A proximal alternating direction method of multipliers for a minimization problem with nonconvex constraints

Abstract

In this paper, a proximal alternating direction method of multipliers is proposed for solving a minimization problem with Lipschitz nonconvex constraints. Such problems are raised in many engineering fields, such as the analytical global placement of very large scale integrated circuit design. The proposed method is essentially a new application of the classical proximal alternating direction method of multipliers. We prove that, under some suitable conditions, any subsequence of the sequence generated by the proposed method globally converges to a Karush-Kuhn-Tucker point of the problem. We also present a practical implementation of the method using a certain self-adaptive rule of the proximal parameters. The proposed method is used as a global placement method in a placer of very large scale integrated circuit design. Preliminary numerical results indicate that, compared with some state-of-the-art global placement methods, the proposed method is applicable and efficient.

The Alternating Direction Method of Multipliers for Finding the Distance between Ellipsoids

“…where κ > 0 and η ∈ (0, 1) are fixed parameters. If inequalities (20) are satisfied, then τ n+1 = βτ n for some β > 1. Otherwise, we put τ n+1 = τ n .…”

“…Remark 4. Let us explain the motivation behind criterion (20) for increasing the penalty parameter. Multiple numerical experiments with fixed penalty parameter τ demonstrated that one needs to increase τ only if the infeasibility measure S i x n i − y n i − c i , i ∈ {1, 2}, does not decrease with iterations.…”

“…As a result, at later stages, when the value S i x n i − y n i − c i is very small but nonzero, the second inequality in (20) might be violated, although there is no need to increase τ at this stage. Therefore, the first inequality in (20) is used as a safeguard to avoid an unnecessary increase of the penalty parameter. A safe choice of parameter κ is κ < ε, where ε > 0 is from the stopping criterion discussed below.…”

“…Let us now turn to a convergence analysis of the ADMM for finding the distance between the boundaries of two ellipsoids. Since inequalities (20) were chosen heuristically without any theoretical foundation, we will analyse the method under the assumption that inequalities (19) are used as a criterion for penalty updates. Although, this criterion performed poorly in our numerical experiments, its analysis provides an insight into the performance of the ADDM in the nonconvex case and the choice of parameters of this method.…”

“…The alternating direction method of multipliers (ADMM) is an efficient method for solving structured convex optimisation problems [5,7,10,11,27] that has found a wide variety of applications [4,6,14,16], including applications to some nonconvex and nonsmooth optimisation problems [9,26]. Although the ADMM was originally developed for structured convex problems, extensions of this method to various nonconvex settings have recently become an active area of research [3,8,12,19,20,28].…”