FORWARD-BACKWARD-HALF FORWARD ALGORITHM FOR SOLVING MONOTONE INCLUSIONS

Tseng's algorithm finds a zero of the sum of a maximally monotone operator and a monotone continuous operator by evaluating the latter twice per iteration. In this paper, we modify Tseng's algorithm for finding a zero of the sum of three operators, where we add a cocoercive operator to the inclusion. Since the sum of a cocoercive and a monotone-Lipschitz operator is monotone and Lipschitz, we could use Tseng's method for solving this problem, but implementing both operators twice per iteration and without taking into advantage the cocoercivity property of one operator. Instead, in our approach, although the continuous monotone operator must still be evaluated twice, we exploit the cocoercivity of one operator by evaluating it only once per iteration. Moreover, when the cocoercive or continuous-monotone operators are zero it reduces to Tseng's algorithm or forward-backward splittings, respectively, unifying in this way both algorithms. In addition, we provide a preconditioned version of the proposed method including non-self-adjoint linear operators in the computation of resolvents and the single-valued operators involved. This approach allows us to also extend previous variable metric versions of the Tseng and forward-backward methods and simplify their conditions on the underlying metrics. We also exploit the case in which non-self-adjoint linear operators are triangular by blocks in the primal-dual product space for solving primal-dual composite monotone inclusions, obtaining Gauss-Seidel-type algorithms which generalize several primal-dual methods available in the literature. Finally we explore applications to the obstacle problem, empirical risk minimization, distributed optimization, and nonlinear programming and we illustrate the performance of the method via some numerical simulations.