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.

In this data article, we report data and numerical results related to the research article entitled ”A trust region algorithm for heterogeneous multiobjective optimization” by Thomann and Eichfelder in SIAM Journal on Optimization. The method MHT which is presented there is designed for multiobjective heterogeneous optimization problems where one of the objective functions is an expensive black-box function, for example given by a time-consuming simulation. Here, we present the data of numerical tests with a set of 78 test problems mainly collected from literature and only complemented by few self-chosen test problems. The presence of expensive functions is artificially introduced in the test problems by defining one of the objective functions as expensive.

This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.