Characterizations of Pseudoconvex Functions

TL;DR

This paper characterizes pseudoconvex functions, including lower semicontinuous ones, by their domain partition into monotone and constant segments, extending classical results and providing new insights into their structure and applications.

Contribution

It extends the characterization of pseudoconvex functions to lower semicontinuous functions using the lower Dini derivative, with a domain partition approach.

Findings

Pseudoconvex functions can be characterized by domain partitions into monotone and constant segments.

Lower semicontinuous pseudoconvex functions have a similar structure with stationary points as global minimizers.

The paper provides applications of these characterizations in optimization and analysis.

Abstract

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if there exist some numbers $\alpha$ and $\beta$ such that $a\le\alpha\le\beta\le b$ and the function is strictly monotone decreasing on $[a,\alpha]$, it is constant on $[\alpha,\beta]$, the function is strictly monotone increasing on $[\beta,b]$ and there is no stationary points outside $[\alpha,\beta]$. This property is very simple. In this paper, we show that a similar result holds for lower semicontinuous functions, which are pseudoconvex with respect to the lower Dini derivative. We prove that a function, defined on some interval, is pseudoconvex if and only if its domain can be split into three parts such that the function is strictly monotone decreasing in the first part, constant in the second one, strictly monotone increasing in the third part, and every stationary point is a global minimizer. Each one or two of these parts may be empty or degenerate into a single point. Some applications are derived.