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Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach

Author

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  • X. J. Zheng

    (Shanghai University
    Fudan University)

  • X. L. Sun

    (Fudan University)

  • D. Li

    (The Chinese University of Hong Kong)

Abstract

We present in this paper an integer diagonalization approach for deriving new lower bounds for general quadratic integer programming problems. More specifically, we introduce a semiunimodular transformation in order to diagonalize a symmetric matrix and preserve integral property of the feasible set at the same time. Via the semiunimodular transformation, the resulting separable quadratic integer program is a relaxation of the nonseparable quadratic integer program. We further define the integer diagonalization dual problem to identify the best semiunimodular transformation and analyze some basic properties of the set of semiunimodular transformations for a rational symmetric matrix. In particular, we present a complete characterization of the set of all semiunimodular transformations for a nonsingular 2×2 symmetric matrix. We finally discuss Lagrangian relaxation and convex relaxation schemes for the resulting separable quadratic integer programming problem and compare the tightness of different relaxation schemes.

Suggested Citation

  • X. J. Zheng & X. L. Sun & D. Li, 2010. "Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 463-489, August.
    • Handle: RePEc:spr:joptap:v:146:y:2010:i:2:d:10.1007_s10957-010-9653-x
    DOI: 10.1007/s10957-010-9653-x
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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. Loiola, Eliane Maria & de Abreu, Nair Maria Maia & Boaventura-Netto, Paulo Oswaldo & Hahn, Peter & Querido, Tania, 2007. "A survey for the quadratic assignment problem," European Journal of Operational Research, Elsevier, vol. 176(2), pages 657-690, January.
    3. de Klerk, E. & Pasechnik, D.V., 2005. "A Linear Programming Reformulation of the Standard Quadratic Optimization Problem," Discussion Paper 2005-24, Tilburg University, Center for Economic Research.
    4. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    5. Djerdjour, Mohamed, 1997. "An enumerative algorithm framework for a class of nonlinear integer programming problems," European Journal of Operational Research, Elsevier, vol. 101(1), pages 104-121, August.
    6. A. T. Phillips & J. B. Rosen, 1990. "Guaranteed ϵ‐approximate solution for indefinite quadratic global minimization," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(4), pages 499-514, August.
    7. D. Quadri & E. Soutif & P. Tolla, 2009. "Exact solution method to solve large scale integer quadratic multidimensional knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 17(2), pages 157-167, February.
    8. de Klerk, E. & Pasechnik, D.V., 2007. "A linear programming reformulation of the standard quadratic optimization problem," Other publications TiSEM c3e74115-b343-4a85-976b-8, Tilburg University, School of Economics and Management.
    9. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, July-Dece.
    10. 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.
    11. Duan Li & Xiaoling Sun & Jun Wang, 2006. "Optimal Lot Solution To Cardinality Constrained Mean–Variance Formulation For Portfolio Selection," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 83-101, January.
    12. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    13. Syam, Siddhartha S., 1998. "A dual ascent method for the portfolio selection problem with multiple constraints and linked proposals," European Journal of Operational Research, Elsevier, vol. 108(1), pages 196-207, July.
    14. Samuel Burer & Dieter Vandenbussche, 2009. "Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound," Computational Optimization and Applications, Springer, vol. 43(2), pages 181-195, June.
    15. B. Kalantari & J. B. Rosen, 1987. "An Algorithm for Global Minimization of Linearly Constrained Concave Quadratic Functions," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 544-561, August.
    16. Le Thi Hoai An & Pham Dinh Tao & Le Dung Muu, 1998. "A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 9-28, March.
    17. Caprara, Alberto, 2008. "Constrained 0-1 quadratic programming: Basic approaches and extensions," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1494-1503, June.
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    1. Eric Newby & M. Montaz Ali, 2016. "Transformation-Based Preprocessing for Mixed-Integer Quadratic Programs," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 1039-1045, March.
    2. Eric Newby & M. M. Ali, 2014. "A Note on Convex Reformulation Schemes for Mixed Integer Quadratic Programs," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 457-469, February.

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