Fenchel decomposition for stochastic mixed-integer programming

Journal of Global Optimization

Volumn 55, Issue 1, 2013, Pages 141-163

  Ntaimo, Lewis ^a  

^a Texas AandM University   (United States)

Author keywords

FD cuts; Fenchel decomposition; Integer programming; Lifting; Stochastic programming

Indexed keywords

ALTERNATIVE ALGORITHMS; COMPUTATIONAL RESULTS; CUTTING PLANE METHODS; CUTTING PLANES; DIRECT SOLVERS; DISJUNCTIVE DECOMPOSITION; DISJUNCTIVE PROGRAMMING; FD CUTS; FINITE CONVERGENCE; HIGHER DIMENSIONS; KNAPSACK PROBLEMS; LIFTING; SPECIAL STRUCTURE; STOCHASTIC MIXED INTEGER PROGRAMMING; TEST INSTANCES; VALID INEQUALITY;

ALGORITHMS; INTEGER PROGRAMMING; STOCHASTIC PROGRAMMING; STOCHASTIC SYSTEMS;

FINITE DIFFERENCE METHOD;

EID: 84872614452     PISSN: 09255001     EISSN: 15732916     Source Type: Journal    
DOI: 10.1007/s10898-011-9817-8     Document Type: Article

References (40)

