A modified barrier–augmented Lagrangian method for constrained minimization

A modified barrier–augmented Lagrangian method for constrained minimization

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Article ID: iaor20003787
Country: Netherlands
Volume: 14
Issue: 1
Start Page Number: 55
End Page Number: 74
Publication Date: Jul 1999
Journal: Computational Optimization and Applications
Authors: , , ,
Keywords: lagrange multipliers
Abstract:

We present and analyze an interior–exterior augmented Lagrangian method for solving constrained optimization problems with both inequality and equality constraints. This method, the modified barrier–augmented Lagrangian (MBAL) method, is a combination of the modified barrier and the augmented Lagrangian methods. It is based on the MBAL function, which treats inequality constraints with a modified barrier term and equalities with an augmented Lagrangian term. The MBAL method alternatively minimizes the MBAL function in the primal space and updates the Lagrange multipliers. For a large enough fixed barrier-penalty parameter the MBAL method is shown to converge Q-linearly under the standard second-order optimality conditions. Q-superlinear convergence can be achieved by increasing the barrier-penalty parameter after each Lagrange multiplier update. We consider a dual problem that is based on the MBAL function. We prove a basic duality theorem for it and show that it has several important properties that fail to hold for the dual based on the classical Lagrangian.

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