Four Operator Splitting via a Forward-Backward-Half-Forward Algorithm with Line Search

In this article we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward-back-half forward splitting and the Tseng's algorithm with line-search. At each iteration, our algorithm defines the step-size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for solving non-linearly constrained composite convex optimization problems in real Hilbert spaces. Finally, we implement our algorithm in a non-linearly constrained least-square problem, and we compare its performance with available methods in the literature.Comment: 17 page

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