A Trust-Region Algorithm for Heterogeneous Multiobjective Optimization

Abstract

This paper presents a new trust region method for multiobjective heterogeneous optimization problems. One of the objective functions is an expensive black-box function, for example given by a time-consuming simulation. For this function derivative information cannot be used and the computation of function values involves high computational effort. The other objective functions are given analytically and derivatives can easily be computed. The method uses the basic trust region approach by restricting the computations in every iteration to a local area and replacing the objective functions by suitable models. The search direction is generated in the image space by using local ideal points. It is proved that the presented algorithm converges to a Pareto critical point. Numerical results are presented and compared to another algorithm. Mathematics subject classifications (MSC2010): 90C29, 90C56, 90C30 * This work was funded by DFG under no. GRK 1567.

Representation of the Pareto front for heterogeneous multi-objective optimization

“…The following lemma gives a characterization of Pareto critical points and comes from multiobjective descent methods [4,9,10]. This characterization is a main tool in the convergence proof of MHT in [19].…”

“…. , m k q (x)) , it can be proven that it holds t k+ ∈ [−1, 0), see [19,Lem.3.3]. Whether x k+ is chosen as next iteration point or not is decided by the trial point acceptance test.…”

“…The trust region algorithm, presented in [18,19,20] and called MHT, is designed to consider such multi-objective heterogeneous optimization problems, where one of the objectives is an expensive black-box function. The simulation only gives function values.…”

“…This leads to single-objective subproblems, which have to be solved, and have the structure of the well-known Tammer-Weidner functional [11]. However, the algorithm MHT proposed in [19,20] calculates only one efficient solution, to be more concrete, accumulation points of the generated sequence of iteration points satisfy a necessary optimality criteria to be a locally optimal solution. The method is not designed to find a representation of the Pareto front.…”

“…In case only one of the objective functions is expensive while the others are cheap, for instance, analytically given, this can be used in the optimization procedure. Using a trust-region approach and the Tammer-Weidner-functional for finding descent directions, in [19] an algorithm was proposed which makes use of the heterogeneity of the objective functions. In this paper, we present three heuristic approaches, which allow to find additional optimal solutions of the multiobjective optimization problem and by that representations at least of parts of the Pareto front.…”