 
Summary: Steepest Descent with
Curvature Dynamical System1,2
F. ALVAREZ
3
AND A. CABOT
4
Communicated by G. Di Pillo
Abstract. Let H be a real Hilbert space and let Æ.,.æ denote the corre
sponding scalar product. Given a C2
function F: HfiR that is bounded
from below, we consider the following dynamical system:
(SDC) _xx(t) + l(x(t))rF(x(t)) = 0, t$0,
where l(x) corresponds to a quadratic approximation to a linear search
technique in the direction rF(x). The term l(x) is connected inti
mately with the normal curvature radius r(x) in the direction rF(x). The
remarkable property of (SDC) lies in the fact that the gradient norm
rF(x(t)) decreases exponentially to zero when tfi+O.
When F is a convex function which is nonsmooth or lacks strong
convexity, we consider a parametric family {Fe, e >0} of smooth strongly
convex approximations of F and we couple this approximation scheme
