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

Within the current literature on multiobjective 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...

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Bibliographic Details
Published in:IEEE transactions on evolutionary computation Vol. 29; no. 3; pp. 661 - 675
Main Authors: Shir, Ofer M., Emmerich, Michael
Format: Journal Article
Language:English
Published: IEEE 01.06.2025
Subjects:
Box-constraints
Computational modeling
convex quadratic models
CPLEX
diversity maximization approach (DMA)
Evolutionary computation
Linear programming
Mathematical models
mathematical programming (MP)
Metaheuristics
MIQP
nonscalarizing
Pareto optimization
population-based meta-heuristics
S-metric selection EMOA (SMS-EMOA)
set-oriented Pareto optimization
Systematics
undecidability
ISSN:1089-778X, 1941-0026
Online Access:Get full text
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Summary:Within the current literature on multiobjective 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 article 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 multiobjective 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 multiobjective 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.
ISSN:1089-778X
1941-0026
DOI:10.1109/TEVC.2024.3374519