Optimality conditions for the constrained Lp-regularization

The -regularization problem with is a nonsmooth and nonconvex problem and has remarkable advantages in the restoration of discrete signals and images. The constrained -regularization problem can improve the image restoration using a priori information. In this paper, we study the optimality conditions for the constrained -regularization problem. We derive the first-order and second-order necessary optimality conditions for the problem. We also give a second-order sufficient condition. The obtained optimality conditions are extensions of the optimality conditions for the smooth constrained optimization. We will also investigate some other interesting properties of the problem. In particular, we will show that a point that satisfies the first-order necessary condition will not be a maximizer of the problem as long as zero is not a solution of the problem.