An SDP method for Fractional Semi-infinite Programming Problems with SOS-convex polynomials

TL;DR

This paper introduces a new SDP relaxation method for fractional semi-infinite polynomial programming problems involving SOS-convex polynomials, enabling computation of convergent bounds and approximate solutions.

Contribution

The paper develops a novel SDP relaxation approach using Lasserre's measure-based representation for fractional semi-infinite SOS-convex polynomial problems, improving bounds and solution extraction.

Findings

Convergent upper bounds for the optimal value are obtained.

A procedure for constructing convergent outer approximations is proposed.

The convergence rate of bounds and approximations is analyzed.

Abstract

In this paper, we study a class of fractional semi-infinite polynomial programming problems involving s.o.s-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre's measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many s.o.s-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations. The convergence rate of the lower bounds and outer approximations are also discussed.