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The robust binomial approach to chance-constrained optimization problems with application to stochastic partitioning of large process networks

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Abstract

In this paper, we study an interpretation of the sample-based approach to chance-constrained programming problems grounded in statistical testing theory. On top of being simple and pragmatic, this approach is theoretically well founded, non parametric and leads to a general method for leveraging existing heuristic algorithms for the deterministic case to their chance-constrained counterparts. Throughout this paper, this algorithm design approach is illustrated on a real world graph partitioning problem which crops up in the field of compilation for parallel systems. Extensive computational results illustrate the practical relevance of the approach, as well as the robustness of the obtained solutions.

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Notes

  1. An assumption that can be in practice checked, to some extent, using static program analysis techniques. An assumption which also relies reasonably on the expertise of test engineers in terms of designing validation cases representative of real-world system operation.

  2. An one line OpenMP pragma will do the trick.

References

  • Aringhieri, R.: Solving chance-constrained programs combining tabu search and simulation. In: Proceedings of the 3rd International Workshop on Experimental and Efficient Algorithms (WEA04). Lecture Notes in Computer Science, vol. 3059, pp. 30–41. Springer, Berlin (2004)

  • Barbu, A., Zhu, S.-C.: Stochastic graph partition: generalizing the Swendsen–Wang method. Technical Report Paper 2003010120, UCLA Department of Statistics (2003)

  • Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Beraldi, P., Ruszczynski, A.: Beam search heuristic to solve stochastic integer problems under probabilistic constraints. Eur. J. Oper. Res. 167(1), 35–47 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Google Scholar 

  • Bianchi, L., Dorigo, M., Gambardella, L., Gutjahr, W.: A survey on metaheuristics for stochastic combinatorial optimization. Nat Comput 8(2), 239–287 (2006)

    Article  MathSciNet  Google Scholar 

  • Bichot, C.H.: A new method, the fusion fission, for the relaxed-way graph partitioning problem, and comparisons with some multilevel algorithms. J. Math. Model. Algorithms 6(3), 319–344 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Bichot, C., Durand, N.: Partitionnement de graphe. Lavoisier, Paris (2010)

    MATH  Google Scholar 

  • Calafiore, G., Campi, M.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102, 25–46 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Calafiore, G., Campi, M.: The scenario approach to robust control design. IEEE Trans. Automat. Control 51(5), 742–753 (2006)

    Article  MathSciNet  Google Scholar 

  • Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manag. Sci. 4(3), 235–263 (1958)

    Article  Google Scholar 

  • David, V., Fraboul, C., Rousselot, J.Y., Siron, P.: Etude et realisation d’une architecture modulaire et reconfigurable: Projet MODULOR. Technical Report, 1/3364/DERI.ONERA (1991)

  • de Farias, D., Van Roy, B.: On constraint sampling in the linear programming approach to approximate linear programming. In: Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 3, pp. 2441–2446 (2003)

  • Demange, M., Paschos, V.: On an approximation measure founded on the links between optimization and polynomial approximation theory. Theor. Comput. Sci. 158, 117–141 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Dentcheva, D., Prékopa, A., Ruszczynski, A.: Concavity and efficient points of discrete distributions in probabilistic programming. Math. Program. 89, 55–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Efron, B., Tibshirani, R.: An Introduction to the Bootstrap. CRC Press, Boca Raton (1994)

    Google Scholar 

  • Elsner, U.: Graph partitioning—a survey. Technical Report, TU Chemnitz SFB393/97-27 (1997)

  • Fan, N., Pardalos, P.: Robust optimization of graph partitioning and critical node detection in analyzing networks. In: Proceedings of the 4th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2010), pp. 170–183 (2010)

  • Fan, N., Zheng, Q., Pardalos, P.: On the two-stage stochastic graph partitioning problem. In: Proceedings of the 5th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2011), pp. 500–509 (2011)

  • Ferreira, C.E., Martin, A., de Souza, C., Weismantel, R., Wolsey, L.: The node capacitated graph partitioning problem: a computational study. Math. Program. 81, 229–256 (1998)

    MATH  Google Scholar 

  • Fiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. In: Proceedings of the 19th Design Automation Conference. DAC ’82, pp. 175–181. IEEE Press, Piscataway (1982)

