# ROM-based multiobjective optimization of elliptic PDEs via numerical continuation

## TL;DR

This paper introduces a ROM-based continuation method for multiobjective PDE-constrained optimization, enabling efficient computation of Pareto sets with multiple objectives by reducing computational costs through surrogate modeling.

## Contribution

It combines reduced basis model reduction with a continuation approach to handle multiple objectives efficiently in PDE-constrained optimization.

## Key findings

- Significant reduction in computational time for multiobjective PDE optimization.
- Ability to handle an arbitrary number of objectives.
- Effective surrogate modeling for expensive PDE evaluations.

## Abstract

Multiobjective optimization plays an increasingly important role in modern applications, where several objectives are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. Since the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging which is particularly problematic when the objectives are costly to evaluate as is the case for models governed by partial differential equations (PDEs). To decrease the numerical effort to an affordable amount, surrogate models can be used to replace the expensive PDE evaluations. Existing multiobjective optimization methods using model reduction are limited either to low parameter dimensions or to few (ideally two) objectives. In this article, we present a combination of the reduced basis model reduction method with a continuation approach using inexact gradients. The resulting approach can handle an arbitrary number of objectives while yielding a significant reduction in computing time.