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Completely positive reformulations of polynomial optimization problems with linear constraints

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).

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References

  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)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  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)

  4. Bomze, I.M.: Copositive optimization-recent developments and applications. Eur. J. Oper. Res. 216, 509–520 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  6. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  8. Burer, S., Dong, H.: Representing quadratically constrained quadratic programs as generalized copositive programs. Oper. Res. Lett. 40, 203–206 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. Chen, B., He, S., Li, Z., Zhang, S.: Maximum block improvement and polynomial optimization. SIAM J. Optim. 22, 87–107 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  10. Dong, H.: Symmetric tensor approximation hierarchies for the completely positive cone. SIAM J. Optim. 23, 1850–1866 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  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)

  12. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  13. Lasserre, J.B.: Moments and sums of squares for polynomial optimization and related problems. J. Glob. Optim. 45, 39–61 (2009a)

    MathSciNet  Article  MATH  Google Scholar 

  14. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications, vol. 1. World Scientific, Singapore (2009b)

    Google Scholar 

  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)

  16. Luo, Z., Qi, L., Ye, Y.: Linear operators and positive semidefiniteness of symmetric tensor spaces. Sci. China Math. 58, 197–212 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  17. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is np-hard. J. Glob. Optim. 1, 15–22 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  18. Peña, J., Vera, J., Zuluaga, L.F.: Completely positive reformulations for polynomial optimization. Math. Program. 151, 405–431 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  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)

    MathSciNet  Article  MATH  Google Scholar 

  20. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, vol. 317. Springer, Berlin (2009)

    MATH  Google Scholar 

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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.

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Xia, W., Zuluaga, L.F. Completely positive reformulations of polynomial optimization problems with linear constraints. Optim Lett 11, 1229–1241 (2017). https://doi.org/10.1007/s11590-017-1123-z

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  • DOI: https://doi.org/10.1007/s11590-017-1123-z

Keywords

  • Polynomial optimization
  • Completely positive tensors
  • Completely positive relaxations