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A DYNAMICAL SYSTEM ASSOCIATED WITH NEWTON'S METHOD FOR PARAMETRIC APPROXIMATIONS OF CONVEX
 

Summary: A DYNAMICAL SYSTEM ASSOCIATED WITH NEWTON'S
METHOD FOR PARAMETRIC APPROXIMATIONS OF CONVEX
MINIMIZATION PROBLEMS
F. ALVAREZ D. AND J.M. P´EREZ C.
Abstract. We study the existence and asymptotic convergence when t + for the trajectories
generated by
2
f(u(t), (t)) u(t) + (t)
2f
x
(u(t), (t)) + f(u(t), (t)) = 0
where {f(·, )} >0 is a parametric family of convex functions which approximates a given convex
function f we want to minimize, and (t) is a parametrization such that (t) 0 when t +.
This method is obtained from the following variational characterization of Newton's method
(Pt ) u(t) Argmin{f(x, (t)) - e-t
f(u0, 0), x : x H}
where H is a real Hilbert space. We find conditions on the approximating family f(·, ) and the
parametrization (t) to ensure the norm convergence of the solution trajectories u(t) towards a par-
ticular minimizer of f. The asymptotic estimates obtained allow us to study the rate of convergence
as well. The results are illustrated through some applications to barrier and penalty methods for

  

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

 

Collections: Mathematics; Engineering