Open Journal of Applied Sciences, 2013, 3, 70-73

doi:10.4236/ojapps.2013.31B1014 Published Online April 2013 (http://www.scirp.org/journal/ojapps)

A Modified Augemented Lagrangian Method for a Class of

Nonlinear Ill-Posed Problems

Mhbm Shariff

Department of Applied Mathematics and Sciences, Khalifa University, Sharjah, UAE

Email: shariff@kustar.ac.ae

Received 2013

ABSTRACT

A class of nonlinear problems with real parameters is defined. Generally, in this class of problems, when the parametric

values are very large, the problems become ill-posed and numerical difficulties are encountered when trying to solve

these problems. In this paper, the nonlinear problems are reformulated to overcome the numerical difficulties associated

with large parametric values. A novel iterative algorithm, which is suitable for large scale problems and can be easily

parallelized, is proposed to solve the reformulated problems. Numerical tests indicate that the proposed algorithm gives

stable solutions. Convergence properties of the proposed algorithm are investigated. In the limiting case, when the cor-

responding constraint is exactly satisfied, the proposed method is equivalent to the standard augmented Lagrangian

method.

Keywords: Iterative Method; Nonlinear; Ill-conditioned; Large Parameter Values; Large-scale

1. Introduction

There are a number of physical problems, where penalty

terms (large parameter) occur naturally in the problems.

For example, in nonlinear isotropic elasticity, the bulk

modulus [1] can be considered as a penalty term for the

incompressible constraint. When the penalty term is very

large the corresponding constraint is nearly satisfied.

Generally, numerical difficulties are encountered, see,

e.g., [2], when trying to solve problems with very large

penalty values.

In this paper we consider a class of parametric prob-

lems that can be reformulated in such a way that the nu-

merical difficulties mentioned above can be overcome.

The reformulated problem generally yields indefinite

system of equations. When such a system is solved using

an existing iterative method, its convergence properties

are generally not as good as an iterative method design

for a symmetric positive definite system. Here, we pro-

posed a novel iterative algorithm to solve the reformu-

lated problem. The proposed algorithm solves a symmet-

ric positive definite system in each iteration. The con-

vergence properties of the proposed algorithm are inves-

tigated and numerical tests are implemented. The algo-

rithm is an extension of the algorithm developed by

Shariff [3] for quadratic problems with (near) linear con-

straints. Our proposed method is suitable for large scale

problems in the sense that it only uses a handful of vec-

tors in the algorithm. The proposed algorithm is easily

parallelized and is especially suitable for sparse system

of equations.

2. A Class of Nonlinear Constrained

Problems

Consider a class of nonlinear problems which is of the

form

Problem (I): Minimize

1

()( ()),

m

i

i

fx εχhx

where

is a real constant (can be considered as a pen-

alty term), 1

and :RR

with the properties

(a)

is twice continuously differentiable and strictly

convex on R

(b) '

(0)0, (0)0

limlim( )l

and

'' (0) 1

''

im( )

(c) .im l

tt

tt t

Examples of the function t

()t

.

are:

2

9

() [2],

2

()cosh()1 [2],

1

ln( 1)(( 1)1)

9

() [4],

9

()ln(1) [5].

t

t

tt

tt

t

tt t

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