For nonconvex optimization problem with both equality and inequality constraints, we introduce a new augmented Lagrangian function and propose the corresponding multiplier algorithm. New iterative strategy on penalty parameter is presented. Different global convergence properties are established depending on whether the penalty parameter is bounded. Even if the iterative sequence {xk} is diverg...

Adaptive augmented Lagrangian methods: algorithms and practical numerical experience Frank E. Curtis, Nicholas I.M. Gould, Hao Jiang & Daniel P. Robinson a Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA b STFC-Rutherford Appleton Laboratory, Numerical Analysis Group, R18, Chilton, OX11 0QX, UK c Department of Applied Mathematics and Statistics, Johns Hop...

In this paper, we consider the linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose an inexact augmented Lagrangian (IAL) framework for solving the problem. The proposed IAL framework requires solving the augmented Lagrangian (AL) subproblem at each iteration less accurately than most of the exist...

The alternating direction of multipliers (ADMM) is a form of augmented Lagrangian algorithm that has experienced a renaissance in recent years due to its applicability to optimization problems arising from “big data” and image processing applications, and the relative ease with which it may be implemented in parallel and distributed computational environments. While it is easiest to describe th...

Tchebychev iteration may be used for acceleration convergence of an iterative algorithm to solve a general linear system equation. Associating it with the Uzawa method, we suggest a new iterative solution method for the Stokes problems. The new algorithm retains the simplicity and robustness of the Uzawa method. So it requires almost no additional cost of computation, in terms of storage or CPU...

We consider solving nonlinear programming problems using an augmented Lagrangian method that makes use of derivative-free generating set search to solve the subproblems. Our approach is based on the augmented Lagrangian framework of Andreani, Birgin, Mart́ınez, and Schuverdt which allows one to partition the set of constraints so that one subset can be left explicit, and thus treated directly wh...

This paper proposes an augmented Lagrangian Hopfield network (ALHN) for combined heat and power economic dispatch (CHPED) problem. The ALHN is the continuous Hopfield neural network based on augmented Lagrangian relaxation as its energy function. In the proposed ALHN, its energy function is augmented by Hopfield terms from Hopfield neural network and penalty factors from augmented Lagrangian re...

Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence...

Abstract. In this paper we present augmented Lagrangians for nonconvex minimization problems with equality constraints. We construct a dual problem with respect to the presented here Lagrangian, give the saddle point optimality conditions and obtain strong duality results. We use these results and modify the subgradient and cutting plane methods for solving the dual problem constructed. Algorit...

An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solvers. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.