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Optimality conditions via exact penalty functions

Vsevolod Ivanov Ivanov

Abstract

In this paper, we obtain optimality conditions for the problem with inequality, equality and closed set constraints in terms of the lower Hadamard derivative. The results are obtained applying exact penalty functions.

Optimality conditions via exact penalty functions

Abstract

In this paper, we obtain optimality conditions for the problem with inequality, equality and closed set constraints in terms of the lower Hadamard derivative. The results are obtained applying exact penalty functions.
Paper Structure (4 sections, 6 theorems, 58 equations)

This paper contains 4 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Optimality and exact penalty function
  3. Optimality conditions with non-strict inequalities
  4. Optimality conditions with strict inequalities

Key Result

Theorem 2.1

Let $\bar{x}$ be a local minimizer of the Problem (P), the set $X$ be closed, the function $f$ be Lipschitz with a constant $L$ on $N_\delta(\bar{x})\cap X$ for some $\delta>0$, the functions $g_i$, $i=1,2,...,m$ be lower semicontinuous, and the functions $h_j$, $j=1,2,...,q$ be Hadamard differentia Then there exists an integer $s$ such that $\bar{x}$ is a local minimizer of $F(x,\gamma)$ on $G$ f

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.1
  • ...and 3 more