Projective Splitting with Backward, Half-Forward and Proximal-Newton Steps

We propose and study the weak convergence of a projective splitting algorithm studied by A. Alotaibi, P. Combettes, and N. Shahzad [Solving coupled composite monotone inclusions by successive Fejér approximations of their Kuhn-Tucker set, SIAM J. Optim. 24/4 (2014) 2076–2095] and by J. Eckstein and B. F. Svaiter [A family of projective splitting methods for the sum of two maximal monotone operators, Math. Programming Ser. B 111/1-2 (2008) 173–199] for solving multi-term composite monotone inclusion problems involving the finite sum of n maximal monotone operators, each of which having an inner four-block structure: sum of maximal monotone, Lipschitz continuous, cocoercive and smooth differentiable operators. We show how to perform backward and half-forward steps [see L. M. Briceno Arias and D. Davis [Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J. Optim. 28/4 (2018) 2839–2871] with respect to the maximal monotone and Lipschitz + cocoercive components, respectively, while performing proximal-Newton steps with respect to smooth differentiable blocks.