A Convergent ADMM Algorithm for Grain Boundary Energy Minimization

TL;DR

The paper addresses constrained nonconvex energy minimization for grain boundary dislocation structures under Frank's formula. It introduces a modified ADMM with increasing penalty parameter $\rho^{(k)}$ that exploits a multi-block separable objective with linear constraints $\sum_{j=1}^J A_j \bm{u}_j = \bm{c}$, where each $A_j \in \mathbb{R}^{6\times 2}$ has full column rank and satisfies $A_j^T A_j = b^2 I_2$, making subproblems strongly convex for large $\rho$. The authors prove convergence to a feasible limit and stationarity of the augmented Lagrangian, and establish quasi-convexity-based uniqueness for the three-Burgers-vectors case; numerical experiments show faster convergence and more accurate constraint satisfaction than the penalty method and ALM. This framework broadens ADMM applicability to certain nonconvex constrained problems with per-block full-column-rank constraints, providing a practical tool for grain boundary energy minimization.

Abstract

In this paper, we study a constrained minimization problem that arise from materials science to determine the dislocation (line defect) structure of grain boundaries. The problems aims to minimize the energy of the grain boundary with dislocation structure subject to the constraint of Frank's formula. In this constrained minimization problem, the objective function, i.e., the grain boundary energy, is nonconvex and separable, and the constraints are linear. To solve this constrained minimization problem, we modify the alternating direction method of multipliers (ADMM) with an increasing penalty parameter. We provide a convergence analysis of the modified ADMM in this nonconvex minimization problem, with settings not considered by the existing ADMM convergence studies. Specifically, in the linear constraints, the coefficient matrix of each subvariable block is of full column rank. This property makes each subvariable minimization strongly convex if the penalty parameter is large enough, and contributes to the convergence of ADMM without any convex assumption on the entire objective function. We prove that the limit of the sequence from the modified ADMM is primal feasible and is the stationary point of the augmented Lagrangian function. Furthermore, we obtain sufficient conditions to show that the objective function is quasi-convex and thus it has a unique minimum over the given domain. Numerical examples are presented to validate the convergence of the algorithm, and results of the penalty method, the augmented Lagrangian method, and the modified ADMM are compared.