Some Augmented Lagrangian Algorithms applied to convex nondiff erentiable optimization problems

摘要

The penalty methods constitute a family of particularly interesting algorithms of the two points of view: the simpleness of principle and the practical efficiency. They are considered as ill-conditioned because the penalty parameter r_k tends to infinity, as k tends infinity. It is what led to us to introduce what we call augmented Lagrangian to regularize the solutions. Namely, to avoid the numerical instability of the classical penalty method. Another favors of the augmented Lagrangian is that by presence of the term λ_i, g_i, the exact solution of the problem can be determined without making aim r_k towards the infinity, contrary to the penalty method, where it has the effect of diverting the packaging of the problem to be resolved. The using of augmented Lagrangian is considered as an improvement of the penalty methods. It avoids having to use too big parameters of penalties. Besides, the fact of adding the quadratic term r_k(g~+)~2 in the Lagrangian will improve the properties of convergence of the algorithms of duality in this paper. In this paper, we study some augmented Lagrangian algorithms applied to convex nondifferentiable optimization problems and prove their convergence.