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We study the asymptotic behavior of the solutions of evolution equations of the form _ u(t) 2 @f(u(t) ;r ( t)), where f(;r )i s ao ne-parameter family of approximations of a convex function f() we wish to minimize. We investigate sucient conditions on the parametrization r(t) ensuring that the integral curves u(t) converge when t !1 towards a particular minimizer u1 of f. The speed of convergence is also investigated, and a result concerning the continuity of the limit point u1 with respect to the parametrization r() is established. The results are illustrated on dierent approximation methods. In par- ticular, we present a detailed application to the logarithmic barrier in linear programming.

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