A DC-Reformulation for Gradient-$L^0$-Constrained Problems

TL;DR

This paper introduces a DC-reformulation approach for optimization problems with $L^0$-type gradient constraints, enabling efficient handling of sparsity in solutions and extending existing methods to gradient-based sparsity scenarios.

Contribution

It extends the difference-of-convex reformulation technique to problems with gradient sparsity constraints, broadening its applicability to piecewise constant solutions.

Findings

Provides a new DC-reformulation for gradient sparsity constraints

Enables solving problems with $L^0$-type gradient constraints efficiently

Extends existing convex optimization techniques to gradient support constraints

Abstract

Cardinality constraints in optimization are commonly of $L^0$-type, and they lead to sparsely supported optimizers. An efficient way of dealing with these constraints algorithmically, when the objective functional is convex, is reformulating the constraint using the difference of suitable $L^1$- and largest-$K$-norms and subsequently solving a sequence of penalized subproblems in the difference-of-convex (DC) class. We extend this DC-reformulation approach to problems with $L^0$-type cardinality constraints on the support of the gradients, i.e., problems where sparsity of the gradient and thus piecewise constant solutions are the target.