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Charnes--Cooper transformation and fractional optimization with SOS-convex polynomials

Chengmiao Yang, Liguo Jiao, Jae Hyoung Lee

TL;DR

Fractional programs with objective $f(x)/(-g(x))$ on a convex feasible set $K$ are addressed when $f,g$ and the constraints are SOS-convex polynomials. The authors leverage the Charnes--Cooper transformation to a perspective formulation, enabling a parameter-free reformulation and SDP/SOS relaxations that preserve optimal value. They establish existence of an optimal solution (Theorem 4.1), strong duality under Slater conditions (Theorem 4.2), and a solution-extraction mechanism (Theorem 4.3), validated by an illustrative example. This work provides an exact, tractable approach for a class of fractional programs, offering an alternative to iterative methods such as Dinkelbach and a foundation for extending to broader polynomial classes.

Abstract

This paper proposes a parameter-free scheme that is based on the Charnes--Cooper transformation for solving a class of fractional programs with SOS-convex polynomials. Under certain conditions, we establish theorems of solution existence,strong duality and solution extraction. An illustrative example is designed to show the obtained results.

Charnes--Cooper transformation and fractional optimization with SOS-convex polynomials

TL;DR

Fractional programs with objective on a convex feasible set are addressed when and the constraints are SOS-convex polynomials. The authors leverage the Charnes--Cooper transformation to a perspective formulation, enabling a parameter-free reformulation and SDP/SOS relaxations that preserve optimal value. They establish existence of an optimal solution (Theorem 4.1), strong duality under Slater conditions (Theorem 4.2), and a solution-extraction mechanism (Theorem 4.3), validated by an illustrative example. This work provides an exact, tractable approach for a class of fractional programs, offering an alternative to iterative methods such as Dinkelbach and a foundation for extending to broader polynomial classes.

Abstract

This paper proposes a parameter-free scheme that is based on the Charnes--Cooper transformation for solving a class of fractional programs with SOS-convex polynomials. Under certain conditions, we establish theorems of solution existence,strong duality and solution extraction. An illustrative example is designed to show the obtained results.
Paper Structure (8 sections, 11 theorems, 70 equations)

This paper contains 8 sections, 11 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation
  4. Real Polynomials
  5. The Charnes--Cooper Transformation
  6. Solving fractional optimization by CCT method
  7. A simple illustrative example
  8. Conclusions

Key Result

Lemma 2.1

Suppose $C \subset \mathbb{R}^{n}$ and $D \subset \mathbb{R}^{n}$ are nonempty disjoint convex sets$,$ i.e.$,$$C \cap D = \emptyset.$ Then there exist $0 \neq a \in \mathbb{R}^n$ and $b\in \mathbb{R}$ such that $a^{T} x \leq b$ for all $x \in C$ and $a^{T} x \geq b$ for all $x \in D.$

Theorems & Definitions (20)

  • Lemma 2.1: Separating hyperplane theorem Boyd2004
  • Proposition 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Remark 3.1: CCT and perspective functions
  • Lemma 3.2
  • proof
  • ...and 10 more