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

Cominetti, R.; Alemany, O.

doi:10.1090/S0002-9947-99-02508-8

Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence
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by R. Cominetti and O. Alemany PDF

Trans. Amer. Math. Soc. 351 (1999), 4847-4860 Request permission

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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