Penalty and Augmented Lagrangian Methods to Solve Nonlinear Optimization Prob- lems

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Addis Ababa University


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.



Unconstrained Optimization, Constrained Optimization, Penalty and Barrier Methods, Augmented Lagrangian Methods