# A cutting plane method from analytic centers for stochastic programming

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## Abstract

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.

## Keywords

Cutting plane Stochastic programming Analytic center Interior-point method## Preview

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## References

- [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]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]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]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]J.F. Benders, “Partitioning procedures for solving mixed-variables programming problems,”
*Numerische Mathematik*4 (1962) 238–252.Google Scholar - [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]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]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]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]R.W. Cottle, “Manifestations of the Schur complement,”
*Linear Algebra and Applications*8 (1974) 189–211.Google Scholar - [11]G.B. Dantzig, “Linear programming under uncertainty,”
*Management Science*1 (1955) 197–206.Google Scholar - [12]G.B. Dantzig,
*Linear Programming and Extensions*(Princeton University Press, Princeton, NJ, 1963).Google Scholar - [13]G.B. Dantzig, “Planning under uncertainty using parallel computing,”
*Annals of Operations Research*14 (1988) 1–16.Google Scholar - [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]G.B. Dantzig and P. Wolfe, “The decomposition algorithm for linear programming,”
*Econometrica*29 (1961) 767–778.Google Scholar - [16]G. de Ghellinck and J.-P. Vial, “A polynomial Newton method for linear programming,”
*Algorithmica*1 (1986) 425–453.Google Scholar - [17]M. Dempster, ed.,
*Stochastic Programming*(Academic Press, New York, 1980).Google Scholar - [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]J.J. Dongarra, C.B. Moler, J.R. Bunch and G.W. Stewart,
*LINPACK User's Guide*(SIAM, Philadelphia, PA, 1979).Google Scholar - [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]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]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]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]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]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]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]C.C. Gonzaga, “Path following methods for linear programming,”
*SIAM Review*34 (1992) 167–227.Google Scholar - [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]N. Karmarkar, “A new polynomial time algorithm for linear programming,”
*Combinatorica*4 (1984) 373–395.Google Scholar - [30]J.E. Kelley, “The cutting plane method for solving convex programs,”
*Journal of the SIAM*8 (1960) 703–712.Google Scholar - [31]L.S. Lasdon,
*Optimization Theory for Large Scale Systems*(Macmillan, New York, 1970).Google Scholar - [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]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]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]D. Mehdi, “Parallel bundle-based decomposition for large-scale structured mathematical programming problems,”
*Annals of Operations Research*22 (1990) 101–127.Google Scholar - [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]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]A. Prékopa and R.J.-B. Wets, eds.,
*Stochastic Programming 84: Part I, Mathematical Programming Study*27 (1986).Google Scholar - [39]A. Prékopa and R.J.-B. Wets, eds.,
*Stochastic Programming 84: Part II, Mathematical Programming Study*28 (1986).Google Scholar - [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]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]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]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]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]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]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]Y. Ye, “A potential reduction algorithm allowing column generation,”
*SIAM Journal on Optimization*2 (1992) 7–20.Google Scholar - [48]“CPLEX User's Guide,” CPLEX Optimization, Inc., Incline Village, NV, 1992.Google Scholar
- [49]“Optimization Subroutine Library, Guide and Reference,” IBM Corp., Kingston, NY, 1991.Google Scholar