Robust Optimization of Unconstrained Binary Quadratic Problems

TL;DR

This paper develops a method to find robust solutions for unconstrained binary quadratic problems, accounting for data uncertainty and computational limitations, with practical applications and theoretical insights.

Contribution

It introduces a novel approach combining experimental design and surface response modeling to identify stable solutions under Q matrix perturbations, including theoretical robustness analysis.

Findings

Robust solutions can be efficiently identified using scenario-based experimental design.

Surface response equations enable quick estimation of bounds for Q variations.

The method demonstrates practical applicability in business decision-making contexts.

Abstract

In this paper we focus on the unconstrained binary quadratic optimization model, maximize x^t Qx, x binary, and consider the problem of identifying optimal solutions that are robust with respect to perturbations in the Q matrix.. We are motivated to find robust, or stable, solutions because of the uncertainty inherent in the big data origins of Q and limitations in computer numerical precision, particularly in a new class of quantum annealing computers. Experimental design techniques are used to generate a diverse subset of possible scenarios, from which robust solutions are identified. An illustrative example with practical application to business decision making is examined. The approach presented also generates a surface response equation which is used to estimate upper bounds in constant time for Q instantiations within the scenario extremes. In addition, a theoretical framework for the robustness of individual x_i variables is considered by examining the range of Q values over which the x_i are predetermined.