Sensitivity analysis for mixed binary quadratic programming

TL;DR

This paper investigates the complexity of sensitivity analysis in mixed binary quadratic programs, introduces dual bounding techniques via copositive reformulations, and provides computational insights into their effectiveness.

Contribution

It develops dual bounds for MBQPs using copositive programming, analyzes duality conditions, and proposes methods for selecting effective dual solutions.

Findings

Sensitivity analysis is NP-hard for MBQPs.

Dual bounds can be obtained via copositive reformulations.

Computational results show the potential of the proposed methods.

Abstract

We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.