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The local minimality of differentiable functions

Tien-Son Pham

Abstract

In this paper we present necessary and sufficient conditions (in terms of Łojasiewicz inequalities) for the stability of local minimum points in smooth unconstrained optimization. In particular, we derive a sufficient condition for which the local minimum property of a given function is determined by its Taylor polynomial of a certain degree.

The local minimality of differentiable functions

Abstract

In this paper we present necessary and sufficient conditions (in terms of Łojasiewicz inequalities) for the stability of local minimum points in smooth unconstrained optimization. In particular, we derive a sufficient condition for which the local minimum property of a given function is determined by its Taylor polynomial of a certain degree.
Paper Structure (5 sections, 9 theorems, 61 equations)

This paper contains 5 sections, 9 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Subdifferentials
  5. The main result and its corollaries

Key Result

Lemma 2.3

For any point $\overline{x} \in \mathbb{R}^n,$ we have

Theorems & Definitions (21)

  • Example 1.1
  • Definition 2.1: subdifferentials
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4: Fermat's rule
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 11 more