Multi-Objective Mixed-Integer Quadratic Models: A Study on Mathematical Programming and Evolutionary Computation

Within the current literature on multi-objective optimization, there is a scarcity of comparisons between equation-based white-box solvers to evolutionary black-box solvers. It is commonly held that when dealing with linear and quadratic models, equation-based deterministic solvers are generally the preferred choice. The present study aims at challenging this hypothesis, and we show that particularly in box-constrained mixed-integer (MI) problems it is worth employing evolutionary methods when the goal is to achieve a good approximation of a Pareto frontier. To do so, this paper compares a mathematical programming approach with an evolutionary method for set-oriented Pareto front approximation of bi-objective quadratic MI optimization problems. The focus is on convex quadratic under-constrained models wherein the decision variables are either tightly or loosely bounded by box-constraints. Through an empirical assessment of families of quadratic models across varying Hessian forms, variable ranges, and condition numbers, the study compares the performance of the CPLEX-based Diversity Maximization Approach to a state-of-the-art evolutionary multi-objective optimization meta-heuristic with MI mutation and crossover operators. We identify and explain strengths and weaknesses of both approaches when dealing with loosely bounded box-constraints, and prove a theorem regarding the potential undecidability of such multi-objective problems featuring unbounded integer decision variables. The empirical results systematically confirm that black-box and white-box solvers can be competitive, especially in the case of loose box-constraints.