# Improved penalty algorithm for Mixed Integer PDE Constrained Optimization Problems

## TL;DR

This paper introduces an improved penalty algorithm combining basin hopping and interior point methods to efficiently solve large-scale mixed-integer PDE-constrained optimization problems, offering an alternative to traditional branch-and-bound methods.

## Contribution

A novel penalty algorithm tailored for large-scale mixed-integer PDE-constrained problems, integrating basin hopping and interior point techniques for enhanced performance.

## Key findings

- Effective on standard stationary test problems
- Versatile extension to convection-diffusion problems
- Demonstrates competitiveness with existing methods

## Abstract

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.