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Completely positive reformulations for polynomial optimization

Mathematical Programming volume 151, pages 405–431 (2015)Cite this article

Abstract

Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dual, the cone of copositive matrices. As a result of this reformulation approach, novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools, and the extensively studied cones of completely positive and of copositive matrices. In particular, this approach has been applied to solve key combinatorial optimization problems. Along this line of research, we consider polynomial optimization problems that are not necessarily quadratic. For this purpose, we use a natural extension of the cone of completely positive matrices; namely, the cone of completely positive tensors. We provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors. As a consequence of this characterization, it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems. Also, we show that the conditions underlying the characterization are conceptually the same, regardless of the degree of the polynomials defining the problem. To illustrate our results, we discuss in further detail special and relevant instances of polynomial optimization problems.

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References

  1. Amaral, P., Bomze, I.M., Júdice, J.: Copositivity and constrained fractional quadratic problems. Math. Program. 146(1–2), 325–350 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anjos, M., Lasserre, J.B. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization Handbook on Semidefinite, Conic and Polynomial Optimization, volume 166 of International Series in Operations Research & Management Science. Springer, Berlin (2012)

  3. Arima, N., Kim, S., Kojima, M. A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming. Technical Report B-468, Dept. of Math. and Comp. Sciences, Tokyo Institute of Technology. http://www.optimization-online.org/DB_FILE/2012/09/3600.pdf (2012)

  4. Bai, L., Mitchell, J.E., Pang, J.: On QPCCs, QCQPs and Copositive Programs. Technical report, Rensselaer Polytechnic Institute. http://eaton.math.rpi.edu/faculty/Mitchell/papers/QCQP_QPCC.html (2012)

  5. Bertsekas, D.P.: Non-linear Programming. Athena Scientific, Belmont, MA (1995)

    Google Scholar 

  6. Bertsimas, D., Popescu, I.: On the relation between option and stock prices: an optimization approach. Oper. Res. 50, 358–374 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bertsimas, D., Popescu, I.: Optimal inequalities in probability theory: a convex optimization approach. SIAM J. Optim. 15(3), 780–804 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Blekherman, G., Parrilo, P., and Thomas, R. (eds): Semidefinite Optimization and Convex Algebraic Geometry, volume 13 of MOS-SIAM Series on Optimization. SIAM, Philadelphia (2012)

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Bomze, I., Jarre, F.: A note on Burer’s copositive representation of mixed-binary QPs. Optim. Lett. 4(3), 465–472 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bomze, I.M.: Copositive optimization. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 561–564. Springer, Berlin (2009)

    Chapter  Google Scholar 

  12. Bomze, I.M.: Copositive Relaxation Beats Lagrangian Dual Bounds In Quadratically And Linearly Constrained QPs. Technical Report NI13064-POP, Isaac Newton Institute (2013)

  13. Bomze, I.M.: Copositive-Based Approximations for Binary and Ternary Fractional Quadratic Optimization. Technical Report NI14043-POP, Isaac Newton Institute (2014)

  14. 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, 301–320 (2000)

    Article  MATH  Google Scholar 

  15. Bomze, I.M., Jarre, F., Rendl, F.: Quadratic factorization heuristics for copositive programming. Math. Program. Comput. 3(1), 37–57 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bundfuss, S.: Copositive Matrices, Copositive Programming, and Applications. Ph.D. thesis, TU Darmstadt (2009)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Burer, S., Kim, S., Kojima, M.: Faster, but weaker, relaxations for quadratically constrained quadratic programs. Comput. Optim. Appl. 59(1–2), 27–45 (2014)

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

    Article  MATH  MathSciNet  Google Scholar 

  22. de Klerk, E., Laurent, M., Parrilo, P.A.: A PTAS for the minimization of polynomials of fixed degree over the simplex. Theor. Comput. Sci. 361(2–3), 210–225 (2006)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Dickinson, P.J., Eichfelder, G., Povh, J.: Erratum to: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets [optim. letters, 2012]. Optim. Lett. 7(6), 1387–1397 (2013)

  25. Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: An inequality for circle packings proved by semidefinite programming. Phys. Rev. A 69, 022308 (2004)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Dong, H., Anstreicher, K.: Separating doubly nonnegative and completely positive matrices. Math. Program. Ser. A 137, 131–153 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. 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 (2010)

