A dynamical approach to convex minimization coupling approximation with the steepest descent method

We study the asymptotic behavior of the solutions to evolution equations of the form 0 ∈u̇(t) + ∂f(u(t),ε(t)); u(0) = u₀, where {f(·,ε):ε>0} is a family of strictly convex functions whose minimum is attained at a unique point x(ε). Assuming that x(ε) converges to a point x* as ε tends to 0, and depending on the behavior of the optimal trajectory x(ε), we derive sufficient conditions on the parametrization ε(t) which ensure that the solution u(t) of the evolution equation also converges to x* when t→ + ∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization.