Handling non-convex or expensive objectives: algorithms for multiobjective optimization without scalarization
In multiobjective optimization, one considers optimization problems with several competing objective functions. For instance, in engineering problems a design often has to be stable and light weighted at the same time. A classical approach to such optimization problems is to formulate suitable parameter-dependent single-objective replacement problems, called scalarizations, such as considering a weighted sum of the objective functions. Then the parameters are varied and the scalarized problems are solved iteratively. This talk is about numerical methods which do avoid such scalarizations to improve the performance of the procedure.
The first algorithm which we present in this talk is for multiobjective optimization problems with non-convex objective functions. Then, methods of global optimization are necessary to solve the replacement problems. Instead of this detour via scalarization, we presents a direct deterministic method for finding a representation of all globally optimal solutions. This branch-and-bound method is based on a subdivision of the feasible set and the consideration of convex underestimators of the objective functions for the determination of lower bounds.
The second algorithm is for so called heterogeneous multiobjective optimization problems, i.e. problems, where one of the functions is assumed to be an expensive black-box function while the other objectives are given analytically. The proposed method uses the basic trust region approach by restricting the computations in every iteration to a local area. The objective functions are replaced by suitable models which reflect the heterogeneity of the functions. Convergence results as well as numerical experiments are presented.