Advertisement

Computational Optimization and Applications

, Volume 47, Issue 1, pp 33–59 | Cite as

Extended duality for nonlinear programming

  • Yixin Chen
  • Minmin Chen
Article

Abstract

Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex.

In this paper, we propose an extended duality theory for nonlinear optimization in order to overcome some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed spaces under mild conditions. Comparing to recent developments in nonlinear Lagrangian functions and exact penalty functions, the proposed theory always requires lesser penalty to achieve zero duality. This is very desirable as the lower function value leads to smoother search terrains and alleviates the ill conditioning of dual optimization.

Based on the extended duality theory, we develop a general search framework for global optimization. Experimental results on engineering benchmarks and a sensor-network optimization application show that our algorithm achieves better performance than searches based on conventional duality and Lagrangian theory.

Keywords

Nonlinear programming Global optimization Duality gap Extended duality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aubin, J.P., Ekeland, I.: Estimates of the duality gap in nonconvex optimization. Math. Oper. Res. 1, 225–245 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ben-Tal, A., Eiger, G., Gershovitz, V.: Global optimization by reducing the duality gap. Math. Program. 63, 193–212 (1994) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Distributed dynamic programming. Trans. Autom. Control AC-27(3), 610–616 (1982) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999) zbMATHGoogle Scholar
  5. 5.
    Burachik, R.S., Rubinov, A.: On the absence of duality gap for Lagrange-type functions. J. Indust. Manag. Optim. 1(1), 33–38 (2005) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Burke, J.V.: Calmness and exact penalization. SIAM J. Control Optim. 29, 493–497 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Conn, A.R., Gould, N.I.M., Toint, Ph.L.: LANCELOT: A Fortran Package for Large-Scale Nonlinear Optimization. Heidelberg, Springer (1992) zbMATHGoogle Scholar
  9. 9.
    Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms. Lecture Notes in Computer Science, vol. 455. Springer, Berlin (1990) zbMATHGoogle Scholar
  10. 10.
    Gill, P.E., Murray, W., Saunders, M.: SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12, 979–1006 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gould, N.I.M., Orban, D., Toint, P.L.: An interior-point 1-penalty method for nonlinear optimization. Technical Report RAL-TR-2003-022, Rutherford Appleton Laboratory Chilton, Oxfordshire, UK, November (2003) Google Scholar
  12. 12.
    Huang, X.X., Yang, X.Q.: A unified augmented Lagrangian approach to duality and exact penalization. Math. Oper. Res. 28(3), 533–552 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Koziel, S., Michalewics, Z.: Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolut. Comput. 7(1), 19–44 (1999) CrossRefGoogle Scholar
  14. 14.
    Luo, Z.Q., Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B, 88(2) (2000) Google Scholar
  15. 15.
    Nedić, A., Ozdaglar, A.: A geometric framework for nonconvex optimization duality using augmented Lagrangian functions. J. Glob. Optim. 40(4), 545–573 (2008) CrossRefGoogle Scholar
  16. 16.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997) Google Scholar
  17. 17.
    Rockafellar, R.T.: Augmented Lagrangian multiplier functions and duality in nonconvex programming. SIAM J. Control Optim. 12, 268–285 (1974) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) zbMATHCrossRefGoogle Scholar
  19. 19.
    Rubinov, A.M., Glover, B.M., Yang, X.Q.: Decreasing functions with applications to penalization. SIAM J. Optim. 10, 289–313 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rubinov, A.M., Glover, B.M., Yang, X.Q.: Modified Lagrangian and penalty functions in continuous optimization. Optimization 46, 327–351 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Tuy, H.: On solving nonconvex optimization problems by reducing the duality gap. J. Glob. Optim. 32, 349–365 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ferrier, G.D., Goffe, W.L., Rogers, J.: Global optimization of statistical functions with simulated annealing. J. Econ. 60(1), 65–99 (1994) zbMATHGoogle Scholar
  23. 23.
    Wang, X., Xing, G., Zhang, Y., Lu, C., Pless, R., Gill, C.: Integrated coverage and connectivity configura-tion in wireless sensor networks. In: Proc. First ACM Conference on Embedded Networked Sensor Systems (2003) Google Scholar
  24. 24.
    Xing, G., Lu, C., Pless, R., Huang, Q.: On greedy geographic routing algorithms in sensing-covered networks. In: Proc. ACM International Symposium on Mobile Ad Hoc Networking and Computing (2004) Google Scholar
  25. 25.
    Xing, G., Lu, C., Pless, R., O’Sullivan, J.A.: Co-Grid: An efficient coverage maintenance protocol for distributed sensor networks. In: Proc. International Symposium on Information Processing in Sensor Networks (2004) Google Scholar
  26. 26.
    Yang, X.Q., Huang, X.X.: A nonlinear Lagrangian approach to constraint optimization problems. SIAM J. Optim. 11, 1119–1144 (2001) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringWashington UniversitySt. LouisUSA

Personalised recommendations