On Solving a Class of Fractional Semi-infinite Polynomial Programming Problems

TL;DR

This paper develops a new approach to solve fractional semi-infinite polynomial programming problems by reformulating them into conic optimization problems and applying semidefinite programming relaxations, enabling efficient solution extraction.

Contribution

It introduces a hierarchy of SDP relaxations for FSIPP problems and establishes conditions for strong duality and solution attainment, advancing optimization methods for these complex problems.

Findings

Hierarchy of SDP relaxations with asymptotic convergence

Reduction of specific FSIPP cases to single or finite SDP problems

Application to multi-objective FSIPP problems for efficient solutions

Abstract

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.