Mathematical Programming

, 115:31

New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability

  • Immanuel M. Bomze
  • Marco Locatelli
  • Fabio Tardella


A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and the maximum clique problem in a graph. In this paper we present several new polynomial-time bounds for StQP ranging from very simple and cheap ones to more complex and tight constructions. The main tools employed in the conception and analysis of most bounds are Semidefinite Programming and decomposition of the objective function into a sum of two quadratic functions, each of which is easy to minimize. We provide a complete diagram of the dominance, incomparability, or equivalence relations among the bounds proposed in this and in previous works. In particular, we show that one of our new bounds dominates all the others. Furthermore, a specialization of such bound dominates Schrijver’s improvement of Lovász’s θ function bound for the maximum size of a clique in a graph.


Standard quadratic optimization Semidefinite programming Quadratic programming Maximum clique Resource allocation 

Mathematics Subject Classification (2000)

90C20 90C26 


  1. 1.
    Anstreicher K. and Burer S. (2005). D.C. Versus copositive bounds for standard QP. J. Glob. Optim. 33: 299–312 CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Bomze I.M. (1998). On standard quadratic optimization problems. J. Glob. Optim. 13: 369–387 CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bomze I.M. (2000). Copositivity aspects of standard quadratic optimization problems. In: Dockner, E., Hartl, R., Luptacik, M. and Sorger, G. (eds) Dynamics, Optimization and Economic Analysis., pp 1–11. Physica, Heidelberg Google Scholar
  4. 4.
    Bomze I.M. (2002). Branch-and-bound approaches to standard quadratic optimization problems. J. Glob. Optim. 22: 17–37 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bomze I.M., Dür M., de Klerk E., Quist A., Roos C. and Terlaky T. (2000). On copositive programming and standard quadratic optimization problems. J. Glob. Optim. 18: 301–320 CrossRefMATHGoogle Scholar
  6. 6.
    Bomze I.M. and de Klerk E. (2002). Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24: 163–185 CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Bomze I.M. and Locatelli M. (2004). Undominated d.c. decompositions of quadratic functions and applications to branch-and-bound approaches. Comput. Optim. Appl. 28(2): 227–245 CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Bomze, I.M., Locatelli, M., Tardella, F.: Efficient and cheap bounds for (Standard) Quadratic Optimization, Technical Report dis tr 2005/2010, Dipartimento di Informatica e Sistemistica “Antonio Ruberti”, Universitá degli Studi di Roma “La Sapienza”, available at (2005)Google Scholar
  9. 9.
    Boyd S. and Vandenberghe L. (2004). Convex Optimization. Cambridge University Press, Cambridge MATHGoogle Scholar
  10. 10.
    Laurent M., Parrillo P.A. and Klerk E. (2006). A PTAS for the minimization of polynomials of fixed degree over the simplex. Theor. Comp. Sci. 361: 210–225 CrossRefMATHGoogle Scholar
  11. 11.
    de Klerk E. and Pasechnik D.V. (2002). Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12: 875–892 CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Diananda P.H. (1967). On non-negative forms in real variables some or all of which are non-negative. Proc. Camb. Philos. Soc. 58: 17–25 MathSciNetGoogle Scholar
  13. 13.
    Dür M. (2002). A class of problems where dual bounds beat underestimation bounds. J. Glob. Optim. 22: 49–57 CrossRefMATHGoogle Scholar
  14. 14.
    Falk J. (1969). Lagrange multipliers and nonconvex programs. SIAM J. Control 7: 312–321 CrossRefMathSciNetGoogle Scholar
  15. 15.
    Frank M. and Wolfe P. (1956). An algorithm for quadratic programming. Naval Res. Logist. Q. 3: 95–110 CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gibbons L.E., Hearn D.W., Pardalos P.M. and Ramana M.V. (1997). Continuous characterizations of the maximum clique problem. Math. Oper. Res. 22: 754–768 MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Gower J.C. (1985). Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl. 67: 81–97 CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Horst R. and Tuy H. (1990). Global Optimization: Deterministic Approaches. Springer, Berlin MATHGoogle Scholar
  19. 19.
    Ibaraki T. and Katoh N. (1988). Resource Allocation Problems: Algorithmic Approaches. MIT Press, Cambridge MATHGoogle Scholar
  20. 20.
    Lasserre J.B. (2001). Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11: 796–817 CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Lemaréchal, C., Oustry, F.: Semidefinite relaxations and Lagranian duality with application to combinatorial optimization, INRIA research report, vol. 3710 (1999)Google Scholar
  22. 22.
    Markowitz H.M. (1952). Portfolio selection. J. Finance 7: 77–91 CrossRefGoogle Scholar
  23. 23.
    Markowitz H.M. (1995). The general mean-variance portfolio selection problem. In: Howison, S.D., Kelly, F.P. and Wilmott, P. (eds) Mathematical Models in Finance, vol. 93–99., pp. Chapman & Hall, London Google Scholar
  24. 24.
    Motzkin T.S. and Straus E.G. (1965). Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17: 533–540 MathSciNetMATHGoogle Scholar
  25. 25.
    Nesterov, Y.E.: Global Quadratic Optimization on the Sets with Simplex Structure, Discussion paper 9915, CORE. Catholic University of Louvain, Belgium (1999)Google Scholar
  26. 26.
    Nowak I. (1999). A new semidefinite programming bound for indefinite quadratic forms over a simplex. J. Glob. Optim. 14: 357–364 CrossRefMATHGoogle Scholar
  27. 27.
    Pena J., Vera J. and Zuluaga L. (2007). Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18: 87–105 MathSciNetMATHGoogle Scholar
  28. 28.
    Rockafellar R.T. (1970). Convex analysis. Princeton University Press, Princeton MATHGoogle Scholar
  29. 29.
    Schrijver A. (1979). A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inf. Theory 25: 425–429 CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Sahinidis N.V. and Tawarmalani M. (2002). Convex extensions and envelopes of lower semi-continuous functions. Math. Prog. 93: 247–263 CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Tardella F. (1990). On the equivalence between some discrete and continuous optimization problems. Ann. Oper. Res. 25: 291–300 CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Tardella F. (2004). Connections between continuous and combinatorial optimization problems through an extension of the fundamental theorem of linear programming. Electron. Notes Discret. Math. 17: 257–262 CrossRefMathSciNetGoogle Scholar
  33. 33.
    Tuy, H.: A general deterministic approach to global optimization via d.c. programming. In: Fermat days 85: Mathematics for optimization, vol. 129. North-Holland Math. Stud., Toulouse/France, pp. 273–303 (1985)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  • Marco Locatelli
    • 2
  • Fabio Tardella
    • 3
  1. 1.University of ViennaViennaAustria
  2. 2.University of TorinoTorinoItaly
  3. 3.Facoltà di EconomiaUniversity of Rome ’La Sapienza’RomaItaly

Personalised recommendations