1. Ahmed S.; Tawarmalani M.; Sahinidis N.V.: A finite branch and bound algorithm for two-stage stochastic integer programs. Math. Program. 100, 355-377 (2004)
2. Araoz J.; Edmonds J.; Griffin V.: Lifting the facets of polyhedra. In: Pulleyblank, W.R. (eds) Progress in Combinatorial Optimization, pp. 3-12. Academic Press, New York (1984)
3. Balas E.: Disjunctive programming: cutting planes from logical conditions. In: Mangasarian, O.L.; Meyer, R.R.; Robinson, S.M. (eds) Nonlinear Programming, vol. 2, Academic Press, New York (1975)
4. Balas E.: Disjunctive programming. Ann. Discret. Math. 5, 3-51 (1979)
5. Balas E.; Ceria E.S, Cornuéjols G.: A lift-and-project cutting plane algorithm for mixed 0-1 integer programs. Math. Program. 58, 295-324 (1993)
6. Benders J.F.: Partitioning procedures for solving mixed-variable programming problems. Numerische Mathematik 4, 238-252 (1962)
7. Birge J.R.; Louveaux F.V.: Introduction to Stochastic Programming. Springer, New York (1997)
8. Blair C.; Jeroslow R.: A converse for disjunctive constraints. J. Optim. Theory Appl. 25, 195-206 (1978)
9. Blair C.; Jeroslow R.: The value function of an integer program. Math. Program. 23, 237-273 (1982)
10. Boyd E.A.: Generating Fenchel cutting planes for knapsack polyhedra. SIAM J. Optim. 3(4), 734-750 (1993)
11. Boyd E.A.: Fenchel cuts for integer programs. Oper. Res. 42(1), 53-64 (1994)
12. Boyd E.A.: Solving 0/1 integer programs with enumeration cutting planes. Ann. Oper. Res. 50, 61-72 (1994)
13. Boyd E.A.: On the convergence of Fenchel cutting planes in mixed-integer programming. SIAM J. Optim. 5(2), 421-435 (1995)
14. Carøe, C.C.: Decomposition in Stochastic Integer Programming. Ph.D. Thesis, Department of Operations Research, University of Copenhagen, Denmark (1998)
15. Carøe C.C.; Tind J.: A cutting-plane approach to mixed 0-1 stochastic integer programs. Eur. J. Oper. 101, 306-316 (1997)
16. ILOG: CPLEX 11.0 ILOG CPLEX Callable Library 11.0 Reference Manual. ILOG, France (2007)
17. Klein Haneveld W.K.; van der Vlerk M.H.: Stochastic integer programming: general models and algorithms. Ann. Oper. Res. 85, 39-57 (1999)
18. Laporte G.; Louveaux F.V.: The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 1, 133-142 (1993)
19. Louveaux F.V.; Schultz R.: Stochastic integer programming. In: Ruszczyn'ski, A.; Shapiro, A. (eds) Stochastic Programming (Handbooks in Operations Research and Management Science, vol. 10), chap. 6, pp. 213-266. Elsevier, Amsterdam (2003)
20. Ntaimo L.; Sen S.: The million-variable 'march' for stochastic combinatorial optimization. J. Glob. Optim. 32(3), 385-400 (2004)
21. Ntaimo L.; Sen S.: A comparative study of decomposition algorithms for stochastic combinatorial optimization. Comp. Optim. Appl. J. 40(3), 299-319 (2008)
22. Ntaimo L.; Tanner M.W.: Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs. J. Glob. Optim. 41(3), 365-384 (2008)
23. Ramos M.T.; Sáez J.: Solving capacitated facility location problems by fenchel cutting planes. J. Oper. Res. Soc. 56, 297-306 (2005)
24. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
25. Ruszczyn'ski, A.; Shapiro, A. (eds): Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10. Elsevier, Amsterdam (2003)
26. Sáez J.: Solving linear programming relaxations associated with Lagrangean relaxations by Fenchel cutting planes. Eur. J. Oper. 12, 609-626 (2000)
27. Schultz R.: Continuity properties of expectation functions in stochastic integer programming. Math. Oper. Res. 18, 578-589 (1993)
28. Schultz R.: Stochastic programming with integer variables. Math. Program. 97, 285-309 (2003)
29. Schultz R.; Stougie L.; van der Vlerk M.H.: Two-stage stochastic integer programming: a survey. Stat. Neerl. 50, 404-416 (1996)
30. Schultz R.; Stougie L.; van der Vlerk M.H.: Solving stochastic programs with integer recourse by enumeration: a framework using Gröbner basis reduction. Math. Program. 83(2), 71-94 (1998)
31. Sen S.: Algorithms for stochastic mixed-integer programming models. In: Aardal, K.; Nemhauser, G.; Weismantel, R. (eds) Handbooks in Operations Research and Management Science, Volume 12: Discrete Optimization, chap. 9, Elsevier, Dordrecht (2005)
32. Sen S.; Higle J.L.: The C3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: set convexification. Math. Program. 104(1), 1-20 (2005)
33. Sen S.; Higle J.L.; Ntaimo L.: A summary and illustration of disjunctive decomposition with set convexification. In: Woodruff, D.L. (eds) Stochastic Integer Programming and Network Interdiction Models, chap. 6, pp. 105-123. Kluwer, Dordrecht (2002)
34. Sen S.; Sherali H.D.: Decomposition with branch-and-cut approaches for two stage stochastic mixed-integer programming. Math. Program. 106(2), 203-223 (2006)
35. Shapiro, A.; Dentcheva, D.; Ruszczyn'ski, A. (eds.): Lectures on Stochastic Programming. SIAM, Philadelphia (2009). http://www2.isye.gatech.edu/ people/faculty/Alex-Shapiro/SPbook.pdf
36. Sherali H.D.; Shetty C.M.: Optimization with disjunctive constraints. Lect. Notes Econ. Math. Syst. 181, 411-430 (1980)
37. Sherali H.D.; Zhu X.: On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables. Math. Program. 108(2-3), 597-616 (2006)
38. Van Slyke R.; Wets R.-B.: L-shaped linear programs with application to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638-663 (1969)
39. van der Vlerk, M.H.: Stochastic integer programming bibliography (2007). http://mally.eco.rug.nl/biblio/sip.html, 1996-2007
40. Wollmer R.M.: Two stage linear programming under uncertainty with 0-1 first stage variables. Math. Program. 19, 279-288 (1980)