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 u(t)+∂f(u(t), (t)); u(0)=u0, where {f(•, ): >0} is a family of strictly convex functions whose minimum is attained at a unique pointx( ). Assuming thatx( ) converges to a pointx* as tends to 0, and depending on the behavior of the optimal trajectoryx( ), we derive sufficient conditions on the parametrization (t) which ensure that the solutionu(t) of the evolution equation also converges tox* whent→+∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization.