Augmented Lagrangian Optimization under Fixed-Point Arithmetic

TL;DR

This paper introduces a novel inexact Augmented Lagrangian Method tailored for convex and nonsmooth optimization problems with fixed-point arithmetic constraints, ensuring convergence and practical implementation on embedded systems.

Contribution

It presents the first fixed-point ALM capable of handling non-smooth problems, data overflow, and iterative primal updates with convergence guarantees.

Findings

The proposed method converges with proven rate bounds.

It effectively prevents data overflow via a projection operation.

Numerical results demonstrate practical utility on a utility maximization problem.

Abstract

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 utility maximization problem are presented that illustrate the proposed method.