Mathematical Programming

, Volume 69, Issue 1–3, pp 45–73 | Cite as

A cutting plane method from analytic centers for stochastic programming

  • O. Bahn
  • O. du Merle
  • J. -L. Goffin
  • J. -P. Vial


The stochastic linear programming problem with recourse has a dual block-angular structure. It can thus be handled by Benders' decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block-angular structure and can be handled by Dantzig-Wolfe decomposition—the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization.


Cutting plane Stochastic programming Analytic center Interior-point method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D.S. Atkinson and P.M. Vaidya, “A scaling technique for finding the weighted analytic center of a polytope,” University of Illinois at Urbana-Champaign, Urbana, IL, 1992.Google Scholar
  2. [2]
    D.S. Atkinson and P.M. Vaidya, “A cutting plane algorithm for convex programming that uses analytic centers,”Mathematical Programming 69 (1) (1995) 1–43 (this issue).Google Scholar
  3. [3]
    O. Bahn, J.-L. Goffin, J.-P. Vial and O. du Merle, “Experimental behavior of an interior point cutting plane algorithm for convex programming: an application to geometric programming,”Discrete Applied Mathematics 49 (1–3) (1994) 3–23.Google Scholar
  4. [4]
    E.M. Beale, “On minimizing a convex function subject to linear inequalities,”Journal of the Royal Statistical Society, Series B 17 (1955) 173–184.Google Scholar
  5. [5]
    J.F. Benders, “Partitioning procedures for solving mixed-variables programming problems,”Numerische Mathematik 4 (1962) 238–252.Google Scholar
  6. [6]
    J.R. Birge and D.F. Holmes, “Efficient solution of two stage stochastic linear programs using interior point methods,”Computational Optimization and Applications 1 (1992) 245–276.Google Scholar
  7. [7]
    J.R. Birge and F.V. Louveaux, “A multicut algorithm for two-stage stochastic linear programs,”European Journal of Operations Research 34 (1988) 384–392.Google Scholar
  8. [8]
    J.R. Birge and L. Qi, “Computing block-angular Karmarkar projections with applications to stochastic programming,”Management Science 34 (1988) 1472–1479.Google Scholar
  9. [9]
    I.C. Choi and D. Goldfarb, “Exploiting special structure in a primal—dual path-following algorithm,”Mathematical Programming 58 (1) (1993) 33–52.Google Scholar
  10. [10]
    R.W. Cottle, “Manifestations of the Schur complement,”Linear Algebra and Applications 8 (1974) 189–211.Google Scholar
  11. [11]
    G.B. Dantzig, “Linear programming under uncertainty,”Management Science 1 (1955) 197–206.Google Scholar
  12. [12]
    G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).Google Scholar
  13. [13]
    G.B. Dantzig, “Planning under uncertainty using parallel computing,”Annals of Operations Research 14 (1988) 1–16.Google Scholar
  14. [14]
    G.B. Dantzig and A. Madansky, “On the solution of two-stage linear programs under uncertainty,” in: J. Neyman, ed.,Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (University of California Press, Berkeley, CA, 1961) pp. 165–176.Google Scholar
  15. [15]
    G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programming,”Econometrica 29 (1961) 767–778.Google Scholar
  16. [16]
    G. de Ghellinck and J.-P. Vial, “A polynomial Newton method for linear programming,”Algorithmica 1 (1986) 425–453.Google Scholar
  17. [17]
    M. Dempster, ed.,Stochastic Programming (Academic Press, New York, 1980).Google Scholar
  18. [18]
    D. den Hertog, “Interior point approach to linear, quadratic and convex programming: algorithms and complexity,” Ph.D. Thesis, Faculty of Mathematics and Informatics, Technical University Delft, 1992.Google Scholar
  19. [19]
    J.J. Dongarra, C.B. Moler, J.R. Bunch and G.W. Stewart,LINPACK User's Guide (SIAM, Philadelphia, PA, 1979).Google Scholar
  20. [20]
    Yu. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics 10 (Springer, New York, 1988).Google Scholar
  21. [21]
    L. Escudero, P.K. Kamesam, A.J. King and R.J.-B. Wets, “Production planning via scenario modelling,”Annals of Operations Research 43 (1993) 311–355.Google Scholar
  22. [22]
    J.-L. Goffin, A. Haurie and J.-P. Vial, “Decomposition and nondifferentiable optimization with the projective algorithm,”Management Science 38 (1992) 284–302.Google Scholar
  23. [23]
    J.-L. Goffin, Z.-Q. Luo and Y. Ye, “Further complexity analysis of a primal-dual column generation algorithm for convex or quasiconvex feasibility problems,” Manuscript, 1993.Google Scholar
  24. [24]
    J.-L. Goffin and J.-P. Vial, “On the computation of weighted analytic centers and dual ellipsoids with the projective algorithm,”Mathematical Programming 60 (1) (1993) 81–92.Google Scholar
  25. [25]
