A dynamical system associated with Newton's method for parametric approximations of convex minimization problems

• Published in: Applied mathematics & optimization Vol. 38; no. 2; pp. 193 - 217
• Main Authors: ALVAREZ D, F, PEREC C, J. M
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01-09-1998
• Subjects: Applied sciences; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; Exact sciences and technology; FUNCTIONS; HILBERT SPACE; LINEAR PROGRAMMING; Mathematical programming; MINIMIZATION; NEWTON METHOD; Operational research and scientific management; Operational research. Management science; VARIATIONAL METHODS; Existence condition; Approximate method; Barrier function; Optimal trajectory; Evolution equation; Linear programming; Asymptotic convergence; Newton method; Convex programming; Penalty method; Optimization
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s002459900088

We study the existence and asymptotic convergence when t{sup {yields}}+{infinity} for the trajectories generated by {nabla}{sup 2}f(u(t),{epsilon}(t))u-dot(t) + {epsilon}-dot(t) {partial_derivative}{sup 2}f/({partial_derivative}{epsilon}{partial_derivative}x) (u(t),{epsilon}(t)) + {nabla}f(u(t),{epsilon}(t)) = 0, where {l_brace}f(c-dot,{epsilon}{r_brace}{sub {l_brace}}{sub {epsilon}}{sub >0{r_brace}} is a parametric family of convex functions which approximates a given convex function f we want to minimize, and {epsilon}(t) is a parametrization such that {epsilon}(t){sup {yields}} 0 when t{sup {yields}}+{infinity} . This method is obtained from the following variational characterization of Newton's method: u(t) element of Argmin{l_brace}f(x,{epsilon}(t))-e{sup -t}<{nabla}f(u{sub 0},{epsilon}{sub 0}),x>: x element of H{r_brace}, (P{sub t}{sup {epsilon}})where H is a real Hilbert space. We find conditions on the approximating family f(.,{epsilon}) and the parametrization {epsilon}(t) to ensure the norm convergence of the solution trajectories u(t) toward a particular 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 linear programming, and to viscosity methods for an abstract noncoercive variational problem. Comparisons with the steepest descent method are also provided.