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The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification

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Abstract

This paper considers the two-stage stochastic integer programming problem, with an emphasis on instances in which integer variables appear in the second stage. Drawing heavily on the theory of disjunctive programming, we characterize convexifications of the second stage problem and develop a decomposition-based algorithm for the solution of such problems. In particular, we verify that problems with fixed recourse are characterized by scenario-dependent second stage convexifications that have a great deal in common. We refer to this characterization as the C 3 (Common Cut Coefficients) Theorem. Based on the C 3 Theorem, we develop a decomposition algorithm which we refer to as Disjunctive Decomposition (D 2). In this new class of algorithms, we work with master and subproblems that result from convexifications of two coupled disjunctive programs. We show that when the second stage consists of 0-1 MILP problems, we can obtain accurate second stage objective function estimates after finitely many steps. This result implies the convergence of the D 2 algorithm.

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References

  1. Balas, E.: Disjunctive programming. Ann. Disc. Math. 5, 3–31 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balas, E.: A modified lift-and-project procedure. Math. Prog. Series B 79, 19–31 (1997)

    Article  MathSciNet  Google Scholar 

  3. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0-1 programs. Math. Prog. 58, 295–324 (1993)

    Article  MathSciNet  Google Scholar 

  4. Balas, E., Ceria, S, Cornuéjols, G.: Mixed 0-1 programming by lift-and-project in a branch-and-cut framework. Management Sci. 42, 1229–1246 (1996)

    MATH  Google Scholar 

  5. Benders, J.F.: Partitioning procedures for solving mixed-variable programming problems. Numerische Math. 4, 238–252 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  6. Blair: Facial disjunctive programs and sequence of cutting planes. Disc. Appl. Math. 2, pp. 173–179 (1980)

  7. Blair, C.: A closed-form representation of mixed-integer program value functions. Math. Prog. 71, 127–136 (1995)

    MathSciNet  Google Scholar 

  8. Blair, C., Jeroslow, R.: A converse for disjunctive constraints. J. Optim. Theor. Appl. 25, 195–206 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  9. Blair, C., Jeroslow, R.: The value function of an integer program, Math. Prog. 23, 237–273 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer (1997)

  11. Caroe, C.C.: Decomposition in Stochastic Integer Programming. Ph.D. thesis, Institute of Mathematical Sciences, Dept. of Operations Research, University of Copenhagen, Denmark (1998)

  12. Caroe, C.C., Tind, J.: A cutting plane approach to mixed 0-1 stochastic integer programs. European J. Oper. Res. 101, 306–316 (1997)

    Article  Google Scholar 

  13. Caroe, C.C., Tind, J.: L-shaped decomposition of two-stage stochastic programs with integer recourse. Math. Prog. 83 (3), 139–152 (1998)

    MathSciNet  Google Scholar 

  14. Conti, P., Traverso, C.: Buchberger algorithm and integer programming. Proceedings of AAECC-9 New Orleans, Springer Verlag LNCS 539, pp. 130–139 (1991)

  15. Glover, F., Laguna, M.: Tabu Search, Kluwer Academic Publishers (1997)

  16. Higle, J.L., Sen, S.: On the convergence of algorithms with implications for stochastic and nondifferentiable optimization. Math. Oper. Res. 17, 112–131 (1992)

    MATH  MathSciNet  Google Scholar 

  17. Higle, J.L., Sen, S.: Epigraphical Nesting: a unifying theory for the convergence of algorithms. J. Optim. Theor. Appl. 84, 339-360 (1995)

    MATH  MathSciNet  Google Scholar 

  18. Jeroslow, R.: A cutting plane game for facial disjunctive programs. SIAM J. Cont. Optim. 18, 264–281 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kelley, J.E.: The cutting plane method for convex programs, J. SIAM 8, 703–712 (1960)

    MathSciNet  Google Scholar 

  20. Klein Haneveld, W.K., Stougie, L., van der Vlerk, M.H.: On the convex hull of the simple integer recourse objective function. Ann. Oper. Res. 56, 209–224 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Klein Haneveld, W.K., Stougie, L., van der Vlerk, M.H.: An algorithm for the construction of convex hulls in simple integer recourse programming. Ann. Oper. Res. 64, 67–81 (1996)

    MATH  MathSciNet  Google Scholar 

  22. Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic integer programming: general models and algorithms. Ann. Oper. Res. 85, 39–57 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Laporte, G., Louveaux, F.V.: The integer L-shaped methods for stochastic integer programs with complete recourse. Oper. Res. Lett. 13, 133–142 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lokketangen, A., Woodruff, D.L.: Progressive hedging and tabu search applied to mixed integer (0,1) multi-stage stochastic programming. J. Heuristics 2, 111–128 (1996)

