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A polynomial case of convex integer quadratic programming problems with box integer constraints

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  • Chunli Liu
  • Jianjun Gao

Abstract

In this paper, we study a special class of convex quadratic integer programming problem with box constraints. By using the decomposition approach, we propose a fixed parameter polynomial time algorithm for such a class of problems. Given a problem with size $$n$$ n being the number of decision variables and $$m$$ m being the possible integer values of each decision variable, if the $$n-k$$ n - k largest eigenvalues of the quadratic coefficient matrix in the objective function are identical for some $$k$$ k $$(0>k>n)$$ ( 0 > k > n ) , we can construct a solution algorithm with a computational complexity of $${\mathcal {O}}((mn)^k)$$ O ( ( m n ) k ) . To achieve such complexity, we decompose the original problem into several convex quadratic programming problems, where the total number of the subproblems is bounded by the number of cells generated by a set of hyperplane arrangements in $$\mathbb {R}^k$$ R k space, which can be efficiently identified by cell enumeration algorithm. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Chunli Liu & Jianjun Gao, 2015. "A polynomial case of convex integer quadratic programming problems with box integer constraints," Journal of Global Optimization, Springer, vol. 62(4), pages 661-674, August.
  • Handle: RePEc:spr:jglopt:v:62:y:2015:i:4:p:661-674
    DOI: 10.1007/s10898-014-0263-2
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    References listed on IDEAS

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    1. Duan Li & Xiaoling Sun & Shenshen Gu & Jianjun Gao & Chunli Liu, 2010. "Polynomially Solvable Cases of Binary Quadratic Programs," Springer Optimization and Its Applications, in: Altannar Chinchuluun & Panos M. Pardalos & Rentsen Enkhbat & Ider Tseveendorj (ed.), Optimization and Optimal Control, pages 199-225, Springer.
    2. Z. Wu & G. Li & J. Quan, 2011. "Global optimality conditions and optimization methods for quadratic integer programming problems," Journal of Global Optimization, Springer, vol. 51(3), pages 549-568, November.
    3. J. M. W. Rhys, 1970. "A Selection Problem of Shared Fixed Costs and Network Flows," Management Science, INFORMS, vol. 17(3), pages 200-207, November.
    4. X. Sun & C. Liu & D. Li & J. Gao, 2012. "On duality gap in binary quadratic programming," Journal of Global Optimization, Springer, vol. 53(2), pages 255-269, June.
    5. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, April.
    6. Gupta, Renu & Bandopadhyaya, Lakshmisree & Puri, M. C., 1996. "Ranking in quadratic integer programming problems," European Journal of Operational Research, Elsevier, vol. 95(1), pages 231-236, November.
    7. Jianjun Gao & Duan Li, 2013. "A polynomial case of the cardinality-constrained quadratic optimization problem," Journal of Global Optimization, Springer, vol. 56(4), pages 1441-1455, August.
    8. Steven Cosares & Dorit S. Hochbaum, 1994. "Strongly Polynomial Algorithms for the Quadratic Transportation Problem with a Fixed Number of Sources," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 94-111, February.
    9. Vassilev, Vassil & Genova, Krassimira, 1994. "An approximate algorithm for nonlinear integer programming," European Journal of Operational Research, Elsevier, vol. 74(1), pages 170-178, April.
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