regarding the singleton set formed by the null element. These results are illustrated with significant applications.

Primal and Dual Characterizations for Farkas type Lemmas in Terms of Closedness Criteria

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 given convex function contains the intersection, say F, of a convex set of a locally convex space X with the inverse image by a continuous linear operator from X to another locally convex space Y of certain convex subset of 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 XxYxR and YxR, or of the "dual" space X'xR, regarding some singleton set of the corresponding space. Moreover, the paper also provides an existence theorem for the feasible set F in terms of the closedness of certain subset of the dual space X'xR regarding the singleton set formed by the null element. These results are illustrated with significant applications.

Paper Structure

This paper contains 8 sections, 25 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Primal closedness characterizations
  3. Dual closedness characterization
  4. Characterization of a convex Farkas lemma for linear infinite systems
  5. Characterization of a concave Farkas lemma
  6. Applications
  7. Application to constrained convex minimization problems
  8. Functional approximation by polynomials

Key Result

Theorem 1

Let $f\in \overline{\mathbb{R}}^{X}$ be proper convex, $C$ (resp. $D$) be a nonempty convex subset of $X$ (resp. $Y$). Then, one has $(\mathcal{A})\Longrightarrow (\mathcal{B})$ (hence, $(\mathcal{A})\Longleftrightarrow (\mathcal{B})$) if and only if $\mathbb{R}_{+}\mathcal{F}$ is closed regarding $

Theorems & Definitions (28)

  • Theorem 1: 1st characterization of Farkas' Lemma
  • Theorem 2: 2nd characterization of Farkas' Lemma
  • Corollary 3
  • Corollary 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Proposition 9: Existence theorem
  • Corollary 10
  • ...and 18 more