Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Eventual convexity of probability constraints with elliptical distributions

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

Probability constraints are often employed to intuitively define safety of given decisions in optimization problems. They simply express that a given system of inequalities depending on a decision vector and a random vector is satisfied with high enough probability. It is known that, even if this system is convex in the decision vector, the associated probability constraint is not convex in general. In this paper, we show that some degree of convexity is still preserved, for the large class of elliptical random vectors, encompassing for example Gaussian or Student random vectors. More precisely, our main result establishes that, under mild assumptions, eventual convexity holds, i.e. the probability constraint is convex when the safety level is large enough. We also provide tools to compute a concrete convexity certificate from nominal problem data. Our results are illustrated on several examples, including the situation of polyhedral systems with random technology matrices and arbitrary covariance structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. For sake of completeness, let us give a short argumentation leading to it. The characteristic function of \(\xi \), has the form

    $$\begin{aligned} \psi _{\xi }(z) = \mathbb {E}(\exp {(i z^\mathsf {T}\xi )}) = \exp {(iz^\mathsf {T}\mu )}\gamma (z^\mathsf {T}\Sigma z), \end{aligned}$$

    for a function \(\gamma :\mathbb {R}\rightarrow \mathbb {R}\), called characteristic generator. As a consequence, the characteristic function of \(L^{-1}(\xi - \mu )\) satisfies \(\psi _{L^{-1}(\xi - \mu )} = \gamma (z^\mathsf {T}z)\). By [13, Theorem 2.1], \(L^{-1}(\xi - \mu )\) then follows a spherical distribution. It now follows from [13, Corollary to Theorem 2.2] that \(L^{-1}(\xi - \mu )\) admits the representation \(L^{-1}(\xi - \mu ) = \mathcal {R}\zeta ,\) which allows us to conclude.

References

  1. Arnold, T., Henrion, R., Möller, A., Vigerske, S.: A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pac. J. Optim. 10, 5–20 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-Linear Parametric Optimization. Birkhäuser, Basel (1982)

    Book  MATH  Google Scholar 

  3. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. Ser. A 88, 411–424 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brascamp, H., Lieb, E.: On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log-concave functions and with an application to the diffusion equations. J. Funct. Anal. 22, 366–389 (1976)

    Article  MATH  Google Scholar 

  5. Bremer, I., Henrion, R., Möller, A.: Probabilistic constraints via SQP solver: application to a renewable energy management problem. CMS 12, 435–459 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Daníelsson, J., Jorgensen, B., Samorodnitsky, G., Sarma, M., de Vries, C.G.: Fat tails, VaR and subadditivity. J. Econom. 172, 283–291 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dentcheva, D.: Optimisation models with probabilistic constraints. In: Shapiro, A., Dentcheva, D., Ruszczyński, A. (eds.) Lectures on Stochastic Programming. Modeling and Theory, MPS-SIAM Series on Optimization, vol. 9, pp. 87–154. SIAM and MPS, Philadelphia (2009)

    Chapter  Google Scholar 

  9. Dentcheva, D., Martinez, G.: Regularization methods for optimization problems with probabilistic constraints. Math. Program. (Ser. A) 138(1–2), 223–251 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dentcheva, D., Prékopa, A., Ruszczyński, A.: Concavity and efficient points for discrete distributions in stochastic programming. Math. Program. 89, 55–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diniz, A. L., Henrion, R.: On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions. Energy Systems 8(1), 149–167 (2017)

  12. Ermoliev, Y., Ermolieva, T., Macdonald, G., Norkin, V.: Stochastic optimization of insurance portfolios for managing exposure to catastrophic risk. Ann. Oper. Res. 99, 207–225 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang, K., Kotz, S., Ng, K.W.: Symmetric Multivariate and Related Distributions, Monographs on Statistics and Applied Probability, vol. 36, 1st edn. Springer-Science, Berlin (1990)

    Book  Google Scholar 

  14. Farshbaf-Shaker MH, Henrion R, Hömberg D (2017) Properties of chance constraints in infinite dimensions with an application to pde constrained optimization. Set Valued Var. Anal. https://doi.org/10.1007/s11228-017-0452-5

  15. Garnier, J., Omrane, A., Rouchdy, Y.: Asymptotic formulas for the derivatives of probability functions and their Monte Carlo estimations. Eur. J. Oper. Res. 198, 848–858 (2009). https://doi.org/10.1016/j.ejor.2008.09.026

    Article  MathSciNet  MATH  Google Scholar 

  16. Geletu, A., Hoffmann, A., Klöppel, M., Li, P.: A tractable approximation of non-convex chance constrained optimization with non-gaussian uncertainties. Eng. Optim. 47(4), 495–520 (2015)

