On R-linear convergence of semi-monotonic inexact augmented Lagrangians for bound and equality constrained quadratic programming problems with application

https://doi.org/10.1016/j.camwa.2013.11.009Get rights and content
Under an Elsevier user license
open archive

Abstract

New convergence results for a variant of the inexact augmented Lagrangian algorithm SMALBE [Z. Dostál, An optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum, Computing 78 (2006) 311–328] for the solution of strictly convex bound and equality constrained quadratic programming problems are presented. The algorithm SMALBE-M presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary bound constrained problems by a multiple of the norm of violation of the equality constraints and a constant which is updated in order to enforce the increase of Lagrangian function. A nice feature of SMALBE-M is its capability to find an approximate solution of important classes of problems in a number of iterations that is independent of the conditioning of the equality constraints. Here we prove the R-linear rate of convergence of the outer loop of SMALBE-M for any positive regularization parameter after the strong active constraints of the solution are identified. The theoretical results are illustrated by solving two benchmarks, including the contact problem of elasticity discretized by two million of nodal variables. The numerical experiments indicate that the inexact solution of auxiliary problems in the inner loop results in a very small increase of the number of outer iterations as compared with the exact algorithm. The results do not assume independent equality constraints and remain valid when the solution is dual degenerate.

Keywords

Quadratic programming
Bound and equality constraints
Inexact augmented Lagrangians
Adaptive precision control
Error bounds
Contact problem

Cited by (0)

This paper was supported by the IT4Innovations Center of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070 supported by Operational Programme ‘Research and Development for Innovations’ funded by the Structural Funds of the European Union and the budget of the Czech Republic and by the Ministry of Education of the Czech Republic under contract No. MSM6198910027.