Strict convex regularizations, proximal points and augmented lagrangians

¹ Departemento de Ciência da Computação, Instituto de Matemática e Estatística – Usp, rua de Matao 1010, Cidade Universitaria, CEP 05315-970 Sao Paulo, SP, Brazil.
² Research of this author is suported by CNPq and PRONEX, Convênio 76.97.1008.00.
³ Departemento de Matemática Aplicada, Instituto de Matemática e Estatística – USP, rua de Matao 1010, Cidade Universitaria, CEP 05315-970 Sao Paulo, SP, Brazil.
⁴ Research of this author is suported by FAPESP - Processo 96/09939-0.

Received: February 1998

Abstract

Proximal Point Methods (PPM) can be traced to the pioneer works of Moreau [16], Martinet [14, 15] and Rockafellar [19, 20] who used as regularization function the square of the Euclidean norm. In this work, we study PPM in the context of optimization and we derive a class of such methods which contains Rockafellar's result. We also present a less stringent criterion to the acceptance of an approximate solution to the subproblems that arise in the inner loops of PPM. Moreover, we introduce a new family of augmented Lagrangian methods for convex constrained optimization, that generalizes the P_E⁺ class presented in [2].