Skip to main content
SpringerLink
Log in

An approximation algorithm for indefinite mixed integer quadratic programming

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper, we give an algorithm that finds an \(\epsilon \)-approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Even though Theorem 4 in [10] does not give \(\psi \) explicitly, a formula for \(\psi \), as a function of the size of the MIQP instance, can be derived from its proof.

References

  1. Bellare, M., Rogaway, P.: The complexity of approximating a nonlinear program. Math. Program. 69, 429–441 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  3. Cook, W., Hartman, M., Kannan, R., McDiarmid, C.: On integer points in polyhedra. Combinatorica 12(1), 27–37 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Math. Program. 47(1–3), 155–174 (1990)

    Article  MATH  Google Scholar 

  5. Dax, A., Kaniel, S.: Pivoting techniques for symmetric Gaussian elimination. Numer. Math. 28, 221–241 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Loera, J., Hemmecke, R., Köppe, M., Weismantel, R.: FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension. Math. Program. Ser. A 118, 273–290 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Del Pia, A.: On approximation algorithms for concave mixed-integer quadratic programming. In: Proceedings of IPCO, Lecture Notes in Computer Science, vol. 9682, pp. 1–13 (2016)

  8. Del Pia, A.: On approximation algorithms for concave mixed-integer quadratic programming. Math. Program. Ser. B 172(1–2), 3–16 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Del Pia, A.: Subdeterminants and concave integer quadratic programming. SIAM J. Optim. 29(4), 3154–3173 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Del Pia, A., Dey, S., Molinaro, M.: Mixed-integer quadratic programming is in NP. Math. Program. Ser. A 162(1), 225–240 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Del Pia, A., Weismantel, R.: Integer quadratic programming in the plane. In: Proceedings of SODA, pp. 840–846 (2014)

  12. Edmonds, J.: Systems of distinct representatives and linear algebra. J. Res. Natl. Bureau Stand. B. Math. Math. Phys. 71B(4), 241–245 (1967)

  13. Eiben, E., Ganian, R., Knop, D., Ordyniak, S.: Solving integer quadratic programming via explicit and structural restrictions. Proceedings of the AAAI Conference on Artificial Intelligence (2019)

  14. Galbraith, S.: Mathematics of Public Key Cryptography. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  15. Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Golub, G., Van Loan, C.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)

    Book  MATH  Google Scholar 

  17. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  18. Hildebrand, R., Oertel, T., Weismantel, R.: Note on the complexity of the mixed-integer hull of a polyhedron. Oper. Res. Lett. 43, 279–282 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hildebrand, R., Weismantel, R., Zemmer, K.: An FPTAS for minimizing indefinite quadratic forms over integers in polyhedra. In: Proceedings of SODA, pp. 1715–1723 (2016)

  20. Hochbaum, D., Shanthikumar, J.: Convex separable optimization is not much harder than linear optimization. J. Assoc. Comput. Mach. 37(4), 843–862 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Khachiyan, L.: Convexity and complexity in polynomial programming. In: Proceedings of the International Congress of Mathematicians, pp. 16–24. Warsaw (1983)

  22. de Klerk, E., Laurent, M., Parrilo, P.: A PTAS for the minimization of polynomials of fixed degree over the simplex. Theoret. Comput. Sci. 361, 210–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kozlov, M., Tarasov, S., Khachiyan, L.: Polynomial solvability of convex quadratic programming. Doklady Akademii Nauk SSSR 248, 1049–1051 (1979). Translated in: Soviet Mathematics Doklady 20 (1979) 1108-1111

  24. Lee, J., Onn, S., Romanchuk, L., Weismantel, R.: The quadratic graver cone, quadratic integer minimization, and extensions. Math. Program. Ser. B 136, 301–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lenstra, H.J.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Murty, K., Kabadi, S.: Some NP-complete problems in quadratic and linear programming. Math. Program. 39, 117–129 (1987)

    Article  MATH  Google Scholar 

  27. Nemirovsky, A., Yudin, D.: Problem Complexity and Method Efficiency in Optimization. Wiley, Chichester (1983). Translated by E.R. Dawson from Slozhnost’ Zadach i Effektivnost’ Metodov Optimizatsii (1979)

  28. Pardalos, P., Vavasis, S.: Quadratic programming with one negative Eigenvalue is NP-hard. J. Glob. Optim. 1(1), 15–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vavasis, S.: Quadratic programming is in NP. Inf. Process. Lett. 36, 73–77 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vavasis, S.: Approximation algorithms for indefinite quadratic programming. Math. Program. 57, 279–311 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vavasis, S.: On approximation algorithms for concave quadratic programming. In: Floudas, C., Pardalos, P. (eds.) Recent Advances in Global Optimization, pp. 3–18. Princeton University Press, Princeton, NJ (1992)

    Google Scholar 

  32. Vavasis, S.: Polynomial time weak approximation algorithms for quadratic programming. In: P. Pardalos (ed.) Complexity in Numerical Optimization. World Scientific (1993)

Download references

Funding

A. Del Pia is partially funded by ONR grant N00014-19-1-2322. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research.

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering and Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, WI, USA

    Alberto Del Pia

Authors
  1. Alberto Del Pia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alberto Del Pia.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pia, A.D. An approximation algorithm for indefinite mixed integer quadratic programming. Math. Program. 201, 263–293 (2023). https://doi.org/10.1007/s10107-022-01907-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-022-01907-3

Keywords

Mathematics Subject Classification

Search

Navigation