TY - JOUR

T1 - Steepest descent evolution equations

T2 - Asymptotic behavior of solutions and rate of convergence

AU - Cominetti, R.

AU - Alemany, O.

PY - 1999

Y1 - 1999

N2 - We study the asymptotic behavior of the solutions of evolution equations of the form u(t) ε -∂f(u(t),r(t)), where f(·,r) is a one-parameter family of approximations of a convex function f(·) we wish to minimize. We investigate sufficient conditions on the parametrization r(t) ensuring that the integral curves u(t) converge when t → ∞ towards a particular minimizer U∞, of J. The speed of convergence is also investigated, and a result concerning the continuity of the limit point u∞ with respect to the parametrization r(·) is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.

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

KW - Convex optimization

KW - Dissipative evolution equations

KW - Penalty and viscosity methods

KW - Steepest descent

UR - http://www.scopus.com/inward/record.url?scp=22844453750&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-99-02508-8

DO - 10.1090/s0002-9947-99-02508-8

M3 - Article

AN - SCOPUS:22844453750

SN - 0002-9947

VL - 351

SP - 4847

EP - 4860

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

ER -