Multiobjective variational quantum optimization for constrained problems: an application to Cash Management

TL;DR

This paper introduces MOVCO, a multi-objective quantum optimization method that effectively handles complex constraints in combinatorial problems, demonstrated on a financial Cash Management case, outperforming penalty-based approaches.

Contribution

The paper presents MOVCO, a novel multi-objective variational quantum algorithm that improves constraint satisfaction and solution quality in complex combinatorial optimization problems.

Findings

MOVCO significantly reduces solution cost compared to penalty methods.

MOVCO better avoids local minima that violate constraints.

Empirical results on Cash Management show improved solution feasibility.

Abstract

Combinatorial optimization problems are ubiquitous in industry. In addition to finding a solution with minimum cost, problems of high relevance involve a number of constraints that the solution must satisfy. Variational quantum algorithms have emerged as promising candidates for solving these problems in the noisy intermediate-scale quantum stage. However, the constraints are often complex enough to make their efficient mapping to quantum hardware difficult or even infeasible. An alternative standard approach is to transform the optimization problem to include these constraints as penalty terms, but this method involves additional hyperparameters and does not ensure that the constraints are satisfied due to the existence of local minima. In this paper, we introduce a new method for solving combinatorial optimization problems with challenging constraints using variational quantum algorithms. We propose the Multi-Objective Variational Constrained Optimizer (MOVCO) to classically update the variational parameters by a multiobjective optimization performed by a genetic algorithm. This optimization allows the algorithm to progressively sample only states within the in-constraints space, while optimizing the energy of these states. We test our proposal on a real-world problem with great relevance in finance: the Cash Management problem. We introduce a novel mathematical formulation for this problem, and compare the performance of MOVCO versus a penalty based optimization. Our empirical results show a significant improvement in terms of the cost of the achieved solutions, but especially in the avoidance of local minima that do not satisfy any of the mandatory constraints.