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Strongly quasiconvex functions: what we know (so far)

Sorin-Mihai Grad, Felipe Lara, Raúl T. Marcavillaca

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.

Strongly quasiconvex functions: what we know (so far)

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.

Paper Structure

This paper contains 16 sections, 49 theorems, 52 equations, 12 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strongly quasiconvex functions
  4. Examples of strongly quasiconvex functions
  5. Properties of strongly quasiconvex functions
  6. Locally strongly quasiconvex functions and other similar notions
  7. Algorithms for minimizing strongly quasiconvex functions
  8. Algorithms for strongly quasiconvex functions
  9. Dynamical systems involving strongly quasiconvex functions
  10. Equilibrium problems involving strongly quasiconvex functions
  11. Future developments
  12. Availability of supporting data
  13. Competing interests
  14. Funding
  15. Authors' contributions
  16. ...and 1 more sections

Key Result

Proposition 9

(cf. ILM) Let the set $\emptyset \neq K \subseteq {\mathbb{R}}^{n}$, and functions $h: \mathbb{R}^n \rightarrow \overline{{\mathbb{R}}}$ and $g: {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}$ such that $K\cap \mathop{\mathrm{dom}}\nolimits h \neq \emptyset$ and $g(K)\subseteq ]0, +\infty[$. When $h$ is then $h/g$ is strongly quasiconvex with modulus $\gamma^{\prime}= {\gamma}/{M} > 0$ on $K$.

Theorems & Definitions (112)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Proposition 9
  • Corollary 10
  • ...and 102 more