Forward-Half-Reflected-Partial inverse-Backward Splitting Algorithm for Solving Monotone Inclusions

In this article, we proposed a method for numerically solving monotone inclusions in real Hilbert spaces that involve the sum of a maximally monotone operator, a monotone-Lipschitzian operator, a cocoercive operator, and a normal cone to a vector subspace. Our algorithm splits and exploits the intrinsic properties of each operator involved in the inclusion. The proposed method is derived by combining partial inverse techniques and the {\it forward-half-reflected-backward} (FHRB) splitting method proposed by Malitsky and Tam (2020). Our method inherits the advantages of FHRB, equiring only one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator. Furthermore, we develop a method for solving primal-dual inclusions involving a mixture of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones. We apply our method to constrained composite convex optimization problems as a specific example. Finally, in order to compare our proposed method with existing methods in the literature, we provide numerical experiments on constrained total variation least-squares optimization problems. The numerical results are promising

Inertial Douglas-Rachford splitting for monotone inclusion problems

We propose an inertial Douglas-Rachford splitting algorithm for finding the set of zeros of the sum of two maximally monotone operators in Hilbert spaces and investigate its convergence properties. To this end we formulate first the inertial version of the Krasnosel'ski\u{\i}--Mann algorithm for approximating the set of fixed points of a nonexpansive operator, for which we also provide an exhaustive convergence analysis. By using a product space approach we employ these results to the solving of monotone inclusion problems involving linearly composed and parallel-sum type operators and provide in this way iterative schemes where each of the maximally monotone mappings is accessed separately via its resolvent. We consider also the special instance of solving a primal-dual pair of nonsmooth convex optimization problems and illustrate the theoretical results via some numerical experiments in clustering and location theory.Comment: arXiv admin note: text overlap with arXiv:1402.529

Forward Primal-Dual Half-Forward Algorithm for Splitting Four Operators

In this article, we propose a splitting algorithm to find zeros of the sum of four maximally monotone operators in real Hilbert spaces. In particular, we consider a Lipschitzian operator, a cocoercive operator, and a linear composite term. In the case when the Lipschitzian operator is absent, our method reduces to the Condat-V\~u algorithm. On the other hand, when the linear composite term is absent, the algorithm reduces to the Forward-Backward-Half-Forward algorithm (FBHF). Additionally, in each case, the set of step-sizes that guarantee the weak convergence of those methods are recovered. Therefore, our algorithm can be seen as a generalization of Condat-V\~u and FBHF. Moreover, we propose extensions and applications of our method in multivariate monotone inclusions and saddle point problems. Finally, we present a numerical experiment in image deblurring problems