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Indefinite linearized augmented Lagrangian method for convex programming with linear inequality constraints

Bingsheng He (hebma***at***nju.edu.cn)
Shengjie Xu (xsjnsu***at***163.com)
Jing Yuan (jyuan***at***xidian.edu.cn)

Abstract: The augmented Lagrangian method (ALM) is a benchmark for tackling the convex optimization problem with linear constraints; ALM and its variants for linearly equality-constrained convex minimization models have been well studied in the literatures. However, much less attention has been paid to ALM for efficiently solving the linearly inequality-constrained convex minimization model. In this paper, we exploit an enlightening reformulation of the most recent indefinite linearized (equality-constrained) ALM, and present a novel indefinite linearized ALM scheme for efficiently solving the convex optimization problem with linear inequality constraints. The proposed method enjoys great advantages, especially for large-scale optimization cases, in two folds mainly: first, it significantly simplifies the optimization of the challenging key subproblem of the classical ALM by employing its linearized reformulation, while keeping low complexity in computation; second, we prove that a smaller proximity regularization term is needed for convergence guarantee, which allows a bigger step-size and can largely reduce required iterations for convergence. Moreover, we establish an elegant global convergence theory of the proposed scheme upon its equivalent compact expression of prediction-correction, along with a worst-case $\mathcal{O}(1/N)$ convergence rate. Numerical results demonstrate that the proposed method can reach a faster converge rate for a higher numerical efficiency as the regularization term turns smaller, which confirms the theoretical results presented in this study.

Keywords: augmented Lagrangian method, convex programming, convergence analysis, inequality constraints, image segmentation

Category 1: Convex and Nonsmooth Optimization

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation:

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Entry Submitted: 05/10/2019
Entry Accepted: 05/10/2019
Entry Last Modified: 05/03/2021

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