Skip to main content

Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization

Abstract

Completely positive (CP) tensors, which correspond to a generalization of CP matrices, allow to reformulate or approximate a general polynomial optimization problem (POP) with a conic optimization problem over the cone of CP tensors. Similarly, completely positive semidefinite (CPSD) tensors, which correspond to a generalization of positive semidefinite (PSD) matrices, can be used to approximate general POPs with a conic optimization problem over the cone of CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for general POPs and compare them with the bounds obtained via a Lagrangian relaxation of the POPs. This shows that existing results in this direction for quadratic POPs extend to general POPs. Also, we provide some tractable approximation strategies for CP and CPSD tensor relaxations. These approximation strategies show that, with a similar computational effort, bounds obtained from them for general POPs can be tighter than bounds for these problems obtained by reformulating the POP as a quadratic POP, which subsequently can be approximated using CP and PSD matrices. To illustrate our results, we numerically compare the bounds obtained from these relaxation approaches on small scale fourth-order degree POPs.

This is a preview of subscription content, access via your institution.

References

  1. Ahmadi, A.A., Majumdar, A.: DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization. In: Information Sciences and Systems (CISS), 2014 48th Annual Conference on, pp. 1–5. IEEE (2014)

  2. Ahmed, F., Dür, M., Still, G.: Copositive programming via semi-infinite optimization. J. Optim. Theory Appl. 159(2), 322–340 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. Anjos, M.F., Lasserre, J.B.: Handbook on Semidefinite, Conic and Polynomial Optimization, vol. 166. Springer, Berlin (2011)

    MATH  Google Scholar 

  4. Arima, N., Kim, S., Kojima, M.: A quadratically constrained quadratic optimization model for completely positive cone programming. SIAM J. Optim. 23(4), 2320–2340 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 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), 884–900 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  6. Bai, L., Mitchell, J.E., Pang, J.-S.: On conic QPCCs, conic QCQPs and completely positive programs. Math. Program. 159(1–2), 109–136 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  7. Biswas, P., Lian, T.-C., Wang, T.-C., Ye, Y.: Semidefinite programming based algorithms for sensor network localization. ACM Trans. Sens. Netw. (TOSN) 2(2), 188–220 (2006)

    Article  Google Scholar 

  8. Bomze, I.M.: Copositive relaxation beats Lagrangian dual bounds in quadratically and linearly constrained quadratic optimization problems. SIAM J. Optim. 25(3), 1249–1275 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. Bomze, I.M., De Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24(2), 163–185 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  10. Bomze, I.M., Dür, M., De Klerk, E., Roos, C., Quist, A.J., Terlaky, T.: On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18(4), 301–320 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  11. Bose, S., Low, S.H., Teeraratkul, T., Hassibi, B.: Equivalent relaxations of optimal power flow. IEEE Trans. Autom. Control 60(3), 729–742 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30–53 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  13. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  14. Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  15. Burer, S., Dong, H.: Representing quadratically constrained quadratic programs as generalized copositive programs. Oper. Res. Lett. 40(3), 203–206 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  16. Cardoso, J.-F.: Blind signal separation: statistical principles. Proc. IEEE 86(10), 2009–2025 (1998)

    Article  Google Scholar 

  17. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4(1), 33–52 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  18. de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  19. de Klerk, E., Pasechnik, D.V.: A linear programming reformulation of the standard quadratic optimization problem. J. Glob. Optim. 37(1), 75–84 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  20. Dickinson, P.J., Povh, J.: Moment approximations for set-semidefinite polynomials. J. Optim. Theory Appl. 159(1), 57–68 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. Dong, H.: Symmetric tensor approximation hierarchies for the completely positive cone. SIAM J. Optim. 23(3), 1850–1866 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  22. Dukanovic, I., Rendl, F.: Copositive programming motivated bounds on the stability and the chromatic numbers. Math. Program. 121(2), 249–268 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  23. Ghaddar, B., Marecek, J., Mevissen, M.: Optimal power flow as a polynomial optimization problem. IEEE Trans. Power Syst. 31(1), 539–546 (2016)

    Article  Google Scholar 

  24. Goemans, M.X.: Semidefinite programming in combinatorial optimization. Math. Program. 79(1–3), 143–161 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93(1), 012304 (2016)

    MathSciNet  Article  Google Scholar 

  26. Jiang, B., Yang, F., Zhang, S.: Tensor and its tucker core: the invariance relationships. (2016) arXiv preprint arXiv:1601.01469

