Steepest Descent with Curvature Dynamical System

Let H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a function that is bounded from below, we consider the following dynamical system:where (x) corresponds to a quadratic approximation to a linear search technique in the direction –(x). The term (x) is connected intimately with the normal curvature radius (x) in the direction (x). The remarkable property of (SDC) lies in the fact that the gradient norm |(x(t))| decreases exponentially to zero when t+.When is a convex function which is nonsmooth or lacks strong convexity, we consider a parametric family {, >0} of smooth strongly convex approximations of and we couple this approximation scheme with the (SDC) system. More precisely, we are interested in the following dynamical system:where (t, x) is a time-dependent function involving a curvature term. We find conditions on the approximating family and on () ensuring the asymptotic convergence of the solution trajectories x() toward a particular solution of the problem min {(x), xH}. Applications to barrier and penalty methods in linear programming and to viscosity methods are given.