Gradient-Based Multiobjective Optimization with Uncertainties

TL;DR

This paper introduces a gradient-based algorithm for multiobjective optimization under uncertainties, ensuring convergence to a superset of the Pareto set and suitable for PDE-constrained problems with model reduction.

Contribution

It develops a novel descent direction condition for uncertain gradients and integrates it into a subdivision algorithm for global solutions.

Findings

Proves convergence to a superset of the Pareto set.

Provides an upper bound for the distance to substationary points.

Applicable to PDE-constrained multiobjective problems with uncertainties.

Abstract

In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated in a subdivison algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorithm is developed with the intention to use it in combination with model order reduction techniques in order to efficiently solve PDE-constrained multiobjective optimization problems.