A Fully Relaxed Tseng’s Splitting Method

This article introduces a fully relaxed variable metric variant of Tseng’s splitting method for finding zeros of the sum of two operators in a real Hilbert space, one maximal monotone and the other Lipschitz or even uniformly continuous and monotone. In the case of the variable metric operator being the identity operator, it allows the relaxation factor to vary freely in any closed subinterval of (0,2) . To ensure its weak convergence, and with the hope of improving performance, two new strategies for choosing the step length are developed. One is based on a novel equality to evaluate when the backtracking can be accepted. The other is based on its corresponding strict equality, but avoids the backtracking and adopts an innovative parabolic update to determine the step length once per iteration. In addition, if the forward operator is Lipschitz continuous, then we can give special strategies for choosing the step length. On top of convergence guarantees, we show that, under a mild growth condition on the sum operator, the method enjoys a locally linear rate of convergence. Numerical results demonstrate the necessity of introducing both the relaxation factor, with values close to 2, and new strategies beyond the backtracking.