  • Fjällström, P.O.: Algorithms for graph partitioning: a survey. Linköping Electron. Articles Comput. Inf. Sci. 3, 10 (1998)

    Google Scholar 

  • Gaivoronski, A., Lisser, A., Lopez, R., Xu, H.: Knapsack problem with probability constraints. J. Glob. Optim. 49, 397–413 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  • Hendrickson, B., Leland, R.: A multilevel algorithm for partitioning graphs. In: Proceedings of the 1995 ACM/IEEE Conference on Supercomputing (CDROM). ACM, New York (1995)

  • Johnson, D., Aragon, C., McGeoch, L., Schevon, C.: Optimization by simulated annealing: an experimental evaluation; part i, graph partitioning. Oper. Res. 37(6), 865–892 (1989)

    Article  MATH  Google Scholar 

  • Johnson, E., Mehrotra, A., Nemhauser, G.L.: Min-cut clustering. Math. Program. 62, 133–151 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1998)

    Article  MathSciNet  Google Scholar 

  • Kernighan, B., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49(1), 291–307 (1970)

    Article  MATH  Google Scholar 

  • Kirkpatrick, S.: Optimization by simulated annealing: quantitative studies. J. Stat. Phys. 34, 975–986 (1984)

    Article  MathSciNet  Google Scholar 

  • Lisser, A., Rendl, F.: Graph partitioning using linear and semidefinite programming. Math. Program. 95, 91–101 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Loughlin, D.H., Ranjithan, S.: Chance-constrained genetic algorithms. In: GECCO-99: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 369–376 (1999)

  • Mehrotra, A., Trick, M.: Cliques and clustering: a combinatorial approach. Oper. Res. Lett. 22, 1–12 (1997)

    MathSciNet  Google Scholar 

  • Pagnoncelli, B.K., Ahmed, S., Shapiro, A., Pardalos, P.M.: Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142, 399–416 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Prekopa, A.: Stochastic Programming. Kluwer Academic Publishers, Dordrecht (1995)

    Book  Google Scholar 

  • Sensen, N.: Lower bounds and exact algorithms for the graph partitioning problem using multicommodity flows. In: Meyer auF der Heide, F. (ed.) Lecture Notes in Computer Science, vol. 2161, pp. 391–403. Springer, Berlin (2001)

    Google Scholar 

  • Sirdey, R., David, V.: Approches heuristiques des problèmes de partitionnement, placement et routage de rèseaux de processus sur architectures parallèles clusterisées. Technical Report, CEA LIST DTSI/SARC/09-470/RS (2009)

  • Stan, O., Sirdey, R., Carlier, J., Nace, D.: A heuristic algorithm for stochastic partitioning of process networks. In: ICSTCC (2012)

  • Tanner, M.W., Beier, E.B.: A general heuristic method for joint chance-constrained stochastic programs with discretely distributed parameters (2007). http://www.optimization-online.org/DB_HTML/2007/08/1755.html

  • Taskin, Z.C., Smith, J.C., Ahmed, S., Schaefer, A.: Cutting plane algorithms for solving a stochastic edge-partition problem. Discret. Optim. 6(4), 420–435 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Vidyasagar, M.: Randomized algorithms for robust controller synthesis using statistical learning theory. In: Learning Control and Hybrid Systems. Lecture Notes in Control and Information Sciences, vol. 241, pp. 3–24. Springer, Berlin (1999)

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Acknowledgments

The authors thank the anonymous referees for several suggestions that led to improvements in the paper.

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Authors and Affiliations

  1. CEA, LIST, Embedded Real Time System Laboratory, Point Courrier 172, 91191 , Gif-sur-Yvette, France

    Oana Stan & Renaud Sirdey

  2. UMR CNRS 6599 Heudiasyc, Université de Technologie de Compiègne, BP 20529, 60205 , Compiègne, France

    Jacques Carlier & Dritan Nace

Authors
  1. Oana Stan
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  2. Renaud Sirdey
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  3. Jacques Carlier
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  4. Dritan Nace
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Correspondence to Oana Stan.

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Stan, O., Sirdey, R., Carlier, J. et al. The robust binomial approach to chance-constrained optimization problems with application to stochastic partitioning of large process networks. J Heuristics 20, 261–290 (2014). https://doi.org/10.1007/s10732-014-9241-6

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  • DOI: https://doi.org/10.1007/s10732-014-9241-6

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