The Sparsest Solution to the System of Absolute Value Equations

Abstract

On one hand, to find the sparsest solution to the system of linear equations has been a major focus since it has a large number of applications in many areas; and on the other hand, the system of absolute value equations (AVEs) has attracted a lot of attention since many practical problems can be equivalently transformed as a system of AVEs. Motivated by the development of these two aspects, we consider the problem to find the sparsest solution to the system of AVEs in this paper. We first propose the model of the concerned problem, i.e., to find the solution to the system of AVEs with the minimum $$l_0$$ -norm. Since $$l_0$$ -norm is difficult to handle, we relax the problem into a convex optimization problem and discuss the necessary and sufficient conditions to guarantee the existence of the unique solution to the convex relaxation problem. Then, we prove that under such conditions the unique solution to the convex relaxation is exactly the sparsest solution to the system of AVEs. When the concerned system of AVEs reduces to the system of linear equations, the obtained results reduce to those given in the literature. The theoretical results obtained in this paper provide an important basis for designing numerical method to find the sparsest solution to the system of AVEs.