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A parameter-free approach for solving SOS-convex semi-algebraic fractional programs

Chengmiao Yang, Liguo Jiao, Jae Hyoung Lee

Abstract

In this paper, we study a class of nonsmooth fractional programs {\rm (FP, for short)} with SOS-convex semi-algebraic functions. Under suitable assumptions, we derive a strong duality result between the problem (FP) and its semidefinite programming (SDP) relaxations. Remarkably, we extract an optimal solution of the problem (FP) by solving one and only one associated SDP problem. Numerical examples are also given.

A parameter-free approach for solving SOS-convex semi-algebraic fractional programs

Abstract

In this paper, we study a class of nonsmooth fractional programs {\rm (FP, for short)} with SOS-convex semi-algebraic functions. Under suitable assumptions, we derive a strong duality result between the problem (FP) and its semidefinite programming (SDP) relaxations. Remarkably, we extract an optimal solution of the problem (FP) by solving one and only one associated SDP problem. Numerical examples are also given.
Paper Structure (9 sections, 9 theorems, 86 equations)

This paper contains 9 sections, 9 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Convex analysis
  4. Real polynomials
  5. Main Results
  6. Duality for the fractional programs with SOS-convex semi-algebraic functions
  7. Extracting optimal solutions
  8. Illustrative examples
  9. Conclusions

Key Result

Lemma 2.1

Jeyakumar2006 Let $f\colon\mathbb{R}^{n} \to \mathbb{R}\cup \{+\infty\}$ and $g\colon \mathbb{R}^{n} \to \mathbb{R}\cup \{+\infty\}$ be proper l.s.c. convex functions. If ${\rm dom\,}f\cap {\rm dom\,}g\neq\emptyset,$ then Moreover$,$ if one of the functions $f$ and $g$ is continuous$,$ then

Theorems & Definitions (19)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.1
  • Definition 2.1
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.2: SOS-convex semi-algebraic functions
  • Remark 3.1
  • Lemma 3.1
  • ...and 9 more