The Linear and Asymptotically Superlinear Convergence Rates of the Augmented Lagrangian Method with a Practical Relative Error Criterion

Xin-Yuan Zhao () and Liang Chen
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Xin-Yuan Zhao: College of Applied Sciences, Beijing University of Technology, Beijing 100022, P. R. China
Liang Chen: School of Mathematics, Hunan University, Changsha 410082, P. R. China

Asia-Pacific Journal of Operational Research (APJOR), 2020, vol. 37, issue 04, 1-16

Abstract: In this paper, we conduct a convergence rate analysis of the augmented Lagrangian method with a practical relative error criterion designed in Eckstein and Silva [Mathematical Programming, 141, 319–348 (2013)] for convex nonlinear programming problems. We show that under a mild local error bound condition, this method admits locally a Q-linear rate of convergence. More importantly, we show that the modulus of the convergence rate is inversely proportional to the penalty parameter. That is, an asymptotically superlinear convergence is obtained if the penalty parameter used in the algorithm is increasing to infinity, or an arbitrarily Q-linear rate of convergence can be guaranteed if the penalty parameter is fixed but it is sufficiently large. Besides, as a byproduct, the convergence, as well as the convergence rate, of the distance from the primal sequence to the solution set of the problem is obtained.

Keywords: Augmented Lagrangian method; relative error criterion; convergence rate (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1142/S0217595920400011

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