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

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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 \(n\) being the number of decision variables and \(m\) being the possible integer values of each decision variable, if the \(n-k\) largest eigenvalues of the quadratic coefficient matrix in the objective function are identical for some \(k\) \((0<k<n)\), we can construct a solution algorithm with a computational complexity of \({\mathcal {O}}((mn)^k)\). 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 \(\mathbb {R}^k\) space, which can be efficiently identified by cell enumeration algorithm.

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References

  1. Allemand, K., Fukuda, K., Liebling, T.M., Steiner, E.: A polynomial case of unconstrained zero–one quadratic optimization. Math. Program. 91, 49–52 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Alperin, H., Nowak, I.: Lagrangian smothing heuristics for max-cut. J. Heuristics 11, 447–463 (2005)

    Article  Google Scholar 

  3. Avis, D., Fukuda, K.: Reverse search for enumeration. Discret. Appl. Math. 65, 21–46 (1996)

    Article  MathSciNet  Google Scholar 

  4. Barahona, F.: A solvable case of quadratic 01 programming. Discret. Appl. Math. 13, 23–26 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertsimas, D., Ye, Y.: Semidefinite relaxations, multivariate normal distributions, and order statistics. In: Du, D.Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, pp. 1–7. Kluwer, Dordrecht (1998)

    Google Scholar 

  6. Borosa, E., Hammera, P.L., Sunb, R., Tavares, G.: A max-flow approach to improved lower bounds for quadratic unconstrained binary optimization (QUBO). Discret. Optim. 5, 501–529 (2008)

    Article  Google Scholar 

  7. Buchheim, C., Caprara, A., Lodi, A.: An effective branch-and-bound algorithm for convex quadratic integer programming. Math. Program. 135, 369–395 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buchheim, C., De Santis, M., Palagi, L., et al.: An exact algorithm for nonconvex quadratic integer minimization using ellipsoidal relaxations. SIAM J. Optim. 23, 1867–1889 (2013)

    Article  MathSciNet  Google Scholar 

  9. Chakradhar, S.T., Bushnell, M.L.: A solvable class of quadratic 0–1 programming. Discret. Appl. Math. 36, 233–251 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cosares, S., Hochbaum, D.S.: Strongly polynomial algorithms for the quadratic transportation problem with a fixed number of sources. Math. Oper. Res. 19, 94–111 (1994)

    Article  MathSciNet  Google Scholar 

  11. Erenguc, S.S., Benson, H.P.: An algorithm for indefinite integer quadratic programming. Comput. Math. Appl. 21, 99–106 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computer and Intractability, a Guide to the Thoery of NP-Completeness. W. H. Freeman Co, San Francisco (1979)

    MATH  Google Scholar 

  13. Gao, J.J., Li, D.: A polynomial case of the cardinality-constrained quadratic optimization problem. J. Glob. Optim 56, 1441–1455 (2013)

    Article  MATH  Google Scholar 

  14. Goemans, M.X., Williamson, D.P.: Improved approximation algorithms formaximum cut ans satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Halikias, G.D., Jaimoukha, I.M., Malik, U., Gungah, S.K.: New bounds on the unconstrained quadratic integer programming problem. J. Glob. Optim. 39, 543–554 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0–1 optimization. Math. Program. 28, 121–155 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hochbaum, D.S., Shamir, R., Shanthikumar, J.G.: A polynomial algorithm for an ineger quadratic non-separable transportation problem. Math. Program. 55, 359–371 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, D., Sun, X.L.: Nonlinear Integer Programming. Springer, New York (2006)

    MATH  Google Scholar 

  19. Li, D., Sun, X.L., Liu, C.L.: An exact solution method for uncoxnstrained quadratic 0–1 programming: a geometric approach. J. Glob. Optim. 52, 797–829 (2012)

    Article  MathSciNet  Google Scholar 

  20. Li, D., Sun, X.L., Gu, S.S., Gao, J.J., Liu, C.L.: Polynomially solvable cases of binary quadratic programs. In: Pardalos, P.M., Enkhbat, R., Tseveendorj, I. (eds.) Optimization and Optimal Control: Theory and Applications, pp. 199–225 (2010)

  21. Pardalos, P.M., Rodgers, G.P.: Computational aspects of branch and bound algorithm for quadratic zeroone programming. Computing 45, 131–144 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Renu, G., Bandopadhyaya, L., Puri, M.C.: Ranking in quadratic integer programming problems. Eur. J. Oper. Res. 95, 231–236 (1996)

    Article  Google Scholar 

  23. Rhys, J.: A selection problem of shared fixed costs and network flows. Manag. Sci. 17, 200–207 (1970)

    Article  MATH  Google Scholar 

  24. Sun, X.L., Liu, C.L., Li, D., Gao, J.J.: On duality gap in binary quadratic optimization. J. Glob. Optim. 53, 255–269 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sleumer, N.: Output-sensitive cell enumeration in hyperplane arrangements. Nordic J. Comput. 6, 137–161 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Vassilev, V., Genova, K.: An approximate algorithm for nonlinear integer programming. Eur. J. Oper. Res. 74, 170–178 (1994)

    Article  Google Scholar 

  27. Wu, Z.Y., Li, G.Q., Quan, J.: Global optimality conditions and optimization methods for quadratic integer programming problems. J. Glob. Optim. 51, 549–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zaslavsky, T.: Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes, vol. 1. Memoirs of the American Mathematical Society. http://www.ams.org/books/memo/0154/ (1975)

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University of Finance and Economics, Shanghai, China

    Chunli Liu

  2. Department of Automation, Shanghai Jiao Tong University, Shanghai, China

    Jianjun Gao

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  1. Chunli Liu
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  2. Jianjun Gao
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Correspondence to Jianjun Gao.

Additional information

This work was supported by the National Natural Science Foundation of China under Grants 11201281, 11271243 and 71201102, by Shanghai Pujiang Program 11PJC059, by Interdiscipline Foundation of Shanghai Jiao Tong University (No. 13JCY10).

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Liu, C., Gao, J. A polynomial case of convex integer quadratic programming problems with box integer constraints. J Glob Optim 62, 661–674 (2015). https://doi.org/10.1007/s10898-014-0263-2

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  • DOI: https://doi.org/10.1007/s10898-014-0263-2

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