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Warm Start and ε-Subgradients in a Cutting Plane Scheme for Block-Angular Linear Programs

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Abstract

This paper addresses the issues involved with an interior point-based decomposition applied to the solution of linear programs with a block-angular structure. Unlike classical decomposition schemes that use the simplex method to solve subproblems, the approach presented in this paper employs a primal-dual infeasible interior point method. The above-mentioned algorithm offers a perfect measure of the distance to optimality, which is exploited to terminate the algorithm earlier (with a rather loose optimality tolerance) and to generate ∈-subgradients. In the decomposition scheme, subproblems are sequentially solved for varying objective functions. It is essential to be able to exploit the optimal solution of the previous problem when solving a subsequent one (with a modified objective). A warm start routine is described that deals with this problem. The proposed approach has been implemented within the context of two optimization codes freely available for research use: the Analytic Center Cutting Plane Method (ACCPM)—interior point based decomposition algorithm and the Higher Order Primal-Dual Method (HOPDM)—general purpose interior point LP solver. Computational results are given to illustrate the potential advantages of the approach applied to the solution of very large structured linear programs.

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References

  1. A. Altman and K.C. Kiwiel, “A note on some cutting plane methods for convex feasibility and minimization problems,” Computational Optimization and Applications, vol. 5, pp. 175–180, 1996.

    Google Scholar 

  2. E. Andersen, J. Gondzio, C. Meszaros, and X. Xu, “Implementation of interior point methods for large scale linear programming,” in Interior Point Methods in Mathematical Programming, T. Terlaky (Ed.), Kluwer Academic Publishers: Dordrecht 1996, pp. 189–252.

    Google Scholar 

  3. O. Bahn, A. Haurie, S. Kypreos, and J.-P. Vial, “A multinational MARKAL model to study joint implementation of carbon dioxide emission reduction measures,” in Joint Implementation of Climate Change Commitments, P. Ghosh and J. Puri (Eds.), Tata Energy Research Institute, 1994.

  4. J.F. Benders, “Partitioning procedures for solving mixed-variables programming problems,” Numerische Mathematik, vol. 4, pp. 238–252, 1962.

    Google Scholar 

  5. Cplex Optimization Inc., Incline Village, NV 89451–9436. Using the Cplex Callable Library, 1995.

  6. G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programming,” Econometrica, vol. 29, pp. 767–778, 1961.

    Google Scholar 

  7. J. Day and A. Haurie, Private communication, Logilab, University of Geneva, 1997.

  8. O. du Merle, J.-L. Goffin, and J.-P. Vial, “On the comparative behavior of Kelley's cutting plane method and the analytic center cutting plane method,” Technical Report, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, March 1996. Computational Optimization and Applications (in press).

  9. D.M. Gay, “Electronic mail distribution of linear programming test problems,” Mathematical Programming Society COAL Newsletter, vol. 13, pp. 10–12, 1985.

    Google Scholar 

  10. J.-L. Goffin, A. Haurie, and J.-P. Vial, “Decomposition and nondifferentiable optimization with the projective algorithm,” Management Science, vol. 38, pp. 284–302, 1992.

    Google Scholar 

  11. J.-L. Goffin, A. Haurie, J.-P. Vial, and D.L. Zhu, “Using central prices in the decomposition of linear programs,” European Journal of Operational Research, vol. 64, pp. 393–409, 1993.

    Google Scholar 

  12. J.-L. Goffin, Z.Q. Luo, and Y. Ye, “On the complexity of a column generation algorithm for convex and quasiconvex feasibility problems,” in Large Scale Optimization: State of the Art, W.W. Hager, D.W. Hearn, and P.M. Pardalos (Eds.), Kluwer Academic Publishers: Dordrecht, 1993, pp. 187–196.

