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A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex... Abstract. We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by \( \nabla^2 f(u(t),\epsilon(t))\dot u(t)+\dot\epsilon(t)\frac{\partial^2 f}{\partial\epsilon\,\partial x}(u(t),\epsilon(t))+\nabla f(u(t),\epsilon(t))=0, \) where \(\{f(\cdot,\epsilon)\}_{\epsilon>0}\) is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method: \(u(t)&\in&\mathop{\rm Argmin} \{f(x,\epsilon(t))-e^{-t}\langle\nabla f(u_0,\epsilon_0),x\rangle : x \in H\} ,\leqno{$(P^\epsilon_t)$}\) where H is a real Hilbert space. We find conditions on the approximating family \(f(\cdot,\epsilon)\) 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

A Dynamical System Associated with Newton's Method for Parametric Approximations of Convex Minimization Problems

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Publisher
Springer Journals
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/s002459900088
Publisher site
See Article on Publisher Site

Abstract

Abstract. We study the existence and asymptotic convergence when t→+∞ for the trajectories generated by \( \nabla^2 f(u(t),\epsilon(t))\dot u(t)+\dot\epsilon(t)\frac{\partial^2 f}{\partial\epsilon\,\partial x}(u(t),\epsilon(t))+\nabla f(u(t),\epsilon(t))=0, \) where \(\{f(\cdot,\epsilon)\}_{\epsilon>0}\) is a parametric family of convex functions which approximates a given convex function f we want to minimize, and ε(t) is a parametrization such that ε(t)→ 0 when t→+∞ . This method is obtained from the following variational characterization of Newton's method: \(u(t)&\in&\mathop{\rm Argmin} \{f(x,\epsilon(t))-e^{-t}\langle\nabla f(u_0,\epsilon_0),x\rangle : x \in H\} ,\leqno{$(P^\epsilon_t)$}\) where H is a real Hilbert space. We find conditions on the approximating family \(f(\cdot,\epsilon)\) 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.

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Oct 1, 1998

Keywords: Key words. Convex minimization, Approximate methods, Continuous methods, Evolution equations, Existence, Optimal trajectory, Asymptotic analysis. AMS Classification. 34G20, 34A12, 34D05, 90C25, 90C31.

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