The sparsest solution of the union of finite polytopes via its nonconvex relaxation

Abstract

Sparse optimization problems have gained much attention since 2004. Many approaches have been developed, where nonconvex relaxation methods have been a hot topic in recent years. In this paper, we study a partially sparse optimization problem, which finds a partially sparsest solution of a union of finite polytopes. We discuss the relationship between its solution set and the solution set of its nonconvex relaxation. In details, by using geometrical properties of polytopes and properties of a family of well-defined nonconvex functions, we show that there exists a positive constant p * ∈ (0, 1] such that for every p ∈ [0, p * ), all optimal solutions to the nonconvex relaxation with the parameter p are also optimal solutions to the original sparse optimization problem. This provides a theoretical basis for solving the underlying problem via its nonconvex relaxation. Moreover, we show that the problem we concerned covers a wide range of problems so that several important sparse optimization problems are its subclasses. Finally, by an example we illustrate our theoretical findings.