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Relaxations for probabilistically constrained stochastic programming problems: review and extensions

  • S.I.:Stochastic Modeling and Optimization, in memory of András Prékopa
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Abstract

We consider probabilistically constrained stochastic programming problems, in which the random variables are in the right-hand sides of the stochastic inequalities defining the joint chance constraints. Problems of that kind arise in a variety of contexts, and are particularly difficult to solve for random variables with continuous joint distributions, because the calculation of the cumulative distribution function and its gradient values involves numerical integration and/or simulation in large dimensional spaces. We revisit known and provide new relaxations extensions to various probability bounding schemes that permit to approximate the feasible set of joint probabilistic constraints. The derived mathematical formulations relax the requirement to handle large multivariate cumulative distribution functions and involve instead the computation of marginal and bivariate cumulative distribution functions. We analyze the convexity of and computational challenges posed by the inferred relaxations

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Notes

  1. The notation \(A_i(x)\) may be used to underline the dependency on the decisions x. We drop the parenthesis (x) to ease the notations as in Prékopa (2003).

References

  • Abraham, J. A. (1979). An improved algorithm for network reliability. IEEE Transactions on Reliability, 28(1), 58–61.

    Article  Google Scholar 

  • Bagnoli, M., & Bergstrom, T. (2005). Log-concave probability and its applications. Economic Theory, 26(2), 445–469.

    Article  Google Scholar 

  • Bennetts, R. G. (1975). On the analysis of fault trees. IEEE Transactions on Reliability, 24(3), 175–185.

    Article  Google Scholar 

  • Charnes, A., Cooper, W. W., & Symonds, G. H. (1958). Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil. Management Science, 4, 235–263.

    Article  Google Scholar 

  • Chen, W., Sim, M., Sun, J., & Teo, C.-P. (2010). From CVaR to uncertainty set: Implications in joint chance-constrained optimization. Operations Research, 58(2), 470–485.

    Article  Google Scholar 

  • Chhetry, D., Kimeldorf, G., & Sampson, A. R. (1989). Concepts of setwise dependence. Probability in the Engineering and Information Sciences, 3, 367–380.

    Article  Google Scholar 

  • Costigan, T. M. (1996). Combination Setwise–Bonferroni-type bounds. Naval Research Logistics, 43, 59–77.

    Article  Google Scholar 

  • Deák, I. (1988). Multidimensional integration and stochastic programming. In Y. Ermoliev & R. J. B. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 187–200). Berlin: Springer.

    Chapter  Google Scholar 

  • Deák, I. (2000). Subroutines for computing normal probabilities of sets—Computer experiences. Annals of Operations Research, 100, 103–122.

    Article  Google Scholar 

  • Dentcheva, D., & Martinez, G. (2013). Regularization methods for optimization problems with probabilistic constraints. Mathematical Programming, 138(1), 223–251.

    Article  Google Scholar 

  • Dentcheva, D., Prékopa, A., & Ruszczyński, A. (2000). Concavity and efficient points of discrete distributions in probabilistic programming. Mathematical Programming, 89, 55–77.

    Article  Google Scholar 

  • Genz, A. (1992). Numerical computation of multivariate normal probabilities. Journal of Computational and Graphical Statistics, 1, 141–149.

    Google Scholar 

  • Genz, A., & Bretz, F. (2009). Computation of multivariate normal and t probabilities. Lecture notes in statistics (Vol. 195). Dordrecht: Springer.

    Book  Google Scholar 

  • Guigues, V., & Henrion, R. (2017). Joint dynamic probabilistic constraints with projected linear decision rules. Optimization Methods & Software, 32(5), 1006–1032.

    Article  Google Scholar 

  • Heidtmann, K. D. (2002). Statistical comparison of two sum-of-disjoint-product algorithms for reliability and safety evaluations. In G. Goos, J. Hartmanis, & J. van Leeuwen (Eds.), Computer safety, reliability and security—Lecture notes in computer science (pp. 70–81). Berlin: Springer.

    Chapter  Google Scholar 

  • Henrion, R., & Möller, A. (2012). A gradient formula for linear chance constraints under Gaussian distribution. Mathematics of Operations Research, 37, 475–488.

    Article  Google Scholar 

  • Hunter, D. (1976). Bounds for the probability of a union. Journal of Applied Probability, 13, 597–603.

    Article  Google Scholar 

  • Kogan, A., & Lejeune, M. A. (2013). Threshold Boolean form for joint probabilistic constraints with random technology matrix. Mathematical Programming, 147, 391–427.

    Article  Google Scholar 

  • Kwerel S. M. (1975). Bounds on the probability of the union and intersection of \(m\) events. Advances in Applied Probability, 431–448.

  • Lejeune, M. A. (2012a). Pattern-based modeling and solution of probabilistically constrained optimization problems. Operations Research, 60(6), 1356–1372.

    Article  Google Scholar 

  • Lejeune, M. A. (2012b). Pattern definition of the \(p\)-efficiency concept. Annals of Operations Research, 200(1), 23–36.

