Continuation Methods for Multi- and Many Objective Optimization Problems
In many applications the problem arises that several objectives have to be optimized leading to a multi-objective optimization problem (MOP). One important haracteristic of a MOP is that its solution set, the so-called Pareto set, is typically not given by a single solution as for ‘classical’ scalar optimization problems. Instead it forms under certain mild assumptions on the model a (k-1)-dimensional object, where k is the number of objectives involved in the problem. After the huge success of the consideration of MOPs in many applications e.g. coming from
engineering or finance in the recent past, and since the decision making process is getting more and more complex in many applications there is a recent trend to consider more objectives. Problems of that kind with four and more objectives are called many objective optimization problems (MaOPs). The numerical treatment of such problems differs from the treatment of MOPs (i.e., k<4) as a finite size representation of the entire Pareto set can not be computed any more with sufficient approximation quality. On the other hand, the computation of single solutions (e.g, via scalarization methods) very unlikely yields the desired 'optimal' trade off solution as the solution of such a problem depends on several factors. In this talk, we present the Pareto Explorer, a global/local exploration tool for the effective treatment of MaOPs. The Pareto Explorer consists of 2 stages: first, a solution is selected via a global searcher such as an volutionary algorithm. In the next step, the search is refined (locally) by steering the search along the solution set into a certain direction given by the decision maker, where this direction can be specified either in decision or in objective space. For this guided search we utilize the recently proposed multi-objective continuation method Pareto Tracer that possesses these steering features. We demonstrate the strength of the novel approach on several examples, including a 14 objective MaOP that arises in the design of a laundry system. Finally, we give some hints of how the ideas of the continuation method can be extended for the design of specialized evolutionary algorithms that aim to perform such exploration steps.