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Finding normal solutions in piecewise linear programming

Finding normal solutions in piecewise linear programming Letf: ℝn → (−∞, ∞] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min x ¦x¦2/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} t ≥0, minimizingf over an ellipsoid, and choosing penalty parameters inL 1QP methods for strictly convex QP. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Finding normal solutions in piecewise linear programming

Kiwiel, K.
Applied Mathematics and Optimization , Volume 32 (3) – Feb 3, 2005
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References (50)

Publisher
Springer Journals
Copyright
Copyright © 1995 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/BF01187901
Publisher site
See Article on Publisher Site

Abstract

Letf: ℝn → (−∞, ∞] be a convex polyhedral function. We show that if any standard active set method for quadratic programming (QP) findsx(t)= arg min x ¦x¦2/2+t f(x) for somet> 0, then its final working set defines a simple equality QP subproblem, whose Lagrange multiplier can be used both for testing ift is large enough forx(t) to coincide with the normal minimizer off, and for increasingt otherwise. The QP subproblem may easily be solved via the matrix factorizations used for findingx(t). This opens up the way for efficient implementations. We also give finite methods for computing the whole trajectory {x(t)} t ≥0, minimizingf over an ellipsoid, and choosing penalty parameters inL 1QP methods for strictly convex QP.

Journal

Applied Mathematics and Optimization – Springer Journals

Published: Feb 3, 2005

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