    Article  MathSciNet  Google Scholar 

  17. Hantoute, A., Henrion, R., Pérez-Aros, P.: Subdifferential characterization of continuous probability functions under gaussian distribution. Submitted preprint: https://arxiv.org/pdf/1705.10160.pdf pp. 1–27 (2017)

  18. Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Comput. Math. Appl. 45, 247–262 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Henrion, R., Möller, A.: A gradient formula for linear chance constraints under Gaussian distribution. Math. Oper. Res. 37, 475–488 (2012). https://doi.org/10.1287/moor.1120.0544

    Article  MathSciNet  MATH  Google Scholar 

  20. Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41, 263–276 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Henrion, R., Strugarek, C.: Convexity of chance constraints with dependent random variables: the use of copulae. In: Bertocchi, M., Consigli, G., Dempster, M. (eds.) Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Market Strategies, International Series in Operations Research and Management Science, pp. 427–439. Springer, New York (2011)

    Chapter  Google Scholar 

  22. Hong, L., Yang, Y., Zhang, L.: Sequential convex approximations to joint chance constrained programed: a monte carlo approach. Oper. Res. 3(59), 617–630 (2011)

    Article  MATH  Google Scholar 

  23. Kataoka, S.: A stochastic programming model. Econometrica 31, 181–196 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kibzun, A., Uryas’ev, S.: Differentiability of probability function. Stoch. Anal. Appl. 16, 1101–1128 (1998). https://doi.org/10.1080/07362999808809581

    Article  MathSciNet  MATH  Google Scholar 

  25. Kogan, A., Lejeune, M.A.: Threshold boolean form for joint probabilistic constraints with random technology matrix. Math. Program. 147(1–2), 391–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kogan, A., Lejeune, M.A., Luedtke, J.: Erratum to: Threshold boolean form for joint probabilistic constraints with random technology matrix. Math. Program. 155(1), 617–620 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Landsman, Z.M., Valdez, E.A.: Tail conditional expectations for elliptical distributions. N. Am. Actuar. J. 7(4), 55–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lejeune, M., Margot, F.: Solving chance-constrained optimization problems with stochastic quadratic inequalities. Oper. Res. 64(4), 939–957 (2016)

  29. Lejeune, M.A.: Pattern-based modeling and solution of probabilistically constrained optimization problems. Oper. Res. 60(6), 1356–1372 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lejeune, M.A., Noyan, N.: Mathematical programming approaches for generating p-efficient points. Eur. J. Oper. Res. 207(2), 590–600 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lubin, M., Bienstock, D., Vielna, J.P.: Two-sided linear chance constraints and extensions. Arxiv 1507(01995), 1–19 (2016)

    Google Scholar 

  32. Luedtke, J.: A branch-and-cut decomposition algorithm for solving chance-constrained mathematical programs with finite support. Math. Program. 146(1–2), 219–244 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19, 674–699 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Marti, K.: Differentiation of probability functions: the transformation method. Comput. Math. Appl. 30, 361–382 (1995). https://doi.org/10.1016/0898-1221(95)00113-1

    Article  MathSciNet  MATH  Google Scholar 

  35. Morgan, D., Eheart, J., Valocchi, A.: Aquifer remediation design under uncertainty using a new chance constraint programming technique. Water Resour. Res. 29, 551–561 (1993)

    Article  Google Scholar 

  36. Naor, A., Romik, D.: Projecting the surface measure of the sphere of \(\ell _p^n\). Ann. I.H. Poincaré 39(2), 241–261 (2003). https://doi.org/10.1016/S0246-0203(02)00008-0

  37. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. Prékopa, A.: Programming under probabilistic constraints with a random technology matrix. Math. Oper. Stat. 5, 109–116 (1974)

    MathSciNet  MATH  Google Scholar 

  40. Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)

    Book  MATH  Google Scholar 

  41. Prékopa, A.: On the concavity of multivariate probability distributions functions. Oper. Res. Lett. 29, 1–4 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Prékopa, A.: Probabilistic programming. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, pp. 267–351. Elsevier, Amsterdam (2003)

    Google Scholar 

  43. Raik, E.: The differentiability in the parameter of the probability function and optimization of the probability function via the stochastic pseudogradient method (Russian). Izvestiya Akad. Nayk Est. SSR Phis. Math. 24(1), 3–6 (1975)

    MathSciNet  MATH  Google Scholar 

  44. Rinott, Y.: On the convexity of measures. Ann. Probab. 4, 1020–1026 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  45. Royset, J., Polak, E.: Extensions of stochastic optimization results to problems with system failure probability functions. J. Optim. Theory Appl. 133(1), 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shapiro, A., Dentcheva, D., Ruszczyński, A.: Lectures on Stochastic Programming, Modeling and Theory, MPS-SIAM Series on Optimization, vol. 9. SIAM and MPS, Philadelphia (2009)

