A First-Order Splitting Method for Solving a Large-Scale Composite Convex Optimization Problem (Journal of Computational Mathematics, 2019)
In this paper, we construct several efficient first-order splitting algorithms for solvinga multi-block composite convex optimization problem. The objective function includes asmooth function with a Lipschitz continuous gradient, a proximable convex function thatmay be nonsmooth, and a finite sum composed of a proximable function and a boundedlinear operator. To solve such an optimization problem, we transform it into the sumof three convex functions by defining an appropriate inner product space. Based on thedual forward-backward splitting algorithm and the primal-dual forward-backward splittingalgorithm, we develop several iterative algorithms that involve only computing the gradientof the differentiable function and proximity operators of related convex functions. Theseiterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally,we employ the proposed iterative algorithms to solve a regularized general prior imageconstrained compressed sensing model that is derived from computed tomography imagereconstruction. Numerical results show that the proposed iterative algorithms outperformthe compared algorithms including the alternating direction method of multipliers, thesplitting primal-dual proximity algorithm, and the preconditioned splitting primal-dualproximity algorithm.