Methods for constrained optimization of expensive mixed-integer multi-objective problems, with application to an internal combustion engine design problem

Engineering design optimization problems increasingly require computationally expensive high-fidelity simulation models to evaluate candidate designs. The evaluation budget may be small, limiting the effectiveness of conventional multi-objective evolutionary algorithms. Bayesian optimization algorithms (BOAs) are an alternative approach for expensive problems but are underdeveloped in terms of support for constraints and non-continuous design variables—both of which are prevalent features of real-world design problems. This study investigates two constraint handling strategies for BOAs and introduces the first BOA for mixed-integer problems, intended for use on a real-world engine design problem. The new BOAs are empirically compared to their closest competitor for this problem—the multi-objective evolutionary algorithm NSGA-II, itself equipped with constraint handling and mixed-integer components. Performance is also analysed on two benchmark problems which have similar features to the engine design problem, but are computationally cheaper to evaluate. The BOAs offer statistically significant convergence improvements of between 5.9% and 31.9% over NSGA-II across the problems on a budget of 500 design evaluations. Of the two constraint handling methods, constrained expected improvement offers better convergence than the penalty function approach. For the engine problem, the BOAs identify improved feasible designs offering 36.4% reductions in nitrogen oxide emissions and 2.0% reductions in fuel consumption when compared to a notional baseline design. The use of constrained mixed-integer BOAs is recommended for expensive engineering design optimization problems.