Convergence of the Peaceman-Rachford Splitting Method for a Class of Nonconvex Programs

Abstract

In this paper, we analyze the convergence of the Peaceman-Rachford splitting method (PRSM) for a type of nonconvex and nonsmooth optimization with linear constraints, whose objective function is the sum of a proper lower semicontinuous function and a strongly convex differential function. When a suitable penalty parameter is chosen and the iterative point sequence is bounded, we show the global convergence of the PRSM. Furthermore, under the assumption that the associated function satisfies the Kurdyka-Łojasiewicz property, we prove the strong convergence of the PRSM. We also provide sufficient conditions guaranteeing the boundedness of the generated sequence. Finally, we present some preliminary numerical results to show the effectiveness of the PRSM and also give a comparison with the Douglas-Rachford splitting method.