A DYNAMICAL-APPROACH TO CONVEX MINIMIZATION COUPLING APPROXIMATION WITH THE STEEPEST DESCENT METHOD

We study the asymptotic behavior of the solutions to evolution equatio ns of the form 0 is an element of u over dot (t) + partial derivative f(u(t),epsilon(t)); u(0) = u(0), where {f(., epsilon): epsilon > 0} is a family of strictly convex functions whose minimum is attained at a unique point x(epsilon). Assuming that x(epsilon) converges to a point x as epsilon tends to 0, and depending on the behavior of the optima l trajectory x(epsilon), we derive sufficient conditions on the parame trization epsilon(t) which ensure that the solution u(t) of the evolut ion equation also converges to x when t --> + infinity. The results a re illustrated on three different penalty and viscosity approximation methods for convex minimization. (C) 1996 Academic Press, Inc.