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Normal cones to sublevel sets of convex and quasi-convex supremum functions

Stephanie Caro, Rafael Correa, Abderrahim Hantoute

Abstract

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate subdifferential of the individual data functions at the nominal point. In contrast, the quasi-convex framework requires the use of the Fréchet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasi-convex optimization problems.

Normal cones to sublevel sets of convex and quasi-convex supremum functions

Abstract

We provide sharp and explicit characterizations of the normal cone to sublevel sets of suprema of arbitrary functions, expressed exclusively in terms of subdifferentials of the data functions. In the convex case, the resulting formulas involve the approximate subdifferential of the individual data functions at the nominal point. In contrast, the quasi-convex framework requires the use of the Fréchet subdifferential of these data functions but evaluated at nearby points. These results are applied to derive optimality conditions for infinite convex and quasi-convex optimization problems.
Paper Structure (5 sections, 14 theorems, 100 equations)

This paper contains 5 sections, 14 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Normal cone to sublevel sets in the convex case
  4. Normal cone to sublevel sets in the quasi-convex case
  5. Conclusion

Key Result

Lemma 1

Let $f:X\rightarrow\overline{\mathbb{R}}$ be a quasi-convex function such that $[f\leq0]\neq\emptyset.$ If either $X=\mathbb{R}^{n}$ and $\operatorname*{ri}(\operatorname*{dom}f)\cap\lbrack f\leq0]\neq\emptyset$, or $f$ is continuous at some point in $[f\leq0],$ then

Theorems & Definitions (15)

  • Lemma 1
  • Corollary 2
  • Proposition 3
  • Theorem 4
  • Remark 1
  • Corollary 5
  • Corollary 6
  • Theorem 7
  • Corollary 8
  • Lemma 9
  • ...and 5 more