Skip to main content
SpringerLink
Log in

Proximity in concave integer quadratic programming

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

Abstract

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of \(n \Delta \) on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and \(\Delta \) denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman and Magagnosc showed that the same upper bound is valid if a more general convex function is minimized, instead of a linear function. No proximity result of this type is known when the objective function is nonconvex. In fact, if we minimize a concave quadratic, no upper bound can be given as a function of n and \(\Delta \). Our key observation is that, in this setting, proximity phenomena still occur, but only if we consider also approximate solutions instead of optimal solutions only. In our main result we provide upper bounds on the distance between approximate (resp., optimal) solutions to a Concave Integer Quadratic Programming problem and optimal (resp., approximate) solutions of its continuous relaxation. Our bounds are functions of \(n, \Delta \), and a parameter \(\epsilon \) that controls the quality of the approximation. Furthermore, we discuss how far from optimal are our proximity bounds.

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

References

  1. Aliev, I., Henk, M., Oertel, T.: Distances to Lattice Points in Knapsack Polyhedra. Mathematical Programming, Series A. Springer, Berlin (2019)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Cook, W., Gerards, A., Schrijver, A., Tardos, E.: Sensistivity theorems in integer linear programming. Math. Program. 34, 251–264 (1986)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. 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)

  7. 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 

  8. Del Pia, A.: Subdeterminants and concave integer quadratic programming. SIAM J. Optim. 29(4), 3154–3173 (2020)

    MathSciNet  MATH  Google Scholar 

  9. 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  Google Scholar 

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

  11. Eisenbrand, F., Weismantel, R.: Proximity results and faster algorithms for integer programming using Steinitz lemma. In: Proceedings of SODA, pp. 808–816 (2018)

  12. Granot, F., Skorin-Kapov, J.: Some proximity and sensitivity results in quadratic integer programming. Math. Program. 47, 259–268 (1990)

    Article  MathSciNet  Google Scholar 

  13. 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  Google Scholar 

  14. 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)

  15. Paat, J., Weismantel, R., Weltge, S.: Distances Between Optimal Solutions of Mixed-integer Programs. Mathematical Programming, Series A. Springer, Berlin (2018)

    MATH  Google Scholar 

  16. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)

    MATH  Google Scholar 

  17. 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 (1992)

    Google Scholar 

  18. Werman, M., Magagnosc, D.: The relationship between integer and real solutions of constrained convex programming. Math. Program. 51, 133–135 (1991)

    Article  MathSciNet  Google Scholar 

  19. Xu, L., Lee, J.: On proximity for \(k\)-regular mixed-integer linear optimization. In: Proceedings of WCGO, pp. 438–447 (2019)

Download references

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

  2. Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA

    Mingchen Ma

Authors
  1. Alberto Del Pia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingchen Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingchen Ma.

Additional information

Publisher's Note

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

This work is supported 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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Del Pia, A., Ma, M. Proximity in concave integer quadratic programming. Math. Program. 194, 871–900 (2022). https://doi.org/10.1007/s10107-021-01655-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-021-01655-w

Keywords

Mathematics Subject Classification

Search

Navigation