Handling Uncertainty in Indicator-Based Multiobjective Optimization

Real-world optimization problems are often sub-ject to uncertainties caused by, e.g., missing information in the problem domain or stochastic models. These uncertainties can take different forms in terms of distribution, bounds, and cen-tral tendency. In the multiobjective context, some approaches have been proposed to take uncertainties into account within the optimization process. Most of them are based on a stochas-tic extension of Pareto dominance that is combined with stan-dard, non-stochastic diversity preservation mechanisms. Fur-thermore, it is often assumed that the shape of the underly-ing probability distribution is known and that for each solu-tion there is a ’true ’ objective value per dimension which is dis-turbed by noise. In this paper, we consider a slightly different scenario where the optimization goal is specified in terms of a quality indicator—a real-valued function that induces a total preorder on the set of Pareto set approximations. We propose a general indicator-model that can handle any type of distribution repre-senting the uncertainty, allows different distributions for differ-ent solutions, and does not assume a ’true ’ objective vector per solution, but in general regards a solution to be inherently as-sociated with an unknown probability distribution in the objec-tive space. To this end, several variants of an evolutionary algo-rithm for a specific quality indicator, namely the ǫ-indicator, are suggested and empirically investigated. The comparison to ex-isting techniques such as averaging or probabilistic dominance ranking indicates that the proposed approach is especially use-ful for high-dimensional objective spaces. Moreover, we intro-duce a general methodology to visualize and analyze Pareto set approximations in the presence of uncertainty which extends the concept of attainment functions. I.