An indefinite proximal Peaceman-Rachford splitting method with a convex combination technique for separable convex programming

Abstract In this paper, we consider the linearly constrained separable convex optimization problem, where the objective function is the sum of two individual extended real-valued functions without coupled variables. Based on the popular convex combination technique and with the help of the indefinite proximal regularization technique, we propose a novel Peaceman-Rachford splitting method (PRSM). The convex combination technique is integrated into the proximal term of the first subproblem, where the proximal matrix could be positive semidefinite so as to ensure the solution existence of the just-mentioned subproblem, while we allow the proximal term in the second subproblem to be indefinite but still guaranteeing the convergence of the proposed method theoretically. It is worth to mention that the range of the convex combination coefficient can be extended from $[0.618,1)$ to $(-1,1)$. Under some mild conditions, we establish the global convergence and ergodic $\mathcal{O}(\frac{1}{N})$ sublinear convergence rate measured by the function value residual and constraint violation, where $N$ denotes the number of iterations. To our knowledge, this is the first time that the convex combination technique has been used to accelerate the convergence of PRSM-based methods. Finally, numerical experiments are conducted to verify the effectiveness of the proposed algorithm in solving LASSO and total variation image restoration problems. Mathematics Subject Classification 65K10 · 90C25 · 90C06