Impacts of constraints and constraint handling strategies for multi-objective mechanical design problems

Multi-objective optimization tools are becoming increasingly popular in mechanical engineering and allow decision-makers to better understand the inevitable trade-offs. Mechanical design problems can however combine properties that make the use of optimization more complex: (i) expensive cost functions; (ii) discrete or step-like behavior of the cost functions; and (iii) non-linear constraints. The latter in particular has a great impact on the convergence and the diversity of the obtained Pareto front. In this paper, we present five bi-objective mechanical design optimization problems with various levels of constraint complexity. They are used to rigorously benchmark two common constraint handling strategies (constrained-dominance and penalty function). The results suggest that both strategies have similar performance, and that as constraints become more intricate, convergence to the best-known Pareto front is not guaranteed. Indeed, analyzing the evolution of the hypervolume along generations reveals that the optimizer can get trapped in local optima. A detailed analysis of the obtained Pareto fronts for the proposed problems allows us to qualify the effects of the different constraints.