Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices

  • João GouveiaEmail author
  • Ting Kei Pong
  • Mina Saee


Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being a linear programming based method proposed by Bundfuss and Dür (SIAM J Optim 20(1):30–53, 2009) and also Yıldırım (Optim Methods Softw 27(1):155–173, 2012), and a semidefinite programming based method proposed by Lasserre (Math Program 144(1):265–276, 2014). In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the completely positive cone. Using projections of this cone we derive new graph-based second-order cone approximation schemes for completely positive programming, leading to both uniform and problem-dependent hierarchies. This offers a compromise between the expressive power of semidefinite programming and the speed of linear programming based approaches. Numerical results on random problems, standard quadratic programs and the stable set problem are presented to illustrate the effectiveness of our approach.


Completely positive cones Inner approximations Scaled diagonal dominant matrices 



  1. 1.
    Abraham, B., Naomi, S.: Completely Positive Matrices. World Scientific, Singapore (2003)zbMATHGoogle Scholar
  2. 2.
    Ahmadi, A.A., Dash, S., Hall, G.: Optimization over structured subsets of positive semidefinite matrices via column generation. Discrete Optim. 24, 129–151 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ahmadi, A.A., Hall, G.: Sum of squares basis pursuit with linear and second order cone programming. Contemp. Math. 685, 27–53 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ahmadi, A.A., Majumdar, A.: Dsos and sdsos optimization: more tractable alternatives to sum of squares and semidefinite optimization. SIAM J. Appl. Algebra Geom. 3(2), 193–230 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boman, E.G., Chen, D., Parekh, O., Toledo, S.: On factor width and symmetric h-matrices. Linear Algebra Appl. 405, 239–248 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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(3), 423–445 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30–53 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Burer, S.: Copositive programming. In: Miguel, A.F., Lasserre, J.B. (eds.) Handbook on Semidefinite Conic and Polynomial Optimization, pp. 201–218. Springer, Berlin (2012)CrossRefGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1 (2014). Accessed Jan 2018
  15. 15.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  16. 16.
    Johnson, D.J., Trick, M.A.: Cliques, Colorings and Satisfiability. 2nd DIMACS Implementation Challenge, 1993, pp. 492–497. American Mathematical Society, Providence (1996)Google Scholar
  17. 17.
    Lasserre, J.B.: New approximations for the cone of copositive matrices and its dual. Math. Program. 144(1), 265–276 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. SIAM J. Optim. 12(4), 875–892 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. Thesis, California Institute of Technology (2000)Google Scholar
  20. 20.
    Pena, 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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Permenter, F., Parrilo, P.: Partial facial reduction: simplified, equivalent sdps via approximations of the psd cone. Math. Program. 171, 1–54 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar
  23. 23.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  24. 24.
    Sloane, N.: Challenge problems: independent sets in graphs. Information Sciences Research Center (2005). Accessed May 2018
  25. 25.
    Yıldırım, E.A.: On the accuracy of uniform polyhedral approximations of the copositive cone. Optim. Methods Softw. 27(1), 155–173 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yıldırım, E.A.: Inner approximations of completely positive reformulations of mixed binary quadratic programs : a unified analysis. Optim. Methods Softw. 32(6), 1163–1186 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonPeople’s Republic of China

Personalised recommendations