Some extensions of the operator splitting schemes based on Lagrangian and primal–dual: a unified proximal point analysis

Abstract

By revisiting some popular operator splitting ideas, we present several classes of splitting schemes based on the Lagrangian, primal–dual and hybrid formulations, from which can be recovered many existing algorithms, including alternating direction method of multipliers and primal–dual hybrid gradient algorithms. In particular, we show that the generalized proximal point framework is at the root of many past and recent splitting algorithms allowing for an elementary convergence analysis of these methods through a unified scheme. The numerical tests on constrained total variation minimization show that the proposed algorithms could offer more freedom in parameter selections and, thus, achieve faster convergence speed.