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Steepest Descent with Curvature Dynamical System1,2
 

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

  

Source: Alvarez, Felipe - Departamento de Ingeniería Matemática, Universidad de Chile

 

Collections: Mathematics; Engineering