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Tractable algorithms for chance-constrained combinatorial problems

Published online by Cambridge University Press:  28 April 2009

Olivier Klopfenstein
Olivier Klopfenstein*
Affiliation:
France Télécom R&D, 38-40 rue du gl Leclerc, 92130 Issy-les-Moulineaux, France; olivier.klopfenstein@orange-ftgroup.com Université de Technologie de Compiègne, Laboratoire Heudiasyc UMR CNRS 6599, 60205 Compiègne Cedex, France
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Abstract

This paper aims at proposing tractable algorithms to find effectively good solutions to large size chance-constrained combinatorial problems. A new robust model is introduced to deal with uncertainty in mixed-integer linear problems. It is shown to be strongly related to chance-constrained programming when considering pure 0–1 problems. Furthermore, its tractability is highlighted. Then, an optimization algorithm is designed to provide possibly good solutions to chance-constrained combinatorial problems. This approach is numerically tested on knapsack and multi-dimensional knapsack problems. The results obtained outperform many methods based on earlier literature.

Keywords

Integer linear programmingchance constraintsrobust optimizationheuristic.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 2 , April 2009 , pp. 157 - 187
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

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References

R. Aringheri, A Tabu search algorithm for solving chance-constrained programs, Note del Polo 73, DTI – University of Milano (2005), available at http://www.crema.unimi.it/Biblioteca/SchedaNota.asp?Nota=92.
Ben-Tal, A. and Nemirovski, A., Robust solutions of linear programming problems contamined with uncertain data. Math. Program. (Ser. A) 88 (2000) 411–424. CrossRef
Beraldi, P. and Ruszczyński, A., A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Softw. 17 (2002) 359–382. CrossRef
Bertsimas, D. and Sim, M., Robust discrete optimization and network flows. Math. Program. (Ser. B) 98 (2003) 49–71. CrossRef
D. Bertsimas and M. Sim, The Price of Robustness. Oper. Res. 52-1 (2004) 35–53.
L. Bianchi, M. Dorigo, L.M. Gambardella and W.J. Gutjahr, Metaheuristics in stochastic combinatorial optimization: a survey, technical report IDSIA-08-06, IDSIA, available at www.idsia.ch/idsiareport/IDSIA-08-06.pdf (2006).
J.R. Birge and F. Louveaux, Introduction to stochastic programming. Springer-Verlag (1997).
Calafiore, G. and Campi, M.C., Uncertain convex programs: randomized solutions and confidence levels. Math. Program. A 102 (2005) 25–46. CrossRef
Calafiore, G. and El Ghaoui, L., Distributionally robust chance-constrained linear programs with applications. J. Optim. Theor. Appl. 130 (2006) 1–22. CrossRef
Charnes, A. and Cooper, W.W., Chance-constrained programming. Manage. Sci. 5 (1959) 73–79. CrossRef
X. Chen, M. Sim and P. Sun, A Robust optimization perspective of stochastic programming. Oper. Res. (2007) 55 1058–1071.
Dantzig, G.B., Linear programming under uncertainty. Manage. Sci. 1 (1955) 179–206. CrossRef
E. Erdoğan and G. Iyengar, Ambiguous chance constrained problems and robust optimization. Math. Program. Ser. B 55 (20057) 37–61.
C. Grinstead and J. Snell, Introduction to probability. American Mathematical Society, Providence, Rhode Island (1994).
W.K. Klein Haneveld and M.H. van der Vlerk, Stochastic integer programming: state of the art (1998), available at http://citeseer.ist.psu.edu/kleinhaneveld98stochastic.html.
Hoeffding, W., Probability inequalities for sums of bounded random variables. Am. Stat. Assoc. J. 58 (1963) 13–30. CrossRef
P. Kall and S.W. Wallace, Stochastic programming. Wiley, Chichester (1994).
O. Klopfenstein and D. Nace, A robust approach to the chance-constrained knapsack problem. to appear in Oper. Res. Lett.
O. Klopfenstein, Tractable algorithms for chance-constrained combinatorial problems, http://perso.rd.francetelecom.fr/klopfenstein/Papers/rmilp_online.pdf (long version of the current paper).
A.M. Mood and F.A. Graybill, Introduction to the theory of statistics. McGraw-Hill Book Company Inc., New York (1963).
A. Nemirovski and A. Shapiro, Convex Approximations of chance constrained programs. SIAM J. Optim. 17-4 (2006) 969–996.
A. Nemirovski and A. Shapiro, Scenario approximations of chance constraints, in Probabilistic and Randomized Methods for Design under Uncertainty, edited by G. Calafiore and F. Dabbene. Springer, London (2005) 3–48.
Y. Nikulin, Robustness in combinatorial optimization and scheduling theory: an annotated bibliography, www.optimization-online.org/DB_FILE/2004/11/995.pdf (2004).
Ruszczyński, A., Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. A 93 (2002) 195–215. CrossRef
Sahinidis, N.V., Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28 (2004) 971–983. CrossRef
S.R. Tayur, R.R. Thomas and N.R. Natraj, An algebraic geometry algorithm for scheduling in the presence of setups and correlated demands. Math. Program. 69-3 (1995) 369–401.