Inexact Semimonotonic Augmented Lagrangians with Optimal Feasibility Convergence for Convex Bound and Equality Constrained Quadratic Programming

Abstract

A variant of the augmented Lagrangian-type algorithm for strictly convex quadratic programming problems with bounds and equality constraints is considered. The algorithm exploits the adaptive precision control in the solution of auxiliary bound constraint problems in the inner loop while the Lagrange multipliers for the equality constraints are updated in the outer loop. The update rule for the penalty parameter is introduced that depends on the increase of the augmented Lagrangian. Global convergence is proved and an explicit bound on the penalty parameter is given. A qualitatively new feature of our algorithm is a bound on the feasibility error that is independent of conditioning of the constraints. When applied to the class of problems with the spectrum of the Hessian matrix in a given interval, the algorithm returns the solution in O(1) matrix-vector multiplications. The results are valid even for linearly dependent constraints. Theoretical results are illustrated by numerical experiments including the solution of an elliptic variational inequality.