Majorized iPADMM for Nonseparable Convex Minimization Models with Quadratic Coupling Terms

Abstract

In this paper, we consider nonseparable convex minimization models with quadratic coupling terms arised in many practical applications. We use a majorized indefinite proximal alternating direction method of multipliers (iPADMM) to solve this model. The indefiniteness of proximal matrices allows the function we actually solved to be no longer the majorization of the original function in each subproblem. While the convergence still can be guaranteed and larger stepsize is permitted which can speed up convergence. For this model, we analyze the global convergence of majorized iPADMM with two different techniques and the sublinear convergence rate in the nonergodic sense. Numerical experiments illustrate the advantages of the indefinite proximal matrices over the positive definite or the semi-definite proximal matrices.

Understanding the convergence of the preconditioned PDHG method: a view of indefinite proximal ADMM

“…Motivated by [36], we intend to investigate PrePDHG from the perspective of proximal ADMM. A known result is that indefinite proximal ADMM (iPADMM), with weaker convergence conditions, outperforms positive semidefinite proximal ADMM (sPADMM) [5,10,11,17,18,22,32,37,49]. In this paper, we restudy the PrePDHG (1.3) from the point view of iPADMM other than sPADMM as done in [36] and give positive answers to the above question.…”

“…Based on the key observation that PrePDHG and iPADMM are equivalent, we next investigate the convergence of PrePDHG (3.1), namely, Algorithm 1, via the well-established convergence results of iPADMM; see [11,17,22,32,37,49] for instance. Here, we mainly use the global and sublinear convergence rate results developed in [17].…”