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Absolute minimizer in convex programming by exponential penalty

Summary: Absolute minimizer in convex programming by
exponential penalty
F. Alvarez
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
{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


Source: Alvarez, Felipe - Departamento de Ingeniería Matemática, Universidad de Chile


Collections: Mathematics; Engineering