Handling Uncertainty in Indicator-Based Multiobjective Optimization

Real-world optimization problems are often subject to uncertainties caused by, e.g., missing information in the prob- lem domain or stochastic models. These uncertainties can take different forms in terms of distribution, bounds, and central tendency. In the multiobjective context, some approaches have been proposed to take uncertainties into account within the opti- mization process. Most of them are based on a stochastic exten- sion of Pareto dominance that is combined with standard, non- stochastic diversity preservation mechanisms. Furthermore, it is often assumed that the shape of the underlying probability distribution is known and that for each solution there is a 'true' objective value per dimension which is disturbed 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 representing the uncertainty, allows different distributions for different so- lutions, and does not assume a 'true' objective vector per solu- tion, but in general regards a solution to be inherently associ- ated with an unknown probability distribution in the objective space. To this end, several variants of an evolutionary algo- rithm for a specific quality indicator, namely the � -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 useful for high-dimensional objective spaces. Moreover, we introduce a general methodology to visualize and analyze Pareto set ap- proximations in the presence of uncertainty which extends the concept of attainment functions.