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Optimization Letters

, Volume 11, Issue 7, pp 1229–1241 | Cite as

Completely positive reformulations of polynomial optimization problems with linear constraints

  • Wei XiaEmail author
  • Luis F. Zuluaga
Original Paper
  • 240 Downloads

Abstract

A polynomial optimization problem (POP) is an optimization problem in which both the objective and constraints can be written in terms of polynomials on the decision variables. Recently, it has been shown that under appropriate assumptions POPs can be reformulated as conic problems over the cone of completely positive tensors; which generalize the set of completely positive matrices. Here, we show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a generalization of the completely positive reformulation of quadratically constrained quadratic programs recently introduced by Bai et al. (Math Program 1–28, 2016).

Keywords

Polynomial optimization Completely positive tensors Completely positive relaxations 

Notes

Acknowledgements

We would like to thank an anonymous referee for providing thoughtful and thorough comments to improve the article. The work of Wei Xia and Luis F. Zuluaga are supported by NSF Grant CMMI-1300193.

References

  1. 1.
    Arima, N., Kim, S., Kojima, M.: A quadratically constrained quadratic optimization model for completely positive cone programming. SIAM J. Optim. 23, 2320–2340 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arima, N., Kim, S., Kojima, M.: Extension of completely positive cone relaxation to moment cone relaxation for polynomial optimization. J. Optim. Theory Appl. 168(3), 1–17 (2016)Google Scholar
  3. 3.
    Bai, L., Mitchell, J.E., Pang, J.: On conic qpccs, conic qcqps and completely positive programs. Math. Program. 159(1–2), 1–28 (2016)Google Scholar
  4. 4.
    Bomze, I.M.: Copositive optimization-recent developments and applications. Eur. J. Oper. Res. 216, 509–520 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bomze, I.M., Schachinger, W., Uchida, G.: Think co (mpletely) positive! matrix properties, examples and a clustered bibliography on copositive optimization. J. Glob. Optim. 52, 423–445 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burer, S.: Copositive programming. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 201–218. Springer, New York (2012)Google Scholar
  8. 8.
    Burer, S., Dong, H.: Representing quadratically constrained quadratic programs as generalized copositive programs. Oper. Res. Lett. 40, 203–206 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, B., He, S., Li, Z., Zhang, S.: Maximum block improvement and polynomial optimization. SIAM J. Optim. 22, 87–107 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dong, H.: Symmetric tensor approximation hierarchies for the completely positive cone. SIAM J. Optim. 23, 1850–1866 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dür, M.: Copositive programming–a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. Springer, Berlin, Heidelberg (2010)Google Scholar
  12. 12.
    Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lasserre, J.B.: Moments and sums of squares for polynomial optimization and related problems. J. Glob. Optim. 45, 39–61 (2009a)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lasserre, J.B.: Moments, Positive Polynomials and Their Applications, vol. 1. World Scientific, Singapore (2009b)Google Scholar
  15. 15.
    Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, pp. 157–270. Springer, Berlin (2009)Google Scholar
  16. 16.
    Luo, Z., Qi, L., Ye, Y.: Linear operators and positive semidefiniteness of symmetric tensor spaces. Sci. China Math. 58, 197–212 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is np-hard. J. Glob. Optim. 1, 15–22 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Peña, J., Vera, J., Zuluaga, L.F.: Completely positive reformulations for polynomial optimization. Math. Program. 151, 405–431 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Qi, L., Xu, C., Xu, Y.: Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm. SIAM J. Matrix Anal. Appl. 35, 1227–1241 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer, Berlin (2009)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Industrial and System EngineeringLehigh UniversityBethlehemUSA

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