    Google Scholar 

  13. J.-L. Goffin, Z.Q. Luo, and Y. Ye, “Complexity analysis of an interior point cutting plane method for convex feasibility problems,” SIAM Journal on Optimization, vol. 6, pp. 638–652, 1996.

    Google Scholar 

  14. J.-L. Goffin and J.-P. Vial, “Shallow, deep and very deep cuts in the analytic center cutting plane method,” Technical Report, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, May 1996. Mathematical Programming (in press).

  15. J. Gondzio, “HOPDM (version 2.12)—a fast LP solver based on a primal-dual interior point method,” European Journal of Operational Research, vol. 85, pp. 221–225, 1995.

    Google Scholar 

  16. J. Gondzio, “Multiple centrality corrections in a primal-dual method for linear programming,” Computational Optimization and Applications, vol. 6, pp. 137–156, 1996.

    Google Scholar 

  17. J. Gondzio, “Warm start of the primal-dual method applied in the cutting plane scheme,” Mathematical Programming, vol. 83, pp. 125–143, 1998.

    Google Scholar 

  18. J. Gondzio, O. du Merle, R. Sarkissian, and J.-P. Vial, “ACCPM—a library for convex optimization based on an analytic center cutting plane method,” European Journal of Operational Research, vol. 94, pp. 206–211, 1996.

    Google Scholar 

  19. J. Gondzio and R. Sarkissian, “Column generation with the primal-dual method,” Technical Report, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, June 1996.

  20. J. Gondzio and T. Terlaky, “A computational view of interior point methods for large scale linear programming,” in Advances in Linear and Integer Programming, J. Beasley (Ed.), Oxford University Press: Oxford, 1996, pp. 103–144.

    Google Scholar 

  21. J.E. Kelley, “The cutting plane method for solving convex programs,” Journal of the SIAM, vol. 8, pp. 703–712, 1960.

    Google Scholar 

  22. K.C. Kiwiel, “Efficiency of the analytic center cutting plane method for convex minimization,” SIAM Journal on Optimization, vol. 7, pp. 336–346, 1997.

    Google Scholar 

  23. I.J. Lustig, R.E. Marsten, and D.F. Shanno, “Interior point methods for linear programming: computational state of the art,” ORSA Journal on Computing, vol. 6, pp. 1–14, 1994.

    Google Scholar 

  24. N. Nakicenovic, A. Gruebler, A. Inaba, S. Messner, S. Nilsson, Y. Nishimura, H-H. Rogner, A. Schaefer, L. Schrattenholzer, M. Strubegger, J. Swisher, D. Victor, and D. Wilson, “Long-term strategies for mitigating global warming,” Energy—The International Journal, vol. 18,no. 5, pp. 409–601, 1993.

    Google Scholar 

  25. Y. Nesterov, “Complexity estimates of some cutting plane methods based on the analytic center,” Mathematical Programming, vol. 69, pp. 149–176, 1995.

    Google Scholar 

  26. J. Renegar, “A polynomial-time algorithm, based on Newton's method, for linear programming,” Mathematical Programming, vol. 40, pp. 59–93, 1988.

    Google Scholar 

  27. J.-P. Vial, “A generic path-following algorithm with a sliding constraint and its application to linear programming and the computation of analytic centers,” Technical Report, Logilab, University of Geneva, 102 Bd Carl-Vogt, CH-1211, February 1996.

  28. S. Wright, Primal-Dual Interior Point Methods, SIAM: Philadelphia, 1997.

    Google Scholar 

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  1. Logilab, HEC, Section of Management Studies, University of Geneva, 102 Bd Carl Vogt, CH-1211, Genève 4, Switzerland

    J. Gondizo & J.-P Vial

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  1. J. Gondizo
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  2. J.-P Vial
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Gondizo, J., Vial, JP. Warm Start and ε-Subgradients in a Cutting Plane Scheme for Block-Angular Linear Programs. Computational Optimization and Applications 14, 17–36 (1999). https://doi.org/10.1023/A:1008748810765

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