    Google Scholar 

  25. Martin, R.K.: Large Scale Linear and Integer Optimization. Kluwer Academic Publishers (1999)

  26. McDaniel, D., Devine, M.: A modified Benders’ partitioning algorithm for mixed integer programming. Management Science 24, 312–319 (1977)

    MATH  Google Scholar 

  27. Norkin, V.I., Ermoliev, Y.M., Ruszczynski, A.: On optimal allocation of indivisibles under uncertainty. Oper. Res. 46 (3), 381–395 (1998)

    Google Scholar 

  28. Nowak, M., Römisch, W.: Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty. to appear in, Ann. Oper. Res. (2000)

  29. Richter, Jr., C.W., Sheblé, G.B.: Building and evaluating Genco bidding strategies and unit commitment schedules with Genetic Algorithms. In: Next Generation of Electric Power Unit Commitment Models, B.F. Hobbs et al, (eds.), pp. 185-209 (2001)

  30. Rockafellar, R.T., Wets R.J-B.: Scenario and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16, 119–147 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rockafellar, R.T., Wets R.J-B.: Variational Analysis, Springer (1997)

  32. Schultz, R.: Continuity properties of expectation functions in stochastic integer programming. Math. Oper. Res. 18, 578–589 (1993)

    MATH  MathSciNet  Google Scholar 

  33. Schultz, R., Stougie, L., van der Vlerk, M.H.: Solving stochastic programs with integer recourse by enumeration: a framework using Grobner basis reduction. Math. Prog. 83 (2), 71–94 (1998)

    Google Scholar 

  34. Sen, S.: Relaxations for probabilistically constrained programs with discrete random variables. Oper. Res. Lett. 11, 81–86 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sen, S., Sherali, H.D.: On the convergence of cutting plane algorithms for a class of nonconvex mathematical programs. Math. Prog. 31, 42–56 (1985)

    MATH  Google Scholar 

  36. Sen, S., Sherali, H.D.: A branch and bound algorithm for extreme point mathematical programming. Disc. Appl. Math. 11, 265–280 (1985)

    Article  MATH  Google Scholar 

  37. Sen, S., Sherali, H.D.: Facet inequalities from simple disjunctions in cutting plane theory. Math. Prog. 34, 72–83 (1986)

    Article  MATH  Google Scholar 

  38. Sen, S., Sherali, H.D.: Nondifferentiable reverse convex programs and facetial cuts via a disjunctive characterization. Math. Prog. 37, 169–183 (1987)

    MATH  Google Scholar 

  39. Sherali, H.D., Adams: A hierarchy of relaxations between the continuous and convex hull representations for 0-1 programming problems. SIAM J. Disc. Math. 3, 411–430 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  40. Sherali, H.D., Fraticelli, B.M.P: A modified Benders’ partitioning approach for problems having discrete subproblems with application to stochastic programs with integer recourse. Working Paper, Virginia Polytechnic Institute and State University, Blacksburg, VA (2000)

  41. Sherali, H.D., Shetty, C.M.: Optimization with Disjunctive Constraints, Lecture Notes in Economics and Math. Systems, Vol. 181, Springer-Verlag, Berlin (1980)

  42. Takriti, S., Birge, J.R., Long, E.: A stochastic model for the unit commitment problem. IEEE Trans. Power Sys. 11, 1497–1508 (1996)

    Article  Google Scholar 

  43. Thomas, R.: A geometric Buchberger algorithm for integer programming. Math. Oper. Res. 20, 864–884, (1995)

    MATH  MathSciNet  Google Scholar 

  44. van der Vlerk, M.H.: Stochastic Programming with Integer Recourse, Thesis Rijksuniversiteit Groningen, Labyrinth Publication, The Netherlands (1995)

  45. van Slyke, R., Wets, R.J-B.: L-Shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  46. Wallace, S.W.: A two-stage stochastic facility location problem with time-dependent supply. In: Numerical Techniques for Stochastic Optimization, (Yu. Ermoliev and R. J-B. Wets (eds.)) pp. 489–513 (1988)

  47. Wets, R.J-B.: Stochastic Programs with fixed recourse: the equivalent deterministic problem. SIAM Rev. 16, 309–339 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  48. Wollmer, R.M.: Two stage linear programming under uncertainty with 0-1 first stage variables. Math. Prog. 19, 279–288 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dept. of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ, 85721, USA

    Suvrajeet Sen & Julia L. Higle

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  1. Suvrajeet Sen
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  2. Julia L. Higle
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This research was funded by NSF grants DMII 9978780 and CISE 9975050.

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Sen, S., Higle, J. The C 3 Theorem and a D 2 Algorithm for Large Scale Stochastic Mixed-Integer Programming: Set Convexification. Math. Program. 104, 1–20 (2005). https://doi.org/10.1007/s10107-004-0566-z

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