On the Behavior of the Mixed-Integer SMS-EMOA on Box-Constrained Quadratic Bi-Objective Models

Existing research lacks studies on how state-of-the-art evolutionary multi-objective algorithms behave when dealing with problems that involve mixed-variable types. To address this gap, we examine how the popular SMS-EMOA performs on a class of problems that involve both continuous and discrete variables. More particularly, we study the algorithmic behavior on a family of mixed-integer (MI) bi-objective optimization problems of partially discretized convex quadratic models. We are considering search by means of a MI Evolution Strategy (ES), and aim to investigate the evolutionary mechanisms as they operate subject to scenarios of box-constrained decision variables. We also account for a white-box approach to solve the models, and qualitatively mention the relative performance of SMS-EMOA with respect to it. We discuss the ES operation within the SMS-EMOA on the current MI case-study, with a particular focus on the step-size adaptation and the success-rate of offspring generation over time. It is evident that progress is made in the initial stage of the optimization, whereas the process tends to stagnate later on. Moreover, for problems whose decision variables are loosely bounded, the step-sizes exhibit effective self-adaptation. We conclude by summarizing the challenges and opportunities when treating MI problems by ES-driven heuristics.