Multiobjective Optimization of Non-Smooth PDE-Constrained Problems

TL;DR

This paper reviews recent advances in multiobjective optimization for non-smooth PDE-constrained problems, addressing challenges like computational complexity and non-smoothness in objectives or system dynamics.

Contribution

It provides an overview of new algorithms and methods developed for handling non-smoothness and computational challenges in multiobjective PDE-constrained optimization.

Findings

Summarizes recent algorithmic developments for non-smooth PDE optimization

Highlights challenges in computing Pareto sets with expensive models

Reports on progress within a specific research project

Abstract

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria 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. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, 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 or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".