Advertisement

On the Efficiency of the ε-Subgradient Methods Over Nonlinearly Constrained Networks

  • E. Mijangos
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 199)

Abstract

The efficiency of the network flow techniques can be exploited in the solution of nonlinearly constrained network flow problems by means of approximate sub-gradient methods. In particular, we consider the case where the side constraints (non-network constraints) are convex. We propose to solve the dual problem by using ε-subgradient methods given that the dual function is estimated by minimizing approximately a Lagrangian function with only network constraints. Such Lagrangian function includes the side constraints. In order to evaluate the efficiency of these ε-subgradient methods some of them have been implemented and their performance computationally compared with that of other well-known codes. The results are encouraging.

keywords

Nonlinear Programming Approximate Subgradient Methods Network Flows 

References

  1. [1]
    D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.Google Scholar
  2. [2]
    D.P. Bertsekas, Nonlinear Programming. 2nd ed. Athena Scientific, Belmont, Massachusetts, 1999.Google Scholar
  3. [3]
    U. Brännlund. On relaxation methods for nonsmooth convex optimization. Doctoral Thesis, Royal Institute of Technology, Stockholm, Sweden, 1993.Google Scholar
  4. [4]
    R. Correa, C. Lemarechal. Convergence of some algorithms for convex minimization. Mathematical Programming, 62:261–275, 1993.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Czyzyk, M.P. Mesnier, J. Moré. The NEOS server. IEEE Computational Science and Engineering, 5(3):68–75, 1998.CrossRefGoogle Scholar
  6. [6]
    DIMACS. The first DIMACS international algorithm implementation challenge: The bench-mark experiments. Technical Report, DIMACS, New Brunswick, NJ, USA, 1991.Google Scholar
  7. [7]
    R. Fletcher and S. Leyffer. User manual for filterSQP, University of Dundee Numerical Analysis Report NA∖181, 1998.Google Scholar
  8. [8]
    R. Fourer, D.M. Gay, B.W. Kernighan. AMPL a modelling language for mathematical programming. Boyd and Fraser Publishing Company, Danvers, MA 01293, USA, 1993.Google Scholar
  9. [9]
    J.L. Goffin, K. Kiwiel. Convergence of a simple subgradient level method. Mathematical Programming, 85:207–211, 1999.MathSciNetCrossRefGoogle Scholar
  10. [10]
    K. Kiwiel. Convergence of approximate and incremental subgradient methods for convex optimization. SIAM Journal on Optimization, 14(3):807–840, 2004.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    E. Mijangos. An implementation of Newton-like methods on nonlinearly constrained networks. Computers and Operations Research, 32(2): 181–199, 2004.MathSciNetCrossRefGoogle Scholar
  12. [12]
    E. Mijangos. An efficient method for nonlinearly constrained networks. European Journal of Operational Research, 161(3):618–635, 2005.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    E. Mijangos, Approximate subgradient methods for nonlinearly constrained network flow problems. To appear in Journal of Optimization Theory and Applications, 128(1), 2006.Google Scholar
  14. [14]
    E. Mijangos, N. Nabona. The application of the multipliers method in nonlinear network flows with side constraints. Technical Report 96/10, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, Barcelona, Spain, 1996.Google Scholar
  15. [15]
    E. Mijangos, N. Nabona. On the first-order estimation of multipliers from Kuhn-Tucker systems. Computers and Operations Research, 28:243–270, 2001.MathSciNetCrossRefGoogle Scholar
  16. [16]
    B.A. Murtagh, M.A. Saunders. Large-scale linearly constrained optimization. Mathematical Programming, 14:41–72, 1978.MathSciNetCrossRefGoogle Scholar
  17. [17]
    B.A. Murtagh, M.A. Saunders. MINOS 5.5. User’s guide. Report SOL 83-20R, Department of Operations Research, Stanford University, Stanford, CA, USA, 1998.Google Scholar
  18. [18]
    A. Nedić, D.P. Bertsekas. Incremental subgradient methods for nondifferentiable optimization. SIAM Journal on Optimization, 12:109–138, 2001.MathSciNetCrossRefGoogle Scholar
  19. [19]
    B.T. Poljak. Minimization of unsmooth functionals, Z. Vyschisl. Mat. i Mat. Fiz., 9:14–29, 1969.Google Scholar
  20. [20]
    N.Z. Shor. Minimization methods for nondifferentiable functions. Springer-Verlag, Berlin, 1985.Google Scholar
  21. [21]
    Ph.L. Toint, D. Tuyttens. On large scale nonlinear network optimization. Mathematical Programming, 48:125–159, 1990.MathSciNetCrossRefGoogle Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • E. Mijangos
    • 1
  1. 1.Department of Applied Mathematics, Statistics and Operations ResearchUniversity of the Basque CountrySpain

Personalised recommendations