Nondifferentiable augmented Lagrangian, \(\varepsilon\)-proximal penalty methods and applications

Abstract

The purpose of this work is to prove results concerning the duality theory and to give detailed study on the augmented Lagrangian algorithms and \(\varepsilon\)-proximal penalty method which are considered, today, as the most strong algorithms to solve nonlinear differentiable and nondifferentiable problems of optimization. We give an algorithm of primal-dual type, where we show that sequences \(\left\{\lambda^k\right\}_k\) and \(\left\{x^k\right\}_k\) generated by this algorithm converge globally, with at least the Slater condition, to \(\bar{\lambda}\) and \(\bar{x}\). Numerical simulations are given.