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Lagrangian Relaxations on Networks by ε-Subgradient Methods

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Abstract

The efficiency of the network flow techniques can be exploited in the solution of nonlinearly constrained network flow problems by means of approximate subgradient methods. The idea is to solve the dual problem by using ε-subgradient methods, where the dual function is estimated by minimizing approximately a Lagrangian function, which relaxes the side constraints and is subject only to network constraints. In this paper, convergence results for some kind of ε-subgradient methods are put forward. Moreover, in order to evaluate the quality of the solution and the efficiency of these methods some of them have been implemented and their performances computationally compared with codes that are able to solve the proposed test problems.

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References

  1. Mijangos, E.: An implementation of Newton-like methods on nonlinearly constrained networks. Comput. Oper. Res. 32(2), 181–199 (2004)

    Article  MathSciNet  Google Scholar 

  2. Mijangos, E.: An efficient method for nonlinearly constrained networks. Eur. J. Oper. Res. 161(3), 618–635 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kiwiel, K.: Convergence of approximate and incremental subgradient methods for convex optimization. SIAM J. Optim. 14(3), 807–840 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Mijangos, E.: Approximate subgradient methods for nonlinearly constrained network flow problems. J. Optim. Theory Appl. 128(1), 167–190 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mijangos, E.: On the efficiency of the ε-subgradient methods over nonlinearly constrained networks. In: System Modeling and Optimization. IFIP Advances in Information and Communication Technology, vol. 199, pp. 101–111 (2006)

    Chapter  Google Scholar 

  6. Nedić, A., Ozdaglar, A.: Approximate primal solutions and rate analysis for dual subgradient methods. SIAM J. Optim. 19(4), 1757–1780 (2009)

    Article  MATH  Google Scholar 

  7. Shor, N.Z.: Minimization Methods for Nondifferentiable Functions. Springer, Berlin (1985)

    Book  Google Scholar 

  8. Ermoliev, Yu.M.: Methods for solving nonlinear extremal problems. Cybernetics 2, 1–17 (1966)

    Article  Google Scholar 

  9. Poljak, B.T.: Introduction to Optimization. Optimization Software Inc., New York (1987)

    Google Scholar 

  10. Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–275 (1993)

    Article  MATH  Google Scholar 

  11. Nedić, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiable optimization. SIAM J. Optim. 12(1), 109–138 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Brännlund, U.: On relaxation methods for nonsmooth convex optimization. Ph.D. Thesis, Royal Institute of Technology, Stockholm, Sweden (1993)

  13. Goffin, J.L., Kiwiel, K.: Convergence of a simple subgradient level method. Math. Program. 85, 207–211 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Byrd, R.H., Nocedal, J., Waltz, R.A.: KNITRO: an integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds.) Large-Scale Nonlinear Optimization. Springer, Berlin (2006), pp. 35–59

    Chapter  Google Scholar 

  16. Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  17. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  18. Toint, Ph.L., Tuyttens, D.: On large scale nonlinear network optimization. Math. Program. 48, 125–159 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)

    MATH  Google Scholar 

  20. Murtagh, B.A., Saunders, M.A.: Large-scale linearly constrained optimization. Math. Program. 14, 41–72 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mijangos, E., Nabona, N.: 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, Spain (1996) (downloadable from website http://www.ehu.es/~mepmifee/)

  22. Czyzyk, J., Mesnier, M.P., Moré, J.: The NEOS server. IEEE Comput. Sci. Eng. 5(3), 68–75 (1998)

    Article  Google Scholar 

  23. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL, a Modelling Language for Mathematical Programming. Boyd & Fraser, Danvers (1993)

    Google Scholar 

  24. DIMACS: The first DIMACS international algorithm implementation challenge: The bench-mark experiments. Technical Report, DIMACS, New Brunswick, NJ, USA (1991)

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics and Operations Research, University of the Basque Country, P.O. Box 644, 48080, Bilbao, Spain

    E. Mijangos

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  1. E. Mijangos
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Correspondence to E. Mijangos.

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Communicated by F. Zirilli.

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Mijangos, E. Lagrangian Relaxations on Networks by ε-Subgradient Methods. J Optim Theory Appl 152, 51–74 (2012). https://doi.org/10.1007/s10957-011-9881-8

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  • DOI: https://doi.org/10.1007/s10957-011-9881-8

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