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.

翻译：本文研究一个源于材料科学的约束最小化问题，该问题旨在确定晶界的位错（线缺陷）结构。该问题目标是在满足弗兰克公式约束条件下，最小化具有位错结构的晶界能量。在此约束最小化问题中，目标函数（即晶界能量）是非凸且可分离的，而约束条件是线性的。为求解此约束最小化问题，我们采用惩罚参数递增策略对交替方向乘子法（ADMM）进行了改进。我们在现有ADMM收敛性研究未涵盖的设定下，对此非凸最小化问题中改进ADMM的收敛性进行了分析。具体而言，在线性约束中，每个子变量块对应的系数矩阵均具有满列秩。该性质使得当惩罚参数足够大时，每个子变量最小化问题具有强凸性，从而有助于ADMM在无需对整个目标函数作凸性假设的情况下实现收敛。我们证明了改进ADMM生成序列的极限满足原始可行性条件，且为增广拉格朗日函数的驻点。此外，我们获得了证明目标函数拟凸性的充分条件，从而确保其在给定定义域内存在唯一最小值。数值算例验证了算法的收敛性，并对惩罚函数法、增广拉格朗日法及改进ADMM的结果进行了比较。