A Partially Inertial Customized Douglas–Rachford Splitting Method for a Class of Structured Optimization Problems

In this paper, we are concerned with a class of structured optimization problems frequently arising from image processing and statistical learning, where the objective function is the sum of a quadratic term and a nonsmooth part, and the constraint set consists of a linear equality constraint and two simple convex sets in the sense that projections onto simple sets are easy to compute. To fully exploit the quadratic and separable structure of the problem under consideration, we accordingly propose a partially inertial Douglas–Rachford splitting method. It is noteworthy that our algorithm enjoys easy subproblems for the case where the underlying two simple convex sets are not the whole spaces. Theoretically, we establish the global convergence of the proposed algorithm under some mild conditions. A series of computational results on the constrained Lasso and constrained total-variation (TV) based image restoration demonstrate that our proposed method is competitive with some state-of-the-art first-order solvers.