  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.
