Derivative-Free Multiobjective Trust Region Descent Method Using Radial Basis Function Surrogate Models

TL;DR

This paper introduces a derivative-free trust region method for multiobjective optimization that uses radial basis function surrogates, enabling efficient handling of expensive and high-dimensional problems with proven convergence.

Contribution

It develops a novel multiobjective trust region algorithm employing radial basis function surrogates, which scale linearly with problem dimension and are proven to converge to Pareto critical points.

Findings

Convergence to Pareto critical points is theoretically proven.

Radial basis function surrogates improve scalability for high-dimensional problems.

Numerical examples demonstrate the method's effectiveness.

Abstract

We present a flexible trust region descend algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. The method is derivative-free in the sense that neither need derivative information be available for the expensive objectives nor are gradients approximated using repeated function evaluations as is the case in finite-difference methods. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar pendants. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points scales linearly with the parameter space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Convergence to Pareto critical points is proven and numerical examples illustrate our findings.