Exact Relaxation for Classes of Minimization Problems with Binary Constraints

TL;DR

This paper introduces an exact relaxation framework for binary-valued functions involving total variation, enabling efficient solutions for various applications like image processing and topology optimization.

Contribution

It presents a novel relaxation approach turning binary constraints into box constraints, with theoretical guarantees and an efficient numerical algorithm.

Findings

Thresholding relaxed solutions yields original binary solutions.

The proposed algorithm converges superlinearly.

Numerical results demonstrate effectiveness in image processing and optimization.

Abstract

Relying on the co-area formula, an exact relaxation framework for minimizing objectives involving the total variation of a binary valued function (of bounded variation) is presented. The underlying problem class covers many important applications ranging from binary image restoration, segmentation, minimal compliance topology optimization to the optimal design of composite membranes and many more. The relaxation approach turns the binary constraint into a box constraint. It is shown that thresholding a solution of the relaxed problem almost surely yields a solution of the original binary-valued problem. Furthermore, stability of solutions under data perturbations is studied, and, for applications such as structure optimization, the inclusion of volume constraints is considered. For the efficient numerical solution of the relaxed problem, a locally superlinearly convergent algorithm is proposed which is based on an exact penalization technique, Fenchel duality, and a semismooth Newton approach. The paper ends by a report on numerical results for several applications in particular in mathematical image processing.