SDP Duals without Duality Gaps for a Class of Convex Minimax Programs

TL;DR

This paper introduces a new semi-definite programming dual for convex minimax problems involving SOS-convex polynomials, establishing strong duality and no duality gap under certain conditions, with applications to robust and fractional programming.

Contribution

It presents a novel dual formulation for convex minimax problems using SOS-convex polynomials, ensuring no duality gap and enabling practical verification via semi-definite programming.

Findings

Established sum of squares polynomial representations for non-negativity.

Proved strong duality results for SOS-convex minimax problems.

Applied results to robust and fractional programming scenarios.

Abstract

In this paper we introduce a new dual program, which is representable as a semi-definite linear programming problem, for a primal convex minimax programming model problem and show that there is no duality gap between the primal and the dual whenever the functions involved are SOS-convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust SOS-convex programming problems under data uncertainty and to minimax fractional programming problems with SOS-convex polynomials. We obtain these results by first establishing sum of squares polynomial representations of non-negativity of a convex max function over a system of SOS-convex constraints. The new class of SOS-convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The SOS-convexity of polynomials can numerically be checked by solving semi-definite programming problems whereas numerically verifying convexity of polynomials is generally very hard.