Advertisement

Journal of Global Optimization

, Volume 49, Issue 3, pp 481–495 | Cite as

Properties of two DC algorithms in quadratic programming

  • Hoai An Le ThiEmail author
  • Tao Pham Dinh
  • Nguyen Dong Yen
Article

Abstract

Some new properties of the Projection DC decomposition algorithm (we call it Algorithm A) and the Proximal DC decomposition algorithm (we call it Algorithm B) Pham Dinh et al. in Optim Methods Softw, 23(4): 609–629 (2008) for solving the indefinite quadratic programming problem under linear constraints are proved in this paper. Among other things, we show that DCA sequences generated by Algorithm A converge to a locally unique solution if the initial points are taken from a neighborhood of it, and DCA sequences generated by either Algorithm A or Algorithm B are all bounded if a condition guaranteeing the solution existence of the given problem is satisfied.

Keywords

Quadratic programming DC algorithm KKT point Local solution DCA sequence Convergence Boundedness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bank B., Guddat J., Klatte D., Kummer B., Tammer K.: Non-Linear Parametric Optimization. Akademie-Verlag, Berlin (1982)Google Scholar
  2. 2.
    Bomze I.M., Danninger G.: A finite algorithm for solving general quadratic problems. J. Glob. Optim. 4, 1–16 (1994)CrossRefGoogle Scholar
  3. 3.
    Bomze I.M.: On standard quadratic optimization problems. J. Glob. Optim. 13, 369–387 (1998)CrossRefGoogle Scholar
  4. 4.
    Cambini R., Sodini C.: Decomposition methods for solving nonconvex quadratic programs via Branch and Bound. J. Glob. Optim. 33, 313–336 (2005)CrossRefGoogle Scholar
  5. 5.
    Contesse L.: Une caracté risation complète des minima locaux en programmation quadratique. Numer. Math. 34, 315–332 (1980)CrossRefGoogle Scholar
  6. 6.
    Cottle R.W., Pang J.-S., Stone R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)Google Scholar
  7. 7.
    Gould, N.I.M., Toint, Ph.L.: A Quadratic Programming Page. http://www.numerical.rl.ac.uk/qp/qp.html
  8. 8.
    Eaves B.C.: On quadratic programming. Manage. Sci. 17, 698–711 (1971)CrossRefGoogle Scholar
  9. 9.
    Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York, London (1980)Google Scholar
  10. 10.
    Lee G.M., Tam N.N., Yen N.D.: Quadratic Programming and Affine Variational Inequalities: A Qualitative Study, Series: Nonconvex Optimization and its Applications, vol. 78. Springer Verlag, New York (2005)Google Scholar
  11. 11.
    Le Thi H.A., Pham Dinh T.: Solving a class of linearly constrained indefinite quadratic programming problems. J. Glob. Optim. 11, 253–285 (1997)CrossRefGoogle Scholar
  12. 12.
    Le Thi H.A., Pham Dinh T.: A combined DC optimization—ellipsoidal branch-and-bound algorithm for solving nonconvex quadratic programming problems. J. Comb. Optim. 2(1), 9–29 (1998)CrossRefGoogle Scholar
  13. 13.
    Le Thi H.A., Pham Dinh T.: A branch and bound method via DC optimization algorithm and ellipsoidal techniques for box constrained nonconvex quadratic programming problems. J. Glob. Optim. 13, 171–206 (1998)CrossRefGoogle Scholar
  14. 14.
    Le Thi H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. Ser. A 87(3), 401–426 (2000)CrossRefGoogle Scholar
  15. 15.
    Le Thi H.A., Pham Dinh T.: A continuous approach for large-scale constrained quadratic zero-one programming. Optimization 45(3), 1–28 (2001)Google Scholar
  16. 16.
    Le Thi H.A., Pham Dinh T.: Large scale molecular optimization from distance matrices by a DC optimization approach. SIAM J. Optim. 14(1), 77–116 (2003)CrossRefGoogle Scholar
  17. 17.
    Le Thi H.A., Pham Dinh T.: The DC (Difference of Convex functions) and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–47 (2005)CrossRefGoogle Scholar
  18. 18.
    Mangasarian O.L.: Locally unique solutions of quadratic programs, linear and nonlinear complementarity problems. Math. Program. 19, 200–212 (1980)CrossRefGoogle Scholar
  19. 19.
    Pham Dinh T., Le Thi H.A.: Convex analysis approach to DC programming: Theory, algorithms and applications. Acta Math. Vietnam. 22, 289–355 (1997)Google Scholar
  20. 20.
    Pham Dinh T., Le Thi H.A.: DC optimization algorithm for solving the trust region problem. SIAM J. Optim. 8(2), 476–505 (1998)CrossRefGoogle Scholar
  21. 21.
    Pham Dinh, T., Le Thi, H.A.: DC (Difference of Convex functions) programming. Theory, algorithms, applications: The state of the art. In: Proceedings of the First International Workshop on Global Constrained Optimization and Constraint Satisfaction (Cocos’02). Valbonne Sophia Antipolis, France October 2–4 (2002)Google Scholar
  22. 22.
    Pham Dinh T., Le Thi H.A., Akoa F.: Combining DCA (DC Algorithms) and interior point techniques for large-scale nonconvex quadratic programming. Optim. Methods Softw. 23(4), 609–629 (2008)CrossRefGoogle Scholar
  23. 23.
    Pham Dinh, T., Nam, N.C., Le Thi, H.A.: An efficient combination of DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs. J. Glob. Optim. (to appear)Google Scholar
  24. 24.
    Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK User’s Guide : Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. October (1997)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  • Hoai An Le Thi
    • 1
    Email author
  • Tao Pham Dinh
    • 2
  • Nguyen Dong Yen
    • 3
  1. 1.Laboratory of Theoretical and Applied Computer Science (LITA)Paul Verlaine university-MetzMetz CedexFrance
  2. 2.Laboratory of Modelling, Optimization and Operations Research (LMI)National Institute for Applied Sciences (INSA)-RouenSaint-Etienne-du-Rouvray CedexFrance
  3. 3.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

Personalised recommendations