 
Summary: Absolute minimizer in convex programming by
exponential penalty
F. Alvarez
Abstract
We consider a nonlinear convex program. Under some general hy
potheses, we prove that approximate solutions obtained by exponential
penalty converge toward a particular solution of the original convex
program as the penalty parameter goes to zero. This particular solu
tion is called the absolute minimizer and is characterized as the unique
solution of a hierarchical scheme of minimax problems.
Keywords. Convexity, minimax problems, penalty methods, nonunique
ness, optimal trajectory, convergence.
AMS 1991 subject classifications. 90C25, 90C31.
1 Introduction
Let us consider a mathematical program of the type:
(P) min
xIRn
{f0(x)  fi(x) 0, i = 1, ..., m} ,
where for each i = 0, ..., m, fi is a convex function. The exponential penalty
method consists in solving for r > 0 small enough the unconstrained problem
