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In this paper we study the bounded perturbation resilience of the extragradient and the subgradient extragradient methods for solving variational inequality (VI) problem in real Hilbert spaces. This is an important property of algorithms which guarantees the convergence of the scheme under summable errors, meaning that an inexact version of the methods can also be considered. Moreover, once an algorithm is proved to be bounded perturbation resilience, superiorizion can be used, and this allows flexibility in choosing the bounded perturbations in order to obtain a superior solution, as well explained in the paper. We also discuss some inertial extragradient methods. Under mild and standard assumptions of monotonicity and Lipschitz continuity of the VI's associated mapping, convergence of the perturbed extragradient and subgradient extragradient methods is proved. In addition we show that the perturbed algorithms converges at the rate of $O(1/t)$. Numerical illustrations are given to demonstrate the performances of the algorithms., Comment: Accepted for publication in The Journal of Inequalities and Applications. arXiv admin note: text overlap with arXiv:1711.01936 and text overlap with arXiv:1507.07302 by other authors

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 possibly nonsmooth but proximable. It is convenient to solve some optimization problems in the form of dual or primal-dual problems. 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 a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum of a composition of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. On the basis of the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing (PICCS) model that is derived from computed tomography (CT) image reconstruction under sparse sampling of projection measurements. Numerical results show that the proposed iterative algorithms outperform other algorithms., Comment: 27 pages, 4 figures, J. Comp. Math. 2018

In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a closed-form expression of their proximity operators and $B$ is a bounded linear operator. This type of optimization problem has wide application in signal recovery and image processing. To make full use of the differentiability function in the optimization problem, we take advantage of two operator splitting methods: the forward-backward splitting method and the three operator splitting method. In the iteration scheme derived from the two operator splitting methods, we need to compute the proximity operator of $g+h \circ B$ and $h \circ B$, respectively. Although these proximity operators do not have a closed-form solution in general, they can be solved very efficiently. We mainly employ two different approaches to solve these proximity operators: one is dual and the other is primal-dual. Following this way, we fortunately find that three existing iterative algorithms including Condat and Vu algorithm, primal-dual fixed point (PDFP) algorithm and primal-dual three operator (PD3O) algorithm are a special case of our proposed iterative algorithms. Moreover, we discover a new kind of iterative algorithm to solve the considered optimization problem, which is not covered by the existing ones. Under mild conditions, we prove the convergence of the proposed iterative algorithms. Numerical experiments applied on fused Lasso problem, constrained total variation regularization in computed tomography (CT) image reconstruction and low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms., Comment: 37 pages, 10 figures

We consider an inertial primal-dual algorithm to compute the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator. With the idea of coordinate descent, we design a stochastic coordinate descent inertial primal-dual splitting algorithm. Moreover, in order to prove the convergence of the proposed inertial algorithm, we formulate first the inertial version of the randomized Krasnosel'skii-Mann iterations algorithm for approximating the set of fixed points of a nonexpansive operator and investigate its convergence properties. Then the convergence of stochastic coordinate descent inertial primal-dual splitting algorithm is derived by applying the inertial version of the randomized Krasnosel'skii-Mann iterations to the composition of the proximity operator., Comment: arXiv admin note: substantial text overlap with arXiv:1604.04172, arXiv:1604.04282; substantial text overlap with arXiv:1407.0898 by other authors

We consider an inertial primal-dual fixed point algorithm (IPDFP) to compute the minimizations of the following Problem (1.1). This is a full splitting approach, in the sense that the nonsmooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the Problem (1.1) to the sum of three convex functions. This work brings together and notably extends several classical splitting schemes, like the primaldual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles A. et al designed for the L1/TV image denoising model. The iterative algorithm is used for solving nondifferentiable convex optimization problems arising in image processing. The experimental results indicate that the proposed IPDFP iterative algorithm performs well with respect to state-of-the-art methods., Comment: arXiv admin note: text overlap with arXiv:1604.04845, arXiv:1604.04172; text overlap with arXiv:1403.3522, arXiv:1407.0898 by other authors

We consider the problem of finding the minimizations of the sum of two convex functions and the composition of another convex function with a continuous linear operator from the view of fixed point algorithms based on proximity operators, which is is inspired by recent results of Chen, Huang and Zhang. With the idea of coordinate descent, we design a stochastic coordinate descent splitting primal- dual fixed point algorithm. Based on randomized krasnosel'skii mann iterations and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithms., Comment: arXiv admin note: substantial text overlap with arXiv:1407.0898 by other authors; substantial text overlap with arXiv:1604.04172

Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Further, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality., Comment: 22 pages, 3 figures