Strongly quasiconvex functions: what we know (so far)

TL;DR

This survey comprehensively reviews the current state of research on strongly quasiconvex functions, including their properties, algorithms for minimization, and potential future research directions.

Contribution

It compiles and summarizes most existing results on strongly quasiconvex functions, highlighting algorithms and identifying gaps for further study.

Findings

Collection of key results on strongly quasiconvex functions

Algorithms for minimizing these functions

Suggestions for future research directions

Abstract

Introduced by Polyak in 1966, the class of strongly quasiconvex functions includes some interesting nonconvex members, like the square root of the Euclidean norm or ratios with a nonnegative strongly convex numerator and a concave and positive denominator. This survey collects the vast majority of the results involving strongly quasiconvex functions available in the literature at the moment, presenting, in particular, algorithms for minimizing such functions, and suggests some directions where additional investigations would be welcome.