    Chapter  Google Scholar 

  30. Eichfelder, G., Povh, J.: On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets. Optim. Lett. 6(2), 1–14 (2012)

    MathSciNet  Google Scholar 

  31. Genin, Y., Hachez, Y., Nesterov, Y., and Dooren, P. V.: Convex optimization over positive polynomials and filter design. In: Proceedings of UKACC Intenational Conference on Control (2000)

  32. Gouveia, J., Parrilo, P.A., Thomas, R.R.: Theta bodies for polynomial ideals. SIAM J. Optim. 20, 2097–2118 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  33. Gvozdenović, N., Laurent, M.: Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science 3509, 136–151 (2005)

  34. Gvozdenović, N., Laurent, M.: The operator \(\psi \) for the chromatic number of a graph. SIAM J. Optim. 19(2), 572–591 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Kim, S., Kojima, M., Toh, K.: A Lagrangian-DDN Relaxation: A Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems. Technical report. http://www.optimizationonlineorg/DBHTML/2013/10/4073.html (2013)

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Lasserre, J.: Bounds on measures satisfying moment conditions. Ann. Appl. Probab. 12, 1114–1137 (2002a)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Laurent, M.: A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0–1 integer programming. Math. Oper. Res. 28(3), 470–496 (2003a)

    Article  MATH  MathSciNet  Google Scholar 

  40. Laurent, M.: Lower bound for the number of iterations in semidefinite relaxations for the cut polytope. Math. Oper. Res. 28(4), 871–883 (2003b)

  41. Laurent, M.: Copositive vs. moment hierarchies for stable sets. Optima 89, 8–10 (2012)

    Google Scholar 

  42. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)

    Article  MATH  Google Scholar 

  43. Natarajan, K., Teo, C.P., Zheng, Z.: Mixed 0–1 linear programs under objective uncertainty: a completely positive representation. Oper. Res. 59(3), 713–728 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. Nesterov, Y.: Structure of Non-negative Polynomials and Optimization Problems. Technical Report 9749, CORE (1997)

  45. Nie, J.: Sum of squares method for sensor network localization. Comput. Optim. Appl. 43(2), 151–179 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  46. Papachristodoulou, A., Peet, M.M., Lall, S.: Analysis of polynomial systems with time delays via the sum of squares decomposition. IEEE Trans. Autom. Control 54(5), 1058–1064 (2009)

    Article  MathSciNet  Google Scholar 

  47. Parrilo, P.: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, Department of Control and Dynamical Systems, California Institute of Technology, Pasadena, CA (2000)

  48. Parrilo, P.A., Jadbabaie, A.: Approximation of the joint spectral radius using sum of squares. Linear Algebra Appl. 428(10), 2385–2402 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  50. Pólya, G.: Üher positive Darstellung von Polynomen. Vierteljschr. Naturforsch. Ges. Zürich 73, 141–145. (also Collected Papers, vol. 2, 309–313, MIT Press, Cambridge, MA, 1974) (1928)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  53. Reznick, B.: On Hilbert’s Construction of Positive Polynomials. Technical report, University of Illinois at Urbana-Champaign. www.math.uiuc.edu/reznick/paper53.pdf (2007)

  54. Rockafellar, T.: Convex Analysis. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  55. Rockafellar, T., Wets, R.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  56. Schmüdgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203–206 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  57. Schrijver, A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory 25(4), 425–429 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  58. Schweighofer, M.: An algorithmic approach to Schmüdgen’s positivstellensatz. J. Pure Appl. Algebra 166(3), 307–319 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  59. Shor, N.: Class of global minimum bounds of polynomial functions. Cybernetics 23, 731–734 (1987)

    Article  MATH  Google Scholar 

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

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Acknowledgments

The authors would like to thank two anonymous referees and the Associate Editor for their constructive and insightful comments which greatly improved this article. We would also like to thank Immanuel Bomze for his valuable comments on an earlier version of this manuscript.

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Authors and Affiliations

  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Javier Peña

  2. Tilburg School of Economics and Management, Tilburg University, Tilburg, The Netherlands

    Juan C. Vera

  3. Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, 18015, USA

    Luis F. Zuluaga

Authors
  1. Javier Peña
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  2. Juan C. Vera
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  3. Luis F. Zuluaga
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Correspondence to Juan C. Vera.

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Peña, J., Vera, J.C. & Zuluaga, L.F. Completely positive reformulations for polynomial optimization. Math. Program. 151, 405–431 (2015). https://doi.org/10.1007/s10107-014-0822-9

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Keywords

  • Polynomial optimization
  • Copositive programming
  • Completely positive tensors
  • Quadratic programming

Mathematics Subject Classification

  • 90C26
  • 90C20