Multicut L-Shaped Algorithm for Stochastic Convex Programming with Fuzzy Probability Distribution

Affiliation(s)

School of Mathematics and Physics North China Electric Power University Baoding, Hebei Province, China.

School of Mathematics and Physics North China Electric Power University Baoding, Hebei Province, China.

ABSTRACT

Two-stage problem of stochastic convex programming with fuzzy probability distribution is studied in this paper. Multicut L-shaped algorithm is proposed to solve the problem based on the fuzzy cutting and the minimax rule. Theorem of the convergence for the algorithm is proved. Finally, a numerical example about two-stage convex recourse problem shows the essential character and the efficiency.

Two-stage problem of stochastic convex programming with fuzzy probability distribution is studied in this paper. Multicut L-shaped algorithm is proposed to solve the problem based on the fuzzy cutting and the minimax rule. Theorem of the convergence for the algorithm is proved. Finally, a numerical example about two-stage convex recourse problem shows the essential character and the efficiency.

Cite this paper

nullHan, M. and MA, X. (2012) Multicut L-Shaped Algorithm for Stochastic Convex Programming with Fuzzy Probability Distribution.*Open Journal of Applied Sciences*, **2**, 219-222. doi: 10.4236/ojapps.2012.24B050.

nullHan, M. and MA, X. (2012) Multicut L-Shaped Algorithm for Stochastic Convex Programming with Fuzzy Probability Distribution.

References

[1] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, M. Berlin: Springer, 1997.

[2] R. M. Van Slyke and R. J. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming,” J. Applied. Vol. 17(4), pp.638–663, Math 1969.

[3] J. F. Benders, “Partitioning procedures for solving mixed-variable programming problems,” J. Numerical Mathematics, Vol. 4. 1962, pp.238-252.

[4] J. M. Higle and S. Sen, “Stochastic decomposition-an algorithm for 2-stage linear programs with recourse,” J. Mathematics of Operations Research, Vol. 16, 1991, pp. 650-669.

[5] S. Sen, “Subgradient decomposition and differentiability of the recourse function of a 2-stage stochastic linear program,” J. Operations Research Letters, Vol. 13, 1993, pp.143-148.

[6] S. Sen, Z. Zhou and K. Huang, “Enhancements of two-stage stochastic decomposition,” J. Computers & Operations Research, Vol. 36, 2009, pp. 2434-2439.

[7] T. W. Archibald, C. S. Buchanan, K. I. M. Mckinnon and L. C. Thomas, “Nested Benders decomposition and dynamic programming for reservoir optimization,” J. Operational Research Society, Vol. 50,1999, pp.468-479.

[8] L. Ntaimo, “Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse,” J. Operations Research, Vol. 58, 2010, pp. 229-243.

[9] F. B. Abdelaziz and H. Masri, “Stochastic programming with fuzzy linear partial information on probability distribution,” J. European Journal of Operational Research, Vol. 162, 2005, pp. 619-629.

[10] H.Y. Tang and Y.F. Zhao, “L-shaped algorithm for two stage problems of stochastic convex programming,” J. Applied. Math. & Computing Vol. B, 13(1-2), 2003, pp. 261-275.

[11] S. Boyd and L. Vandenberghe, “Convex optimization,” J. Published in the United States of America by Cambridge University Press, New York, Vol. 6, 2004, pp.68-86.

[1] J. R. Birge and F. Louveaux, Introduction to Stochastic Programming, M. Berlin: Springer, 1997.

[2] R. M. Van Slyke and R. J. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming,” J. Applied. Vol. 17(4), pp.638–663, Math 1969.

[3] J. F. Benders, “Partitioning procedures for solving mixed-variable programming problems,” J. Numerical Mathematics, Vol. 4. 1962, pp.238-252.

[4] J. M. Higle and S. Sen, “Stochastic decomposition-an algorithm for 2-stage linear programs with recourse,” J. Mathematics of Operations Research, Vol. 16, 1991, pp. 650-669.

[5] S. Sen, “Subgradient decomposition and differentiability of the recourse function of a 2-stage stochastic linear program,” J. Operations Research Letters, Vol. 13, 1993, pp.143-148.

[6] S. Sen, Z. Zhou and K. Huang, “Enhancements of two-stage stochastic decomposition,” J. Computers & Operations Research, Vol. 36, 2009, pp. 2434-2439.

[7] T. W. Archibald, C. S. Buchanan, K. I. M. Mckinnon and L. C. Thomas, “Nested Benders decomposition and dynamic programming for reservoir optimization,” J. Operational Research Society, Vol. 50,1999, pp.468-479.

[8] L. Ntaimo, “Disjunctive decomposition for two-stage stochastic mixed-binary programs with random recourse,” J. Operations Research, Vol. 58, 2010, pp. 229-243.

[9] F. B. Abdelaziz and H. Masri, “Stochastic programming with fuzzy linear partial information on probability distribution,” J. European Journal of Operational Research, Vol. 162, 2005, pp. 619-629.

[10] H.Y. Tang and Y.F. Zhao, “L-shaped algorithm for two stage problems of stochastic convex programming,” J. Applied. Math. & Computing Vol. B, 13(1-2), 2003, pp. 261-275.

[11] S. Boyd and L. Vandenberghe, “Convex optimization,” J. Published in the United States of America by Cambridge University Press, New York, Vol. 6, 2004, pp.68-86.