ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

Stefan Volkwein,Stefan Banholzer,Lena Reichle,Bennet Gebken

doi:10.3390/mca26020032

Abstract

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.

Highlights

Multiobjective optimization plays an important role in many applications, e.g., in industry, medicine, or engineering
In order to deal with the inexactness introduced by the surrogate model, we combine the firstorder optimality conditions for multiobjective optimization problems with error estimates for the Reduced Basis (RB)-discrete empirical interpolation method (DEIM) method and derive an additional condition for the descent direction [9] to get a tight superset of the Pareto set
We present a way to solve multiobjective parameter optimization problems of semilinear elliptic partial differential equation (PDE) by combining an RB approach and DEIM with the set-oriented method based on gradient evaluations

Summary

Introduction

Multiobjective optimization plays an important role in many applications, e.g., in industry, medicine, or engineering. Based on inexact gradient evaluations of the objective functions with an RB approach and a discrete empirical interpolation method (DEIM) [19,20] for semi-linear elliptic PDEs. In order to deal with the inexactness introduced by the surrogate model, we combine the firstorder optimality conditions for multiobjective optimization problems with error estimates for the RB-DEIM method and derive an additional condition for the descent direction [9] to get a tight superset of the Pareto set. In order to deal with the inexactness introduced by the surrogate model, we combine the firstorder optimality conditions for multiobjective optimization problems with error estimates for the RB-DEIM method and derive an additional condition for the descent direction [9] to get a tight superset of the Pareto set This approach allows us to better control the quality of the result by controlling the errors for the objective functions independently.

Multiobjective Optimization

Descent Direction with Exact Gradients

Descent Direction with Inexact Gradients

Modified Subdivision Algorithm for Inexact Gradients

Problem Formulation

The Parameter Dependent Semi-Linear Elliptic PDE

Reduced Formulation and Adjoint Approach

Convergence Analysis

Numerical Experiments

Method

Findings

Conclusions