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On Characterizations of $σ$-Quasiconvexity

Nguyen Xuan Duy Bao, Nguyen Mau Nam

TL;DR

The paper addresses the problem of characterizing differentiable (strong) quasiconvexity via first-order gradient conditions. It provides corrected, self-contained proofs for classical results on quasiconvexity and establishes full gradient-based equivalences for $\sigma$-quasiconvexity, resolving a gap left in recent literature and reconstructing a characterization first stated in the 1970s. By proving (a) => (b) => (c) for the $\sigma$-quasiconvex case and highlighting that (c) => (a) remains open, it sharpens the first-order theory of strong quasiconvexity and clarifies the limits of current gradient-based criteria. The results yield a concise framework linking quasiconvexity, generalized monotonicity of gradients, and first-order conditions, with implications for variational analysis and optimization theory.

Abstract

We revisit classical gradient characterizations of quasiconvexity and provide corrected proofs that close gaps in earlier arguments. For the differentiable case of $σ$-quasiconvexity, we establish the full equivalence between several first-order conditions, resolving a remaining implication left open in the recent literature. Our approach yields a concise, self-contained proof of a classical characterization originally stated in the 1970s and sharpens the first-order theory for strong quasiconvexity.

On Characterizations of $σ$-Quasiconvexity

TL;DR

The paper addresses the problem of characterizing differentiable (strong) quasiconvexity via first-order gradient conditions. It provides corrected, self-contained proofs for classical results on quasiconvexity and establishes full gradient-based equivalences for -quasiconvexity, resolving a gap left in recent literature and reconstructing a characterization first stated in the 1970s. By proving (a) => (b) => (c) for the -quasiconvex case and highlighting that (c) => (a) remains open, it sharpens the first-order theory of strong quasiconvexity and clarifies the limits of current gradient-based criteria. The results yield a concise framework linking quasiconvexity, generalized monotonicity of gradients, and first-order conditions, with implications for variational analysis and optimization theory.

Abstract

We revisit classical gradient characterizations of quasiconvexity and provide corrected proofs that close gaps in earlier arguments. For the differentiable case of -quasiconvexity, we establish the full equivalence between several first-order conditions, resolving a remaining implication left open in the recent literature. Our approach yields a concise, self-contained proof of a classical characterization originally stated in the 1970s and sharpens the first-order theory for strong quasiconvexity.

Paper Structure

This paper contains 3 sections, 3 theorems, 26 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Differentiable characterizations for quasiconvexity
  3. Differentiable characterizations for strongly quasiconvexity

Key Result

theorem 1

Let $f\colon X \to \mathbb{R}$ be a Fréchet continuously differentiable function. Then the following properties are equivalent:

Theorems & Definitions (8)

  • definition 1: $\sigma$-strong quasiconvexity
  • theorem 1
  • proof
  • lemma 1
  • proof
  • remark 1
  • theorem 2
  • proof