Journal of Global Optimization 28: 153–173, 2004. 153
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
On augmented Lagrangians for Optimization
Problems with a Single Constraint
R. N. GASIMOV and A. M. RUBINOV
1
Department of Industrial Engineering, Osmangazi University, Bademlik 26030, Eski¸ sehir, Turkey;
E-mail: gasimovr@ogu.edu.tr;
2
SITMS, University of Ballarat Victoria 3353, Australia;
E-mail: a.rubinov@ballarat edu.au
(Received and accepted in revised form 21 January 2003)
Abstract. We examine augmented Lagrangians for optimization problems with a single (either
inequality or equality) constraint. We establish some links between augmented Lagrangians and
Lagrange-type functions and propose a new kind of Lagrange-type functions for a problem with
a single inequality constraint. Finally, we discuss a supergradient algorithm for calculating optimal
values of dual problems corresponding to some class of augmented Lagrangians.
Key words: Augmented Lagrangians, Lagrange-type functions, Supergradient method
1. Introduction
Classical Lagrange and penalty functions can be applied only for examination of
some special classes of constrained optimization problems. Some generalizations
of these functions have recently been studied. Currently there are two main types
of such a generalization. One of them is the augmented Lagrangian, which is based
on an augmentation of the classical Lagrange function by a certain augmenting
function (see [6, 12] and references therein).
The fundamental of the other approach to generalization of Lagrangians is a
nonlinear convolution of the objective and constraint functions (see [7, 9, 10, 12]
and references therein). Such a convolution leads to nonlinear Lagrange-type func-
tions. We establish some links between the two mentioned approaches.
It is well-known that each constrained optimization problem can be reformu-
lated as a problem with a single inequality-constraint. Many complicated con-
structions become much simpler and more understandable for single-constrained
problems. In this paper we examine the augmented Lagrangians with certain aug-
menting functions for problems with a single (either inequality or equality) con-
straint. In particular we study the so-called sharp Lagrangian [6] for such problems.
The simple structure of the sharp augmenting function z =z allows us to give
an explicit description of the sharp augmented Lagrangian. By using this result
we propose a new type of nonlinear Lagrangians for problems with an inequality
constraint, for which the dual function can be easily expressed through the dual