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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Steepest descent evolution equations:
asymptotic behavior of solutions
and rate of convergence


Authors: R. Cominetti and O. Alemany
Journal: Trans. Amer. Math. Soc. 351 (1999), 4847-4860
MSC (1991): Primary 34C35, 34D05; Secondary 49M10, 49M30
DOI: https://doi.org/10.1090/S0002-9947-99-02508-8
Published electronically: August 30, 1999
MathSciNet review: 1675174
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the asymptotic behavior of the solutions of evolution equations of the form $\dot u(t)\in -\partial f(u(t),r(t))$, where $f(\cdot,r)$ is a one-parameter family of approximations of a convex function $f(\cdot)$ we wish to minimize. We investigate sufficient conditions on the parametrization $r(t)$ ensuring that the integral curves $u(t)$ converge when $t\rightarrow\infty$ towards a particular minimizer $u_\infty$ of $f$. The speed of convergence is also investigated, and a result concerning the continuity of the limit point $u_\infty$ with respect to the parametrization $r(\cdot)$ is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.


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Additional Information

R. Cominetti
Affiliation: Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile.
Email: rcominet@dim.uchile.cl

O. Alemany
Affiliation: Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile.

DOI: https://doi.org/10.1090/S0002-9947-99-02508-8
Keywords: Dissipative evolution equations, steepest descent, penalty and viscosity methods, convex optimization
Received by editor(s): February 5, 1997
Published electronically: August 30, 1999
Additional Notes: This work was completed while the first author was visiting Laboratoire d’Econometrie, Ecole Polytechnique, Paris. Partially supported by Comisión Nacional de Investigación Científica y Tecnológica de Chile under Fondecyt grant 1961131
Article copyright: © Copyright 1999 American Mathematical Society

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