A trust region reduced basis Pascoletti-Serafini algorithm for multi-objective PDE-constrained parameter optimization

TL;DR

This paper introduces a novel trust-region reduced basis Pascoletti-Serafini algorithm for efficiently solving non-convex multi-objective PDE-constrained parameter optimization problems, combining model reduction with guaranteed convergence.

Contribution

It presents a new combined approach using trust-region strategies and reduced basis methods to efficiently solve complex PDE-constrained multi-objective optimization problems without offline procedures.

Findings

Significant reduction in computational cost for PDE-constrained optimization

Guaranteed convergence of the proposed method

Numerical examples demonstrate high efficiency and accuracy

Abstract

In the present paper non-convex multi-objective parameter optimization problems are considered which are governed by elliptic parametrized partial differential equations (PDEs). To solve these problems numerically the Pascoletti-Serafini scalarization is applied and the obtained scalar optimization problems are solved by an augmented Lagrangian method. However, due to the PDE constraints, the numerical solution is very expensive so that a model reduction is utilized by using the reduced basis (RB) method. The quality of the RB approximation is ensured by a trust-region strategy which does not require any offline procedure, where the RB functions are computed in a greedy algorithm. Moreover, convergence of the proposed method is guaranteed. Numerical examples illustrate the efficiency of the proposed solution technique.