Nonlinear programs (P) can be solved by embedding problem P into one parametric problem P(t), where P(1) and P are equivalent and P(0), has an evident solution. Some embeddings fulfill that the solutions of the corresponding problem P(t) can be interpreted as the points computed by the Augmented Lagrange Method on P. In this paper we study the Augmented Lagrangian embedding proposed in. | Yugoslav Journal of Operations Research Volume 20 (2010), Number 2, 183-196 DOI: A NOTE ON EMBEDDINGS FOR THE AUGMENTED LAGRANGE METHOD Gemayqzel BOUZA-ALLENDE Universidad de La Habana gema@ Jurgen GUDDAT Humboldt-Universitat zu Berlin guddat@ Received: November 2007 / Accepted: December 2010 Abstract: Nonlinear programs (P) can be solved by embedding problem P into one parametric problem P(t), where P(1) and P are equivalent and P(0), has an evident solution. Some embeddings fulfill that the solutions of the corresponding problem P(t) can be interpreted as the points computed by the Augmented Lagrange Method on P. In this paper we study the Augmented Lagrangian embedding proposed in [6]. Roughly speaking, we investigated the properties of the solutions of P(t) for generic nonlinear programs P with equality constraints and the characterization of P(t) for almost every quadratic perturbation on the objective function of P and linear on the functions defining the equality constraints. Keywords: Augmented Lagrangian Method, JJT-regular, generalized critical points, generic set. AMS Subject Classification: 90C31, 49M30. 1. INTRODUCTION We consider the well known nonlinear optimization problem: ( P) min f ( x), , x ∈ M (1) G., Bouza-Allende, J., Guddat / A Note on Embeddings 184 { M = x ∈ Rn hi ( x ) = 0,i =1,., m , g j ( x ) ≥ 0, j =1,., s. } f , h1 ,., hm , g1 ,., g s ∈ C k ( R n , R) Problem P can be solved by algorithms such as the barrier, the penalty and the Augmented Lagrangian method. However, the convergence can be guaranteed under strong assumptions. Since 1980, embedding methods have been proposed for solving nonlinear programming problems. This approach embeds P into one-parametric problem P(t) and applies a path-following on the set of solutions of P(t) for obtaining a solution of P. In order to have at least a local characterization of this curve, Jongen et al. have defined .
