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Canonical Duality Theory pp 315–338Cite as

Global Optimal Solution to Quadratic Discrete Programming Problem with Inequality Constraints

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Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 37))

Abstract

This paper presents a canonical dual method for solving a quadratic discrete value selection problem subjected to inequality constraints. By using a linear transformation, the problem is first reformulated as a standard quadratic 0–1 integer programming problem. Then, by the canonical duality theory, this challenging problem is converted to a concave maximization over a convex feasible set in continuous space. It is proved that if this canonical dual problem has a solution in its feasible space, the corresponding global solution to the primal problem can be obtained directly by a general analytical form. Otherwise, the problem could be NP-hard. In this case, a quadratic perturbation method and an associated canonical primal-dual algorithm are proposed. Numerical examples are illustrated to demonstrate the efficiency of the proposed method and algorithm.

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Notes

  1. 1.

    The inequality \(\det \mathbf{G}(\varvec{\varsigma }) \ne 0\) is not a constraint since the Lagrange multiplier for this inequality is identical zero.

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Acknowledgements

The research is supported by US Air Force Office of Scientific Research under grants FA2386-16-1-4082 and FA9550-17-1-0151.

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Authors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Ballarat, VIC, 3353, Australia

    Ning Ruan & David Yang Gao

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  1. Ning Ruan
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  2. David Yang Gao
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Correspondence to Ning Ruan .

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Editors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    David Yang Gao

  2. Department of Computer, Control, and Managment Engineering, Sapienza University of Rome, Rome, Roma, Italy

    Vittorio Latorre

  3. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    Ning Ruan

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Ruan, N., Gao, D.Y. (2017). Global Optimal Solution to Quadratic Discrete Programming Problem with Inequality Constraints. In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_16

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