Um teorema de equivalencia entre metodos lagrangeano aumentado e algoritmos de pontos proximais

Abstract

Apresentase uma prova geral do Teorema de Equivalencia que relaciona o Método de Ponto Proximal e o Método Lagrangeano Aumentado a qual inclui quase todos os casos existentes na literatura. Também mostramos novos resultados de unicidade a respeito da sequencia de multiplicadores (seqüência dual) gerada pelo algoritmo Lagrangeano Aumentado. Abstract : We present here a general proof of the well known Equivalence Theorem which relates the Proximal Point and the Augmented Lagrangian methods. This p.roof includes almost all the cases existent in the literature. Following a logical path we first do a detailed study of the Proximal Point method and its evolution since its introduction in 1970 until now when applied to the specific problem of minimizing a closed proper convex function. This subjet is very dispersed and so we have attempted to put order and simplicity in it by giving sufficient conditions for the Proximal Point algorithm to be well defined. Next we have described the Generalized Augmented Lagrangian method and we have proved the well-definiteness of the corresponding algorithm. Without making use of the relationship between the Proximal Point and the Augmented Lagrangian methods we have obtained new uniqueness results on the sequence of multipliers (dual sequence) generated by the Augmented Lagrangian algorithm. In order to prove the Theorem of Equivalence we defined the kernel to be used in the Proximal Point method as the summation of the conjugares of the penalties of the Augmented Lagrangian method and then we proved the well-definiteness of the associated Proximal Point algorithm when applied to the dual problem. Finally we proved that the sequences {µk} generated by each of the methods are indeed the same.