Penalty and Augmented Lagrangian Methods to Solve Nonlinear Optimization Prob- lems
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Date
2016-06
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Addis Ababa University
Abstract
Penalty and augmented Lagrangian methods are procedures for approximating constrained
optimization problems by unconstrained optimization problems to and the approximate so-
lution of the given constrained problem. The idea of replacing a constrained optimization
problem by a sequence of unconstrained problems parametrized by a scalar parameter has
played a fundamental role in the formulation of algorithms. Even though, penalty meth-
ods are very natural and general; unfortunately, they suffer from a serious drawback: to
approximate well the solution to constrained problem, we have to work with large penalty
parameters, and this inevitably makes the problem of unconstrained minimization of the pe-
nalized objective very ill-conditioned. Their slow rates of convergence due to ill-conditioning
of the associated Hessian led researchers to pursue other approaches, augmented Lagrangian
methods. The main advantage of the augmented Lagrangian methods is that they allow
to approximate well the solution to a constrained problem by solutions of unconstrained
(and penalized) auxiliary problems without pushing the penalty parameter to inifnity; as a
result, the auxiliary problems remain reasonably conditioned even when we are seeking for
high-accuracy solutions. In this paper we will see how all this works.
Keywords: Unconstrained optimization, constrained optimization, penalty and barrier
methods, augmented Lagrangian methods.
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Keywords
Unconstrained Optimization, Constrained Optimization, Penalty and Barrier Methods, Augmented Lagrangian Methods