The complexity results of the sparse optimization problems and reverse convex optimization problems

Abstract

In this paper, we study complexity results of sparse optimization problems and reverse convex optimization problems. These problems are very important subjects of optimization problems. We prove that the complexity result of the sparsity constraint problem and sparse solution problem are all NP-hard in the strong sense and even testing feasibility of the sparsity constraint is NP-complete in the strong sense. Then the sparse optimization problem is NP-hard in the strong sense. We also prove that the reverse convex problem is NP-hard in the strong sense by transforming the sparsity constraint into a reverse convex constraint.