A Peaceman–Rachford Splitting Method with Monotone Plus Skew-Symmetric Splitting for Nonlinear Saddle Point Problems

Abstract

This paper is devoted to solving the linearly constrained convex optimization problems by Peaceman–Rachford splitting method with monotone plus skew-symmetric splitting on KKT operators. This approach generalizes the Hermitian and skew-Hermitian splitting method, an unconditionally convergent algorithm for non-Hermitian positive definite linear systems, to the nonlinear scenario. The convergence of the proposed algorithm is guaranteed under some mild assumptions, e.g., the strict convexity on objective functions and the consistency on constraints, even though the Lions–Mercier property is not fulfilled. In addition, we explore an inexact version of the proposed algorithm, which allows solving the subproblems approximately with some inexactness criteria. Numerical simulations on an image restoration problem demonstrate the compelling performance of the proposed algorithm.