Mathematical Programming

, Volume 141, Issue 1–2, pp 319–348

A practical relative error criterion for augmented Lagrangians

Full Length Paper Series A

DOI: 10.1007/s10107-012-0528-9

Cite this article as:
Eckstein, J. & Silva, P.J.S. Math. Program. (2013) 141: 319. doi:10.1007/s10107-012-0528-9


This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical since it is readily testable given only a gradient (or subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for proximal point algorithms: in particular, it uses a single relative tolerance parameter, rather than a summable parameter sequence. Our analysis first describes an abstract version of the criterion within Rockafellar’s general parametric convex duality framework, and proves a global convergence result for the resulting algorithm. Specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented Lagrangian method with a novel inexact solution condition for the subproblems. Finally, we present computational results drawn from the CUTE test set—including many nonconvex problems—indicating that the approach works well in practice.

Mathematics Subject Classification

90C25 90C30 

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  1. 1.Department of Management Science and Information Systems and RUTCORRutgers UniversityPiscatawayUSA
  2. 2.Department of Computer ScienceUniversity of São PauloSão PauloBrazil

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