Skip to main content
Log in

Generalized Differentiation of Probability Functions: Parameter Dependent Sets Given by Intersections of Convex Sets and Complements of Convex Sets

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

In this work we consider probability functions working on parameter dependent sets that are given as an intersection of convex sets and their complements. Such an underlying structure naturally arises when having to handle bilateral inequality systems in various applications, such as energy. We provide conditions under which the probability functions are sub-differentiable as well as a constraint qualification condition under which the probability function turns out to be smooth. Our analysis allows for (nearly) arbitrary distributions of random vectors.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bogachev, V.I.: Measure Theory, vol. I, II. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  4. 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 

  5. Campi, M.C., Garatti, S.: A sampling-and-discarding approach to chance-constrained optimization: feasibility and optimality. J. Optim. Theory Appl. 148(2), 257–280 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chieu, N.H.: Limiting subdifferentials of indefinite integrals. J. Math. Anal. Appl. 341(1), 241–258 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke, F.H.: Optimisation and Nonsmooth Analysis. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (1987)

  8. Correa, R., Hantoute, A., Pérez-Aros, P.: Characterizations of the subdifferential of convex integral functions under qualification conditions. J. Funct. Anal. 277(1), 227–254 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Correa, R., Hantoute, A., Pérez-Aros, P.: Subdifferential calculus rules for possibly nonconvex integral functions. SIAM J. Control. Optim. 58(1), 462–484 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  10. Correa, R., Hantoute, A., Pérez-Aros, P.: Qualification conditions-free characterizations of the \(\varepsilon \)-subdifferential of convex integral functions. Appl. Math. Optim. 83, 1709–1737 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dentcheva, D.: Optimisation models with probabilistic constraints. In: Shapiro, A., Dentcheva, D., Ruszczyński, A. (eds.) Lectures on Stochastic Programming. Modeling and Theory, volume 9 of MPS-SIAM series on optimization, pp. 87–154. SIAM and MPS, Philadelphia (2009)

  12. Dentcheva, D., Lai, B., Ruszczyński, A.: Dual methods for probabilistic optimization problems. Math. Methods Oper. Res. 60(2), 331–346 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dentcheva, D., Martinez, G.: Augmented Lagrangian method for probabilistic optimization. Ann. Oper. Res. 200(1), 109–130 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Farshbaf-Shaker, M.H., Henrion, R., Hömberg, D.: Properties of chance constraints in infinite dimensions with an application to PDE constrained optimization. Set Valued Var. Anal. 26(4), 821–841 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

  20. Hantoute, A., Henrion, R., Pérez-Aros, P.: Subdifferential characterization of continuous probability functions under Gaussian distribution. Math. Program. 174(1–2), 167–194 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Heitsch, H.: On probabilistic capacity maximization in a stationary gas network. Optimization 69(3), 575–604 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Henrion, R.: Optimierungsprobleme mit wahrscheinlichkeitsrestriktionen: Modelle, struktur, numerik. Lecture Notes, pp. 1–53 (2016)

  23. Henrion, R., Römisch, W.: Lipschitz and differentiability properties of quasi-concave and singular normal distribution functions. Ann. Oper. Res. 177, 115–125 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hong, L.J., 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 

  25. Kibzun, A.I., Uryas’ev, S.: Differentiability of probability function. Stoch. Anal. Appl. 16, 1101–1128 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1–2), 31–56 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. 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 

  29. Lejeune, M.A.: Pattern definition of the \(p\)-efficiency concept. Ann. Oper. Res. 200, 23–36 (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. Liu, X., Küçükyavuz, S., Luedtke, J.: Decomposition algorithm for two-stage chance constrained programs. Math. Program. Series B 157(1), 219–243 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Luedtke, J.: An integer programming and decomposition approach to general chance-constrained mathematical programs. In: Eisenbrand F., Shepherd F.B. (eds.) Integer Programming and Combinatorial Optimization, volume 6080 of Lecture Notes in Computer Science, pp. 271–284. Springer (2010)

  33. 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 

  34. 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 

  35. Luedtke, J., Ahmed, S., Nemhauser, G.L.: An integer programming approach for linear programs with probabilistic constraints. Math. Program. 122(2), 247–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Marti, K.: Differentiation of probability functions: the transformation method. Comput. Math. Appl. 30, 361–382 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mordukhovich, B.S.: Variational analysis and generalized differentiation I. Basic Theory, volume 330 of Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (2006)

