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

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.