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Global ellipsoidal approximations and homotopy methods for solving convex analytic programs

Global ellipsoidal approximations and homotopy methods for solving convex analytic programs This paper deals with some problems of algorithmic complexity arising when solving convex programming problems by following the path of analytic centers (i.e., the trajectory formed by the minimizers of the logarithmic barrier function). We prove that in the case ofm convex quadratic constraints we can obtain in a simple constructive way a two-sided ellipsoidal approximation for the feasible set (intersection ofm ellipsoids), whose tightness depends only onm. This can be used for the early identification of those constraints which are active at the optimum, and it also explains the efficiency of Newton's method used as a corrector when following the central path. Various parametrizations of the central path are studied. This also leads to an extrapolation (predictor) algorithm which can be regarded as a generalization of the method of conjugate gradients. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Global ellipsoidal approximations and homotopy methods for solving convex analytic programs

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References (27)

Publisher
Springer Journals
Copyright
Copyright © 1990 by Springer-Verlag New York Inc.
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/BF01445161
Publisher site
See Article on Publisher Site

Abstract

This paper deals with some problems of algorithmic complexity arising when solving convex programming problems by following the path of analytic centers (i.e., the trajectory formed by the minimizers of the logarithmic barrier function). We prove that in the case ofm convex quadratic constraints we can obtain in a simple constructive way a two-sided ellipsoidal approximation for the feasible set (intersection ofm ellipsoids), whose tightness depends only onm. This can be used for the early identification of those constraints which are active at the optimum, and it also explains the efficiency of Newton's method used as a corrector when following the central path. Various parametrizations of the central path are studied. This also leads to an extrapolation (predictor) algorithm which can be regarded as a generalization of the method of conjugate gradients.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 23, 2005

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