  38. Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham (2018)

    Book  MATH  Google Scholar 

  39. Mordukhovich, B.S., Pérez-Aros, P.: New extremal principles with applications to stochastic and semi-infinite programming. Math. Program. 189(1), 527–553 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. 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 

  42. Pérez-Aros, P.: Ergodic approach to robust optimization and infinite programming problems. Set-Valued Var. Anal. 29, 409–423 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  43. Pflug, G., Weisshaupt, H.: Probability gradient estimation by set-valued calculus and applications in network design. SIAM J. Optim. 15, 898–914 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  45. 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)

  46. 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 

  47. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, volume 317 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2009)

  48. Royset, J.O., Polak, E.: Implementable algorithm for stochastic optimization using sample average approximations. J. Optim. Theory Appl. 122(1), 157–184 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Royset, J.O., 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 

  50. Rudin, W.: Real and Complex Analysis. Higher Mathematics Series, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  51. Uryas’ev, S.: Derivatives of probability functions and integrals over sets given by inequalities. J. Comput. Appl. Math. 56(1–2), 197–223 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  53. van Ackooij, W.: A discussion of probability functions and constraints from a variational perspective. Set-Valued Var. Anal. 28(4), 585–609 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  54. van Ackooij, W., Aleksovska, I., Munoz Zuniga, M.: (sub-)differentiability of probability functions with elliptical distributions. Set Valued Var. Anal. 26(4), 887–910 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. 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)

    Article  MathSciNet  MATH  Google Scholar 

  56. 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)

    Article  MathSciNet  MATH  Google Scholar 

  57. 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 

  58. 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 

  59. van Ackooij, W., Henrion, R., Pérez-Aros, P.: Generalized gradients for probabilistic/robust (probust) constraints. Optimization 69(7–8), 1451–1479 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  60. van Ackooij, W., Javal, P., Pérez-Aros, P.: Derivatives of probability functions acting on parameter dependent unions of polyhedra. Set-Valued Var. Anal. 1–33 (2021)

  61. van Ackooij, W., Malick, J.: Second-order differentiability of probability functions. Optim. Lett. 11(1), 179–194 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  62. van Ackooij, W., Pérez-Aros, P.: Generalized differentiation of probability functions acting on an infinite system of constraints. SIAM J. Optim. 29(3), 2179–2210 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  63. van Ackooij, W., Pérez-Aros, P.: Gradient formulae for nonlinear probabilistic constraints with non-convex quadratic forms. J. Optim. Theory Appl. 185(1), 239–269 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  64. van Ackooij, W., Pérez-Aros, P.: Gradient formulae for probability functions depending on a heterogenous family of constraints. Open J. Math. Optim. 2, 1–29 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The second author was partially supported by ANID grant: Fondecyt Regular 1200283 and Fondecyt Regular 1190110 and ANID grant: MATH-AmSud 20-MATH-08 CODE: MATH190003.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wim van Ackooij.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Subdifferentials

Appendix A: Subdifferentials

In our work we will make use of generalized (sub)-differentiation. For completeness we will provide below a brief description of their respective definitions. For further details we refer to e.g., [38, 47].

Definition A.1

(Tangent and normal cones) Let \(X \subseteq \mathbb {R}^n\) be a closed set and \({{\bar{x}}} \in X\) be given. The Bouligand/Severi tangent or contingent cone and Fréchet normal cone to X at \({{\bar{x}}}\) are respectively defined as:

$$\begin{aligned} \mathop {\mathrm{T}}\nolimits _X({{\bar{x}}})&:= \left\{ d \in \mathbb {R}^n \; : \; \exists t_n \downarrow 0, \; \exists x_n \in X, \lim _{n\rightarrow \infty } t_n^{-1}(x_n - {{\bar{x}}}) = d\right\} \\ \mathop {\mathrm{N}}\nolimits ^\mathtt{F}_X({{\bar{x}}})&:= \left\{ x^* \in \mathbb {R}^n \; : \left\langle x^*, d \right\rangle \le 0 \;, \; \forall d \in \mathop {\mathrm{T}}\nolimits _X({{\bar{x}}}) \right\} \nonumber \\&\;= \left\{ x^* \in \mathbb {R}^n \; : \limsup _{X \ni x' \rightarrow {\bar{x}}} \left\langle x^*, x' - {{\bar{x}}} \right\rangle \left\| x'-{{\bar{x}}}\right\| ^{-1} \le 0 \right\} . \end{aligned}$$