    Book  MATH  Google Scholar 

  47. Uryas’ev, S.: Derivatives of probability functions and integrals over sets given by inequalities. J. Comput. Appl. Math. 56(1–2), 197–223 (1994). https://doi.org/10.1016/0377-0427(94)90388-3

    Article  MathSciNet  MATH  Google Scholar 

  48. Uryas’ev, S.: Derivatives of probability functions and some applications. Ann. Oper. Res. 56, 287–311 (1995)

    Article  MathSciNet  Google Scholar 

  49. Uryas’ev, S.: Derivatives of probability and integral functions: General theory and examples. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 658–663. Springer, New York (2009)

    Chapter  Google Scholar 

  50. van Ackooij, W.: Eventual convexity of chance constrained feasible sets. Optimization (J. Math. Program. Oper. Res.) 64(5), 1263–1284 (2015). https://doi.org/10.1080/02331934.2013.855211

    Article  MathSciNet  MATH  Google Scholar 

  51. van Ackooij, W.: Convexity statements for linear probability constraints with gaussian technology matrices and copulæ correlated rows. ResearchGate pp. 1–19 (2017). https://doi.org/10.13140/RG.2.2.11723.69926

  52. van Ackooij, W., Berge, V., de Oliveira, W., Sagastizábal, C.: Probabilistic optimization via approximate p-efficient points and bundle methods. Comput. Oper. Res. 77, 177–193 (2017). https://doi.org/10.1016/j.cor.2016.08.002

    Article  MathSciNet  MATH  Google Scholar 

  53. van Ackooij, W., Frangioni, A., de Oliveira, W.: Inexact stabilized Benders’ decomposition approaches: with application to chance-constrained problems with finite support. Comput. Optim. Appl. 65(3), 637–669 (2016). https://doi.org/10.1007/s10589-016-9851-z

    Article  MathSciNet  MATH  Google Scholar 

  54. van Ackooij, W., Henrion, R.: Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM J. Optim. 24(4), 1864–1889 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  55. van Ackooij, W., Henrion, R.: (Sub-)gradient formulae for probability functions of random inequality systems under Gaussian distribution. SIAM J. Uncertain. Quantif. 5(1), 63–87 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  56. van Ackooij, W., Henrion, R., Möller, A., Zorgati, R.: Joint chance constrained programming for hydro reservoir management. Optim. Eng. 15, 509–531 (2014)

    MathSciNet  MATH  Google Scholar 

  57. van Ackooij, W., Malick, J.: Second-order differentiability of probability functions. Optim. Lett. 11(1), 179–194 (2017). https://doi.org/10.1007/s11590-016-1015-7

    Article  MathSciNet  MATH  Google Scholar 

  58. van Ackooij, W., de Oliveira, W.: Level bundle methods for constrained convex optimization with various oracles. Comput. Optim. Appl. 57(3), 555–597 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  59. van Ackooij, W., de Oliveira, W.: Convexity and optimization with copulæ structured probabilistic constraints. Optim. J. Math. Program. Oper. Res. 65(7), 1349–1376 (2016). https://doi.org/10.1080/02331934.2016.1179302

    Article  MATH  Google Scholar 

  60. van Ackooij, W., Sagastizábal, C.: Constrained bundle methods for upper inexact oracles with application to joint chance constrained energy problems. SIAM J. Optim. 24(2), 733–765 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zadeh, Z.M., Khorram, E.: Convexity of chance constrained programming problems with respect to a new generalized concavity notion. Ann. Oper. Res. 196(1), 651–662 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are thankful for the constructive comments from two anonymous referees and the associate editor. We would like to acknowledge the financial support of PGMO (Gaspard Monge Program for Optimization and operations research) of the Hadamard Mathematic Foundation, through the project “Advanced nonsmooth optimization methods for stochastic programming”.

Author information

Authors and Affiliations

  1. OSIRIS, EDF R&D, 7, Boulevard Gaspard Monge, Palaiseau Cedex, 91120, France

    Wim van Ackooij

  2. CNRS, LJK, Grenoble, 38000, France

    Jérôme Malick

Authors
  1. Wim van Ackooij
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jérôme Malick
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wim van Ackooij.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

van Ackooij, W., Malick, J. Eventual convexity of probability constraints with elliptical distributions. Math. Program. 175, 1–27 (2019). https://doi.org/10.1007/s10107-018-1230-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-018-1230-3

Keywords

Mathematics Subject Classification

Search

Navigation