Advertisement

A computational comparison of some branch and bound methods for indefinite quadratic programs

  • Riccardo CambiniEmail author
  • Claudio Sodini
Original Paper

Abstract

The aim of this paper is to discuss different branch and bound methods for solving indefinite quadratic programs. In these methods the quadratic objective function is decomposed in a d.c. form and the relaxations are obtained by linearizing the concave part of the decomposition. In this light, various decomposition schemes have been considered and studied. The various branch and bound solution methods have been implemented and compared by means of a deep computational test.

Keywords

Quadratic programming Branch and bound d.c. decomposition 

Mathematics Subject Classification (2000)

90C20 90C26 90C31 

JEL Classification

C61 C63 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barrientos O and Correa R (2000). An algorithm for global minimization of linearly constrained quadratic functions. J Global Optim 16: 77–93 CrossRefGoogle Scholar
  2. Best MJ and Ding B (1997). Global and local quadratic minimization. J Global Optim 10: 77–90 CrossRefGoogle Scholar
  3. Best MJ and Ding B (2000). A decomposition method for global and local quadratic minimization. J Global Optim 16: 133–151 CrossRefGoogle Scholar
  4. Bomze IM and Danninger G (1994). A finite algorithm for solving general quadratic problems. J Global Optim 4: 1–16 CrossRefGoogle Scholar
  5. Bomze IM (2002). Branch-and-bound approaches to standard quadratic optimization problems. J Global Optim 22: 17–37 CrossRefGoogle Scholar
  6. Bomze IM and Locatelli M (2004). Undominated d.c. decompositions of quadratic functions and applications to branch-and-bound approaches. Comput Optim Appl 28: 227–245 CrossRefGoogle Scholar
  7. Cambini R and Sodini C (2005). Decomposition methods for solving nonconvex quadratic programs via branch and bound. J Global Optim 33(3): 313–336 CrossRefGoogle Scholar
  8. Churilov L and Sniedovich M (1999). A concave composite programming perspective on D.C. programming. In: Eberhard, A, Hill, R, Ralph, D and Glover, BM (eds) Progress in optimization, Applied Optimization, vol 30, pp. Kluwer, Dordrecht Google Scholar
  9. De Angelis PL, Pardalos PM and Toraldo G (1997). Quadratic Programming with Box Constraints. In: Bomze, IM (eds) Developments in global optimization, pp 73–93. Kluwer, Dordrecht, Google Scholar
  10. Floudas CA and Visweswaran V (1995). Quadratic Optimization. In: Horst, R and Pardalos, PM (eds) Handbook of global optimization, nonconvex optimization and its applications, vol 2, pp 217–269. Kluwer, Dordrecht, Google Scholar
  11. Gantmacher FR (1960). The theory of matrices, vol 1. Chelsea, New York Google Scholar
  12. Hansen P, Jaumard B, Ruiz M and Xiong J (1993). Global minimization of indefinite quadratic functions subject to box constraints. Naval Res Logistics 40: 373–392 CrossRefGoogle Scholar
  13. Hiriart-Urruty JB (1985) Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and duality in optimization, Lecture Notes in Economics and Mathematical Systems, vol 256. Springer, BerlinGoogle Scholar
  14. Horst R (1976). An algorithm for nonconvex programming problems. Math Program 10(3): 312–321 CrossRefGoogle Scholar
  15. Horst R (1980). A note on the convergence of an algorithm for nonconvex programming problems. Math Program 19(2): 237–238 CrossRefGoogle Scholar
  16. Horst R and Tuy H (1990). Global optimization. Springer, Berlin Google Scholar
  17. Horst R, Pardalos PM and Thoai NV (1995). Introduction to global optimization, nonconvex optimization and its applications, vol 3. Kluwer, Dordrecht Google Scholar
  18. Horst R and Van Thoai N (1996). A new algorithm for solving the general quadratic programming problem. Comput Optim Appl 5(1): 39–48 CrossRefGoogle Scholar
  19. Konno H (1976). Maximization of a convex quadratic function under linear constraints. Math Program 11(2): 117–127 CrossRefGoogle Scholar
  20. Le Thi Hoai An and Pham Dinh Tao (1997) Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J Global Optim 11:253–285Google Scholar
  21. Muu LD and Oettli W (1991). An algorithm for indefinite quadratic programming with convex constraints. Oper Res Lett 10(6): 323–327 CrossRefGoogle Scholar
  22. Pardalos PM, Glick JH and Rosen JB (1987). Global minimization of indefinite quadratic problems. Computing 39(4): 281–291 CrossRefGoogle Scholar
  23. Pardalos PM (1991). Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput Math Appl 21: 87–97 CrossRefGoogle Scholar
  24. Phong Thai Quynh, Le Thi Hoai An and Pham Dinh Tao (1995) Decomposition branch and bound method for globally solving linearly constrained indefinite quadratic minimization problems. Oper Res Lett 17(5):215–220Google Scholar
  25. Rosen JB and Pardalos PM (1986). Global minimization of large scale constrained concave quadratic problems by separable programming. Math Program 34: 163–174 CrossRefGoogle Scholar
  26. Thoai NV (2005). General Quadratic Programming. In: Audet, C, Hansen, P and Savard, G (eds) Essays and survey in global optimization, pp 107–129. Springer, Berlin, CrossRefGoogle Scholar
  27. Tuy H (1995). D.C. Optimization: theory, methods and algorithms. In: Horst, R and Pardalos, PM (eds) Handbook of global optimization, nonconvex optimization and its applications, vol 2, pp 149–216. Kluwer, Dordrecht, Google Scholar
  28. Tuy H (1998). Convex analysis and global optimization, nonconvex optimization and its applications, vol 22. Kluwer, Dordrecht Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Statistics and Applied Mathematics, Faculty of EconomicsUniversity of PisaPisaItaly

Personalised recommendations