Seminario "Set Oriented Methods for the Numerical Treatment of Multi-Objective Optimization Problems"
In a variety of engineering applications---as well as in other fields---one is faced with the problem that several objectives have to be optimized concurrently leading to a multi-objective optimization problem (MOP).
As a general example, two common goals in product design are certainly to maximize the quality of the product and to minimize its cost. Since these two goals are typically contradicting, it comes as no surprise that the solution set---the so-called Pareto set---of a MOP does in general not consist of one single solution but rather of an entire set of solutions.
Typically, the Pareto set forms a (k-1)-dimensional object, where k is the number of objectives involved in the problem.
In this talk, we present several set oriented methods which aim for the approximation of the Pareto set: Subdivision Techniques which are of global nature but are restricted to moderated dimensions n of the parameter space, and Recovering Techniques which are of local nature but allow for an efficient treatment of higher dimensional models (n >> 1000).
Further on, we present a local search technique, Directed Search, which is designed for a hybridization with set oriented methods such as the ones mentioned above or heuristics like evolutionary strategies. This method allows to steer the search both toward and along the Pareto set and is applicable to higher dimensional models. In addition, the Directed Search method can be used if gradient information is at hand or not. In the latter case, the information given by the neighboring individuals of a population are used to calculate the search direction.
Finally, we conclude the talk with the presentation of numerical results coming from academic examples as well as from real world problems.