Towards identification of solutions of interest for multi-objective problems considering both objective and variable space information

Abstract

In multi/many-objective optimization, a decision maker (DM) may often be interested in examining only a small set of solutions instead of the entire Pareto optimal front (PF). Such solutions are referred to as solutions of interest (SOI) in some recent studies. A number of methods have been proposed to identify SOIs in an offline or online setting using measures based on reflex angle, bend angle, expected marginal utility, etc. However, these measures only account for the desirable trade-offs in the objective space. On the other hand, the variable space information is often critical in practical scenarios as it relates directly to the implemented design. For example, a DM may additionally require that the obtained solutions are robust, i.e., insensitive to variable perturbations, or look significantly different in the variable space, thereby offering multiple equivalent designs to achieve similar performance. These require formulation of new measures and search strategies that simultaneously consider both objective and variable spaces while identifying SOIs. In this paper, we develop an approach that can identify a given number of SOIs for DM’s consideration for three different scenarios: (a) purely based on objective space, (b) simultaneous consideration of objectives and robustness, and (c) simultaneous considerations of objectives and equivalent designs. Towards this end, we first define the relevant quantitative measures and illustrate their use for offline selection for a few 2–3 objective test problems. Thereafter, we design an online algorithm that can identify the SOIs and bias the search towards the SOIs based on the scenarios listed above. Lastly, we also present results on two practical examples: a 2-objective welded beam and a 5-objective wind-turbine design problem.