State Constrained Optimization with Partial Differential Equations via Generalized Gradients

TL;DR

This paper introduces a flexible framework for solving PDE-constrained optimization problems with various constraints using generalized gradients, enabling handling of discontinuities and complex constraints with promising numerical results.

Contribution

The paper develops a novel approach employing infinite-valued penalization and Clarke subgradients for state-constrained PDE optimization, applicable to diverse and discontinuous constraints.

Findings

Effective handling of complex constraints including box and average value constraints.

Numerical validation on elliptic PDEs demonstrating competitive performance.

Framework accommodates discontinuous data in constraints.

Abstract

We consider optimization problems constrained by partial differential equations (PDEs) with additional constraints placed on the solution of the PDEs. We develop a general and versatile framework using infinite-valued penalization functions and Clarke subgradients and apply this to problems with box constraints as well as more general constraints arising in applications, such as constraints on the average value of the state in subdomains. The framework also allows for problems with discontinuous data in the constraints. We present numerical results of this algorithm for the elliptic case and compare with other state-constrained algorithms.