A Strictly Contractive Peaceman-Rachford Splitting Method with Logarithmic-Quadratic Proximal Regularization for Convex Programming

Recently, a strictly contractive Peaceman-Rachford splitting method (PRSM) was proposed for a separable convex minimization model whose variables are subject to some linear constraints and two additional generic constraints. In general, the strictly contractive PRSM requires to solve two constrained minimization subproblems at each iteration. In this paper, we consider the case where the additional constraints on variables are positive orthants and apply the well-developed logarithmic-quadratic proximal (LQP) regularization to regularize the subproblems of the strictly contractive PRSM. A new algorithm in combination with the strictly contractive PRSM and the LQP regularization is thus proposed. The new algorithm only needs to solve two unconstrained subproblems at each iteration. An inexact version allowing the unconstrained subproblems to be solved approximately subject to certain inexactness criterion is also studied. We prove the global convergence and establish a worst-case convergence rate measured by the iteration complexity for both the exact and inexact versions of the new algorithm.