Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with dierent smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

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. Motivated by block partitioned problems arising from group sparsity representation and generalized noncooperative potential games, this paper presents a basic decomposition method for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an optimization problem is given by the sum of two nonseparable functions minus a sum of separable, pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1 , i.e., differentiable with a Lipschitz gradient, while the other summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By taking advantage of the (negative) pointwise maximum structure in the objective, the developed algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth problems. This aim is accomplished by combining the alternating direction method of multipliers (ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall problem into subproblems each involving the individual blocks. To arrive at a stationary solution of the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex function in each pointwise maximum. In order to lessen the potential computational burden in each iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is established.

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