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Canonical Dual Solutions for Fixed Cost Quadratic Programs

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Optimization and Optimal Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 39))

Summary

This chapter presents a canonical dual approach for solving a mixed-integer quadratic minimization problem with fixed cost terms. We show that this well-known NP-hard problem in \(\mathbb{R}^{2n}\) can be transformed into a continuous concave maximization dual problem over a convex feasible subset of \(\mathbb{R}^{2n}\) with zero duality gap. The resulting canonical dual problem can be solved easily, under certain conditions, by traditional convex programming methods. Both existence and uniqueness of global optimal solutions are discussed. Application to a decoupled mixed-integer problem is illustrated and analytic solutions for both a global minimizer and a global maximizer are obtained. Examples for both decoupled and general nonconvex problems are presented. Furthermore, we discuss connections between the proposed canonical duality theory approach and the classical Lagrangian duality approach. An open problem is proposed for future study.

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Author information

Authors and Affiliations

  1. Graduate School of Information Technology and Mathematical Sciences, University of Ballarat, Mt Helen, Victoria, 3353, Australia

    David Yang Gao

  2. Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    Ning Ruan & Hanif D. Sherali

Authors
  1. David Yang Gao
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  2. Ning Ruan
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  3. Hanif D. Sherali
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Corresponding author

Correspondence to David Yang Gao .

Editor information

Editors and Affiliations

  1. South Kensington Campus, London, SW7 2AZ, United Kingdom

    Altannar Chinchuluun

  2. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  3. The School of Economic Studies, Dept.of Mathematics & Computer Science, National University of Mongolia, Ulaanbaatar, 210646, Mongolia

    Rentsen Enkhbat

  4. Labo. PRISM, Université Versailles, av. des Etats-Unis 45, Versailles CX, 78035, France

    Ider Tseveendorj

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Gao, D.Y., Ruan, N., Sherali, H.D. (2010). Canonical Dual Solutions for Fixed Cost Quadratic Programs. In: Chinchuluun, ., Pardalos, P., Enkhbat, R., Tseveendorj, I. (eds) Optimization and Optimal Control. Springer Optimization and Its Applications(), vol 39. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89496-6_7

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