The Non-convex Sparse Problem with Nonnegative Constraint for Signal Reconstruction

The problem of finding a sparse solution for linear equations has been investigated extensively in recent years. This is an NP-hard combinatorial problem, and one popular method is to relax such combinatorial requirement into an approximated convex problem, which can avoid the computational complexity. Recently, it is shown that a sparser solution than the approximated convex solution can be obtained by solving its non-convex relaxation rather than by solving its convex relaxation. However, solving the non-convex relaxation is usually very costive due to the non-convexity and non-Lipschitz continuity of the original problem. This difficulty limits its applications and possible extensions. In this paper, we will consider the non-convex relaxation problem with the nonnegative constraint, which has many applications in signal processing with such reasonable requirement. First, this optimization problem is formulated and equivalently transformed into a Lipschitz continuous problem, which can be solved by many existing optimization methods. This reduces the computational complexity of the original problem significantly. Second, we solve the transformed problem by using an efficient and classical limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. Finally, some numerical results show that the proposed method can effectively find a nonnegative sparse solution for the given linear equations with very low computational cost.