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A first-order splitting method for solving large scale composite convex optimization problem

Research paper by Yu-Chao Tang, Guo-Rong Wu

Indexed on: 23 Nov '17Published on: 23 Nov '17Published in: arXiv - Mathematics - Optimization and Control



Abstract

The forward-backward operator splitting algorithm is one of the most important methods for solving the optimization problem of the sum of two convex functions, where one is differentiable with a Lipschitz continuous gradient and the other is possible nonsmooth but it is proximable. It is convenient to solve some optimization problems through the form of dual problem or the form of primal-dual problem. Both methods are mature in theory. In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with Lipschitz continuous gradient, a proximable convex function may be nonsmooth, and a finite sum of a composition of a proximable function with a bounded linear operator. In order to solve such optimization problem, by defining an appropriate inner product space, we transform the optimization problem into the form of the sum of three convex functions. Based on the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms, which consist of only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion free and are completely splitting. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing (PICCS) model, which derived from computed tomography (CT) images reconstruction under sparse sampling of projection measurements. The numerical results show the outperformance of our proposed iterative algorithms by comparing to other algorithms.