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.