A penalty scheme to solve constrained non-convex optimization problems in $BV(\Omega)$

TL;DR

This paper introduces a penalty scheme for solving constrained non-convex optimization problems in the space of functions of bounded variation, providing theoretical guarantees and numerical validation.

Contribution

It proposes a novel regularization and penalization method for non-convex BV problems with inequality constraints, establishing stationarity and optimality conditions.

Findings

Weak limit points are stationary for the original problem.

Optimality conditions include Lagrange multipliers.

Numerical experiments confirm theoretical results.

Abstract

We investigate non-convex optimization problems in $BV(\Omega)$ with two-sided pointwise inequality constraints. We propose a regularization and penalization method to numerically solve the problem. Under certain conditions, weak limit points of iterates are stationary for the original problem. In addition, we prove optimality conditions for the original problem that contain Lagrange multipliers to the inequality constraints. Numerical experiments confirm the theoretical findings.