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

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.