SIAM Journal on Numerical Analysis
Analysis and implementation of a dual algorithm for constrained optimization
Journal of Optimization Theory and Applications
Inexact and preconditioned Uzawa algorithms for saddle point problems
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Computational Optimization and Applications
A Globally Convergent Linearly Constrained Lagrangian Method for Nonlinear Optimization
SIAM Journal on Optimization
SIAM Journal on Scientific Computing
Optimal Quadratic Programming Algorithms: With Applications to Variational Inequalities
Optimal Quadratic Programming Algorithms: With Applications to Variational Inequalities
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New convergence results for a variant of the inexact augmented Lagrangian algorithm SMALBE [Z. Dostal, 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.