A linearly convergent self-adaptive gradient projection algorithm for sparse signal reconstruction in compressive sensing

For sparse signal reconstruction (SSR) problem in compressive sensing (CS), by the splitting technique, we first transform it into a continuously differentiable convex optimization problem, and then a new self-adaptive gradient projection algorithm is proposed to solve the SSR problem, which has fast solving speed and pinpoint accuracy when the dimension increases. Global convergence of the proposed algorithm is established in detail. Without any assumptions, we establish global $R-$linear convergence rate of the proposed algorithm, which is a new result for constrained convex (rather than strictly convex) quadratic programming problem. Furthermore, we can also obtain an approximate optimal solution in a finite number of iterations. Some numerical experiments are made on the sparse signal recovery and image restoration to exhibit the efficiency of the proposed algorithm. Compared with the state-of-the-art algorithms in SSR problem, the proposed algorithm is more accurate and efficient.