Fast proximity-gradient algorithms for structured convex optimization problems

Abstract

We introduce in this paper fixed-point proximity-gradient algorithms for solving a class of structured convex optimization problems arising from image restoration. The objective function of such optimization problems is the sum of three convex functions. We study in this paper the scenario where one of the convex functions involved is differentiable with a Lipschitz continuous gradient and another convex function is composed by an affine transformation. We first characterize the solutions of the optimization problem as fixed-points of a mapping defined in terms of the gradient of the differentiable function and the proximity operators of the other two functions. Then, a fixed-point proximity-gradient iterative scheme is developed based on the fixed-point equation which characterizes the solutions. We establish the convergence of the proposed iterative scheme by the notion of averaged nonexpansive operators. Moreover, we obtain that in general the proposed iterative scheme has O(1k) convergence rate in the ergodic sense and the sense of partial primal–dual gap. Under stronger assumptions on the convex functions involved the proposed iterative scheme will converge linearly. We in particular derive fixed-point proximity-gradient algorithms from the proposed iterative scheme. The quasi-Newton and the overrelaxation strategies are designed to accelerate the algorithms. Numerical experiments for the computerized tomography reconstruction problem demonstrate that the proposed algorithms perform favorably and the quasi-Newton as well as the overrelaxation strategies significantly accelerate the convergence of the algorithms.