    Article  Google Scholar 

  • Lejeune, M. A., & Margot, F. (2016). Solving chance-constrained optimization problems with stochastic quadratic inequalities. Operations Research, 64(4), 939–957.

    Article  Google Scholar 

  • Lejeune, M. A., & Noyan, N. (2010). Mathematical programming generation of \(p\)-efficient points. European Journal of Operational Research, 207(2), 590–600.

    Article  Google Scholar 

  • Lejeune, M. A., & Ruszczyński, A. (2007). An efficient trajectory method for probabilistic inventory-production-distribution problems. Operations Research, 55(3), 1–17.

    Google Scholar 

  • Margot, F. (2010). Symmetry in integer linear programming. In M. Junger, T. M. Liebling, D. Naddef, G. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, & L. A. Wolsey (Eds.), 50 years of integer programming (pp. 647–686). Berlin: Springer.

    Google Scholar 

  • Miller, B. L., & Wagner, H. M. (1965). Chance constrained programming with joint constraints. Operations Research, 13, 930–945.

    Article  Google Scholar 

  • Mohtashami Borzadaran, G. R., & Mohtashami Borzadaran, H. A. (2011). Log-concavity property for some well-known distributions. Surveys in Mathematics and its Applications, 6, 203–219.

    Google Scholar 

  • Nemirovski, A., & Shapiro, A. (2006). Convex approximations of chance constrained programs. SIAM Journal on Optimization, 17(4), 969–996.

    Article  Google Scholar 

  • Prékopa A. (1970). On probabilistic constrained programming. In Proceedings of the Princeton symposium on mathematical programming (pp. 113–138). Princeton University Press.

  • Prékopa, A. (1973). Contributions to the theory of stochastic programming. Mathematical Programming, 4, 202–221.

    Article  Google Scholar 

  • Prékopa, A. (1990). Dual method for a one-stage stochastic programming problem with random RHS obeying a discrete probability distribution. Zeithchrift fur Operations Research, 34, 441–461.

    Google Scholar 

  • Prékopa, A. (1995). Stochastic programming. Dordrecht-Boston: Kluwer.

    Book  Google Scholar 

  • Prékopa, A. (1999). The use of discrete moment bounds in probabilistic constrained stochastic programming models. Annals of Operations Research, 85, 21–38.

    Article  Google Scholar 

  • Prékopa, A. (2001). On the concavity of multivariate probability distribution functions. Operations Research Letters, 29, 1–4.

    Article  Google Scholar 

  • Prékopa, A. (2003). Probabilistic programming models. In A. Ruszczyński & A. Shapiro (Eds.), Stochastic programming: Handbook in operations research and management science (Vol. 10, pp. 267–351). Amsterdam: Elsevier Science.

    Chapter  Google Scholar 

  • Prékopa, A., Ganczer, S., Deák, I., & Patyi, K. (1980). The STABIL stochastic programming model and its experimental application to the electrical energy sector of the Hungarian economy. In M. A. Dempster (Ed.), Stochastic programming (pp. 369–385). London: Academic Press.

    Google Scholar 

  • Ruszczynski, A. (2002). Probabilistic programming with discrete distribution and precedence constrained knapsack polyhedra. Mathematical Programming, 93(2), 195–215.

    Article  Google Scholar 

  • Saxena, A., Goyal, V., & Lejeune, M. A. (2010). MIP reformulations of the probabilistic set covering problem. Mathematical Programming, 121(1), 1–31.

    Article  Google Scholar 

  • Slepian, D. (1962). On the one-sided barrier problem for Gaussian noise. Bell System Technical Journal, 41, 463–501.

    Article  Google Scholar 

  • Szántai, T. (1988). A computer code for solution of probabilistic-constrained stochastic programming problems. In Y. Ermoliev & R. J. B. Wets (Eds.), Numerical techniques for stochastic optimization (pp. 229–235). New York: Springer.

    Chapter  Google Scholar 

  • Wallace, S. W., & Ziemba, W. T. (2005). Applications of stochastic programming. Philadelphia, PA: MPS-SIAM Series on Optimization.

    Book  Google Scholar 

  • Worsley, K. J. (1982). An improved Bonferroni inequality and applications. Biometrika, 69(2), 297–302.

    Article  Google Scholar 

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Acknowledgements

M. Lejeune acknowledges the partial support of the Office of Naval Research [Grant N000141712420].

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Authors and Affiliations

  1. Department of Decision Sciences, George Washington University, Washington, DC, USA

    Miguel A. Lejeune

  2. Rutgers University, Piscataway, NJ, USA

    A. Prékopa

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  1. Miguel A. Lejeune
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  2. A. Prékopa
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Correspondence to Miguel A. Lejeune.

Additional information

The paper was written in large part before Professor András Prékopa passed away.

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Lejeune, M.A., Prékopa, A. Relaxations for probabilistically constrained stochastic programming problems: review and extensions. Ann Oper Res (2018). https://doi.org/10.1007/s10479-018-2934-8

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  • DOI: https://doi.org/10.1007/s10479-018-2934-8

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