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Generic continuity of the perturbed minima of certain parametric optimization problems

Hristina Topalova, Nadia Zlateva

Abstract

We show that in a quite general framework, the parameterized optimization problem can be so perturbed as to be generically well-posed. As an application, we provide a contribution to Stechkin theory.

Generic continuity of the perturbed minima of certain parametric optimization problems

Abstract

We show that in a quite general framework, the parameterized optimization problem can be so perturbed as to be generically well-posed. As an application, we provide a contribution to Stechkin theory.

Paper Structure

This paper contains 5 sections, 12 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Perturbation method
  3. Epi-continuity
  4. Proof of the main result
  5. On Stečkin theory

Key Result

Theorem 1.2

Let $(P,\mu)$ and $(X,d)$ be complete metric spaces. Let also $P$ be separable. Let $(f_p)_{p\in P}$ be a family of proper, lower semicontinuous and bounded below functions from $X$ to $\mathbb{R}\cup\{+\infty\}$ which are uniformly epi-continuous, that is, they satisfy conditions eq:cond-1 and eq:c Also, the functions are continuous on $P_g$.

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 2.1: Variational Principle
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:VP1']}
  • ...and 16 more