The Mordukhovich or limiting normal cone to X at \({{\bar{x}}}\) is defined as:

$$\begin{aligned} \mathop {\mathrm{N}}\nolimits ^\mathtt{M}_X({{\bar{x}}}) := \left\{ x^* \in \mathbb {R}^n \; : \; \exists (x_n, x_n^*) \rightarrow ({{\bar{x}}}, x^*), x_n \in X, x_n^* \in \mathop {\mathrm{N}}\nolimits ^\mathtt{F}_X(x_n)\right\} . \end{aligned}$$

With the help of these cones, we can define three notions of subdifferentials through the usual construction involving normal cones to epigraphs. This is done as follows:

Definition A.2

Let \(f : \mathbb {R}^n \rightarrow \mathbb {R}\) be a given lower semi-continuous function, then at \({{\bar{x}}} \in \mathbb {R}^n\), its Fréchet subdifferential, Mordukhovich or limiting subdifferential and singular subdifferential are defined as:

$$\begin{aligned} \partial ^\mathtt{F}f({{\bar{x}}})&= \left\{ x^* \; : (x^*, -1) \in \mathop {\mathrm{N}}\nolimits ^\mathtt{F}_{\mathop {\mathrm{epi}}f}({{\bar{x}}}, f({{\bar{x}}})) \right\} \end{aligned}$$
(31a)
$$\begin{aligned} \partial ^\mathtt{M}f({{\bar{x}}})&= \left\{ x^* \; : (x^*, -1) \in \mathop {\mathrm{N}}\nolimits ^\mathtt{M}_{\mathop {\mathrm{epi}}f}({{\bar{x}}}, f({{\bar{x}}})) \right\} \end{aligned}$$
(31b)
$$\begin{aligned} \partial ^\mathtt{\infty }f({{\bar{x}}})&= \left\{ x^* \; : (x^*, 0) \in \mathop {\mathrm{N}}\nolimits ^\mathtt{M}_{\mathop {\mathrm{epi}}f}({{\bar{x}}}, f({{\bar{x}}})) \right\} , \end{aligned}$$
(31c)

where \(\mathop {\mathrm{epi}}f\) refers to the epigraph of f.

For a function \(f : \mathbb {R}^n \rightarrow \mathbb {R}\) that is locally Lipschitzian, Clarke subdifferential enjoys favorable calculus rules. Among the many alternative definitions, let us provide the following characterization, valid in finite dimensions:

$$\begin{aligned} \partial ^\mathtt{C}f({{\bar{x}}}) := \mathop {\mathrm{co}}\left\{ \lim _{\ell \rightarrow \infty } \nabla f(x_{\ell })\; : x_{\ell } \rightarrow {{\bar{x}}}, f \;\text{ is } \text{ differentiable } \text{ at }\; x_{\ell }\right\} , \end{aligned}$$
(32)

resulting from [7, Theorem 2.5.1]. Here \(\mathop {\mathrm{co}}\) stands for the convex hull of a given set. It is moreover known that a function f is locally Lipchitzian at a given \({{\bar{x}}}\) if and only if \(\partial ^\mathtt{\infty }f({{\bar{x}}}) = \left\{ 0\right\} \), in which case moreover \(\partial ^\mathtt{C}f({{\bar{x}}}) = \mathop {\mathrm{co}}\partial ^\mathtt{M}f({{\bar{x}}})\) (see e.g., Theorems 9.13, 8.49 [47]).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

van Ackooij, W., Pérez-Aros, P. Generalized Differentiation of Probability Functions: Parameter Dependent Sets Given by Intersections of Convex Sets and Complements of Convex Sets. Appl Math Optim 85, 2 (2022). https://doi.org/10.1007/s00245-022-09844-5

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09844-5

Keywords

Mathematics Subject Classification

Navigation