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A Primal-Dual Augmented Lagrangian

Philip E. Gill (pgill***at***ucsd.edu)
Daniel P. Robinson (daniel.robinson***at***comlab.ox.ac.uk)

Abstract: Nonlinearly constrained optimization problems can be solved by minimizing a sequence of simpler unconstrained or linearly constrained subproblems. In this paper, we discuss the formulation of subproblems in which the objective is a primal-dual generalization of the Hestenes-Powell augmented Lagrangian function. This generalization has the crucial feature that it is minimized with respect to both the primal and the dual variables simultaneously. A benefit of this approach is that the quality of the dual variables is monitored explicitly during the solution of the subproblem. Moreover, each subproblem can be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of conventional primal methods are proposed: a primal-dual bound constrained Lagrangian method and a primal-dual l_1 linearly constrained Lagrangian method.

Keywords: Nonlinear programming, nonlinear inequality constraints, augmented Lagrangian methods, bound constrained Lagrangian methods, linearly constrained Lagrangian methods, primal-dual methods.

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: UC San Diego Department of Mathematics Technical Report NA-08-02, April 2008

Download: [Postscript][PDF]

Entry Submitted: 05/01/2008
Entry Accepted: 05/01/2008
Entry Last Modified: 05/18/2008

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