    C.C. Gonzaga, “Large steps path-following methods for linear programming, Part I: Barrier function method,”SIAM Journal on Optimization 1 (1991) 268–279.Google Scholar
  26. [26]
    C.C. Gonzaga, “Large steps path-following methods for linear programming, Part II: Potential reduction method,”SIAM Journal on Optimization 1 (1991) 280–292.Google Scholar
  27. [27]
    C.C. Gonzaga, “Path following methods for linear programming,”SIAM Review 34 (1992) 167–227.Google Scholar
  28. [28]
    P. Kall, “Computational methods for solving two-stage stochastic linear programming problems,”Zeitschrift für Angewandte Mathematik und Physik 30 (1979) 261–271.Google Scholar
  29. [29]
    N. Karmarkar, “A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.Google Scholar
  30. [30]
    J.E. Kelley, “The cutting plane method for solving convex programs,”Journal of the SIAM 8 (1960) 703–712.Google Scholar
  31. [31]
    L.S. Lasdon,Optimization Theory for Large Scale Systems (Macmillan, New York, 1970).Google Scholar
  32. [32]
    E. Loute, “A revised simplex method for block structured linear programs,” Ph.D. Thesis, Université Catholique de Louvain, Louvain-la-Neuve, 1976.Google Scholar
  33. [33]
    E. Loute and J.-P. Vial, “A parallelisable block Cholesky factorization for staircase linear programming problems,” Technical Report 1992.15, Department of Management Studies, Faculté des S.E.S., University of Geneva, 1992.Google Scholar
  34. [34]
    I.J. Lustig, J.M. Mulvey and T.J. Carpenter, “The formulation of stochastic programs for interior point methods,”Operations Research 39 (1991) 757–770.Google Scholar
  35. [35]
    D. Mehdi, “Parallel bundle-based decomposition for large-scale structured mathematical programming problems,”Annals of Operations Research 22 (1990) 101–127.Google Scholar
  36. [36]
    J.E. Mitchell and M.J. Todd, “Solving combinatorial optimization problems using Karmarkar's algorithm,”Mathematical Programming 56 (3) (1992) 245–284.Google Scholar
  37. [37]
    Yu. Nesterov, “Complexity estimates of some cutting plane methods based on the analytic barrier,”Mathematical Programming 69 (1) (1995) 149–176 (this issue).Google Scholar
  38. [38]
    A. Prékopa and R.J.-B. Wets, eds.,Stochastic Programming 84: Part I, Mathematical Programming Study 27 (1986).Google Scholar
  39. [39]
    A. Prékopa and R.J.-B. Wets, eds.,Stochastic Programming 84: Part II, Mathematical Programming Study 28 (1986).Google Scholar
  40. [40]
    J. Renegar, “A polynomial-time algorithm based on Newton's method, for linear programming,”Mathematical Programming 40 (1) (1988) 59–93.Google Scholar
  41. [41]
    S.M. Robinson, “Bundle-based decomposition: conditions for convergence,” in: H. Attouch, J.-P. Aubin, F. Clarke and I. Ekeland, eds.,Analyse Non Linéaire (Gauthier-Villars, Paris, 1989) pp. 435–447.Google Scholar
  42. [42]
    C. Roos and J.-P. Vial, “A polynomial method of approximate centers for linear programming,”Mathematical Programming 54 (3) (1992) 295–305.Google Scholar
  43. [43]
    G. Sonnevend, “New algorithms in convex programming based on a notion of ‘centre’ (for systems of analytic inequalities) and on rational extrapolation,” in: K.H. Hoffmann, J.B. Hiriat-Urruty, C. Lemarechal and J. Zowe, eds.,Trends in Mathematical Optimization, Proceedings of the Fourth French-German Conference on Optimization, Irsee, 1986, International Series of Numerical Mathematics 84 (Birkhäuser, Basel, 1988) pp. 311–327.Google Scholar
  44. [44]
    R. Van Slyke and R.J.-B. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming,”SIAM Journal on Applied Mathematics 17 (1969) 638–663.Google Scholar
  45. [45]
    R.J.-B. Wets, “Large-scale linear programming techniques in stochastic programming,” in: Yu. Ermoliev and R.J.-B. Wets, eds.,Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics 10 (Springer, New York, 1988) pp. 65–93.Google Scholar
  46. [46]
    R.J.-B. Wets, “Stochastic programming,” in: G.L. Nemhauser, A.H.G. Rinnooy Kan and M.J. Todd, eds.,Optimization (North-Holland, Amsterdam, 1989) pp. 583–629.Google Scholar
  47. [47]
    Y. Ye, “A potential reduction algorithm allowing column generation,”SIAM Journal on Optimization 2 (1992) 7–20.Google Scholar
  48. [48]
    “CPLEX User's Guide,” CPLEX Optimization, Inc., Incline Village, NV, 1992.Google Scholar
  49. [49]
    “Optimization Subroutine Library, Guide and Reference,” IBM Corp., Kingston, NY, 1991.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1995

Authors and Affiliations

  • O. Bahn
    • 1
  • O. du Merle
    • 1
  • J. -L. Goffin
    • 2
  • J. -P. Vial
    • 1
  1. 1.Département d'Économie Commerciale et IndustrielleUniversité de GenèveGenèveSwitzerland
  2. 2.GERAD, Faculty of ManagementMcGill UniversityMontrealCanada

Personalised recommendations