  27. Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26(2), 143–154 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  28. Kuang, X., Ghaddar, B., Naoum-Sawaya, J., Zuluaga, L.: Alternative LP and SOCP hierarchies for ACOPF problems. IEEE Trans. Power Syst. 32, 2828–2836 (2016)

    Article  Google Scholar 

  29. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  30. Lasserre, J.B.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12(3), 756–769 (2002a)

    MathSciNet  Article  MATH  Google Scholar 

  31. Lasserre, J.B.: Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res. 27(2), 347–360 (2002b)

    MathSciNet  Article  MATH  Google Scholar 

  32. Lavaei, J., Low, S.H.: Zero duality gap in optimal power flow problem. IEEE Trans. Power Syst. 27(1), 92–107 (2012)

    Article  Google Scholar 

  33. Ling, C., Nie, J., Qi, L., Ye, Y.: Biquadratic optimization over unit spheres and semidefinite programming relaxations. SIAM J. Optim. 20(3), 1286–1310 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  34. Löfberg, J.: Yalmip: A toolbox for modeling and optimization in matlab. In: IEEE International Symposium on Computer Aided Control Systems Design, pp. 284–289. IEEE (2004)

  35. Luo, Z., Qi, L., Ye, Y.: Linear operators and positive semidefiniteness of symmetric tensor spaces. Sci. China Math. 58(1), 197–212 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  36. Malek, A., Hosseinipour-Mahani, N.: Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique. Kybernetika 51(5), 890–908 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Mariere, B., Luo, Z.-Q., Davidson, T.N.: Blind constant modulus equalization via convex optimization. IEEE Trans. Signal Process. 51(3), 805–818 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  38. Mavridou, T., Pardalos, P., Pitsoulis, L., Resende, M.G.: A grasp for the biquadratic assignment problem. Eur. J. Oper. Res. 105(3), 613–621 (1998)

    Article  MATH  Google Scholar 

  39. Nesterov, Y.: Structure of non-negative polynomials and optimization problems. Technical report, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) (1997)

  40. Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, Citeseer (2000)

  41. Peña, J., Vera, J., Zuluaga, L.F.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  42. Peña, J., Vera, J.C., Zuluaga, L.F.: A certificate of non-negativity for polynomials over unbounded sets. Lehigh University (2014a)

  43. Peña, J., Vera, J.C., Zuluaga, L.F.: Completely positive reformulations for polynomial optimization. Math. Program. 151(2), 405–431 (2014b)

    MathSciNet  Article  MATH  Google Scholar 

  44. Povh, J., Rendl, F.: A copositive programming approach to graph partitioning. SIAM J. Optim. 18(1), 223–241 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  45. Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6(3), 231–241 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  46. QELA, R.E.B.E., KLINZ, B.: On the biquadratic assignment problem. In: Quadratic Assignment and Related Problems: DIMACS Workshop, May 20–21, 1993, vol. 16, p. 117. American Mathematical Soc (1994)

  47. Shor, N.: Class of global minimum bounds of polynomial functions. Cybern. Syst. Anal. 23(6), 731–734 (1987)

    Article  MATH  Google Scholar 

  48. Song, Y., Qi, L.: Tensor complementarity problem and semi-positive tensors. J. Optim. Theory Appl. 169, 1–10 (2015)

    MathSciNet  Google Scholar 

  49. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49–95 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  50. Ye, Y.: Approximating quadratic programming with bound and quadratic constraints. Math. Program. 84(2), 219–226 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  51. Zheng, X.J., Sun, X.L., Li, D.: Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation. Math. Program. 129(2), 301–329 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  52. Zuluaga, L.F., Vera, J., Peña, J.: LMI approximations for cones of positive semidefinite forms. SIAM J. Optim. 16(4), 1076–1091 (2006)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Sam Burer for his comments on an earlier version of this manuscript. Also, the authors would like to thank anonymous referees for their thorough and thoughtful comments. The authors of this article were supported by the NSF CMMI Grant 1300193.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaolong Kuang.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kuang, X., Zuluaga, L.F. Completely positive and completely positive semidefinite tensor relaxations for polynomial optimization. J Glob Optim 70, 551–577 (2018). https://doi.org/10.1007/s10898-017-0558-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-017-0558-1

Keywords

  • Copositive programming
  • Convex relaxation
  • Completely positive tensor
  • Completely positive semidefinite tensor