In this paper, we propose an inexact Augmented Lagrangian Method (ALM) for the optimization of convex and nonsmooth objective functions subject to linear equality constraints and box constraints where errors are due to fixed-point data. To prevent data overflow we also introduce a projection operation in the multiplier update. We analyze theoretically the proposed algorithm and provide convergence rate results and bounds on the accuracy of the optimal solution. Since iterative methods are often needed to solve the primal subproblem in ALM, we also propose an early stopping criterion that is simple to implement on embedded platforms, can be used for problems that are not strongly convex, and guarantees the precision of the primal update. To the best of our knowledge, this is the first fixed-point ALM that can handle non-smooth problems, data overflow, and can efficiently and systematically utilize iterative solvers in the primal update. Numerical simulation studies on a logistic regression problem are presented that illustrate the proposed method.

在本文中，我们提出了一种不精确的增强拉格朗日方法（ALM），用于优化受线性等式约束和框约束（其中误差是由于定点数据引起）的凸和非光滑目标函数的优化。为了防止数据溢出，我们还在乘法器更新中引入了投影操作。我们从理论上分析了提出的算法，并提供了收敛速度的结果和最优解精度的界限。由于通常需要使用迭代方法来解决ALM中的原始子问题，因此我们还提出了一种早期停止标准，该标准易于在嵌入式平台上实施，可用于非强凸的问题，并保证原始更新的精度。据我们所知，这是第一个可以处理非平滑问题的定点ALM，数据溢出，并且可以在原始更新中高效，系统地利用迭代求解器。进行了逻辑回归问题的数值模拟研究，说明了所提出的方法。