The forward-backward operator splitting algorithm is one of the mostimportant methods for solving the optimization problem of the sum of two convexfunctions, where one is differentiable with a Lipschitz continuous gradient andthe other is possible nonsmooth but it is proximable. It is convenient to solvesome optimization problems through the form of dual problem or the form ofprimal-dual problem. Both methods are mature in theory. In this paper, weconstruct several efficient first-order splitting algorithms for solving amulti-block composite convex optimization problem. The objective functionincludes a smooth function with Lipschitz continuous gradient, a proximableconvex function may be nonsmooth, and a finite sum of a composition of aproximable function with a bounded linear operator. In order to solve suchoptimization problem, by defining an appropriate inner product space, wetransform the optimization problem into the form of the sum of three convexfunctions. Based on the dual forward-backward splitting algorithm and theprimal-dual forward-backward splitting algorithm, we develop several iterativealgorithms, which consist of only computing the gradient of the differentiablefunction and proximity operators of related convex functions. These iterativealgorithms are matrix-inversion free and are completely splitting. Finally, weemploy the proposed iterative algorithms to solve a regularized general priorimage constrained compressed sensing (PICCS) model, which derived from computedtomography (CT) images reconstruction under sparse sampling of projectionmeasurements. The numerical results show the outperformance of our proposediterative algorithms by comparing to other algorithms.
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