Title: Lagrangian Transformation in Convex Optimization


		         Roman A.Polyak
		 Department of System Engineering
		      and Operations Research &
		Mathematical Sciences Department,
		   George  Mason University,
		         Fairfax VA 22030
		         rpolyak@gmu.edu
		  http://mason.gmu.edu/~rpolyak/


Abstract:          
We consider a class of strictly concave and smooth enough
scalar functions of a scalar argument with particular properties.
In contrast to the Nonlinear Rescaling theory (see[1])we used the
functions to transform the terms of the Classical Lagrangian rather
then constraints. The transformation is scaled by a vector of scaling
parameters one for each constraint.
The Lagrangian Transformation(LT) has several special properties, which
we used to develop a general LT multipliers method.
Each step of the LT method consists of unconstrained minimization of
the LT in primal space following by both the vector of Lagrange
multipliers and the scaling parameters vector update.
The scaling parameters are updated inverse proportional to the square
of the Lagrange multipliers. Due to the special way we update the
Lagrange multipliers and the scaling parameters the LT method has some
very important properties.
We will discuss the primal ,dual as well as primal-dual aspects of the
LT method.

1.R.Polyak "Nonlinear Rescaling vs. smoothing technique in convex
optimization" Mathematical Programming ,series A vol 92 n2 pp198-235.