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Cuts and Semidefinite Relaxations for Nonconvex Quadratic Problems

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Generalized Convexity and Generalized Monotonicity

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 502))

Abstract

In this paper, we propose a positive semidefinite relaxation for indefinite quadratic problems with box constraints. We show that a cutting plane procedure can be applied for generating tighter positive semidefinite relaxations. Computational results indicate that the relative gap obtained by the relaxations is less than 1% when the number of variables is up to 100. We also show a branch and bound procedure for obtaining an exact solution of the indefinite problems. Our branching scheme exploits the special structure of box constraints as well as the bounding procedure in which a pair of primal and dual solutions of the associated positive semidefinite relaxation is incorporated.

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Author information

Authors and Affiliations

  1. Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8552, Japan

    Yasutoshi Yajima

  2. University of Florida, Gainesville, FL, 32611-6595, USA

    Motakuri V. Ramana & Panos M. Pardalos

Authors
  1. Yasutoshi Yajima
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  2. Motakuri V. Ramana
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  3. Panos M. Pardalos
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of the Aegean, 83200, Karlovassi/Samos, Greece

    Nicolas Hadjisavvas

  2. CODE and Department of Economics, Universitat Autonoma de Barcelona, 08193, Bellaterra, Spain

    Juan Enrique Martínez-Legaz

  3. Department of Mathematics Faculty of Sciences, Université de Pau, 64000, Pau, France

    Jean-Paul Penot

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© 2001 Springer-Verlag Berlin Heidelberg

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Yajima, Y., Ramana, M.V., Pardalos, P.M. (2001). Cuts and Semidefinite Relaxations for Nonconvex Quadratic Problems. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56645-5_3

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  • DOI: https://doi.org/10.1007/978-3-642-56645-5_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41806-1

  • Online ISBN: 978-3-642-56645-5

  • eBook Packages: Springer Book Archive

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