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Data-driven chance constrained stochastic program

Abstract

In this paper, we study data-driven chance constrained stochastic programs, or more specifically, stochastic programs with distributionally robust chance constraints (DCCs) in a data-driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters. We consider a family of density-based confidence sets based on a general \(\phi \)-divergence measure, and formulate DCC from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set. We derive an equivalent reformulation for DCC and show that it is equivalent to a classical chance constraint with a perturbed risk level. We also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general \(\phi \)-divergence measures. In several special cases, our results can be strengthened such that we can derive closed-form expressions for the perturbed risk levels. In addition, we show that the conservatism of DCC vanishes as the size of historical data goes to infinity. Furthermore, we analyze the relationship between the conservatism of DCC and the size of historical data, which can help indicate the value of data. Finally, we conduct extensive computational experiments to test the performance of the proposed DCC model and compare various \(\phi \)-divergence measures based on a capacitated lot-sizing problem with a quality-of-service requirement.

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Notes

  1. 1.

    Our work is related to [19], which develops equivalent reformulations of ambiguous expectation-constrained programs. To the best of our knowledge, [19] first appeared online after our work became publicized.

References

  1. 1.

    Ahmed, S.: Convex relaxations of chance constrained optimization problems. Optim. Lett. 8(1), 1–12 (2014). doi: 10.1007/s11590-013-0624-7

  2. 2.

    Ahmed, S., Papageorgiou, D.: Probabilistic set covering with correlations. Oper. Res. 61(2), 438–452 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ahmed, S., Shapiro, A.: Solving chance-constrained stochastic programs via sampling and integer programming. In: Chen, Z.-L., Raghavan, S. (eds.) Tutorials in Operations Research, pp. 261–269. INFORMS, Catonsville (2008)

    Google Scholar 

  4. 4.

    Ben-Tal, A., den Hertog, D., De Waegenaere, A., Melenberg, B., Rennen, G.: Robust solutions of optimization problems affected by uncertain probabilities. Manage. Sci. 59(2), 341–357 (2013)

    Article  Google Scholar 

  5. 5.

    Beraldi, P., Ruszczyński, A.: A branch and bound method for stochastic integer problems under probabilistic constraints. Optim. Methods Softw. 17(3), 359–382 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin (1997)

    MATH  Google Scholar 

  7. 7.

    Calafiore, G.C., Campi, M.C.: Uncertain convex programs: randomized solutions and confidence levels. Math. Program. 102(1), 25–46 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Calafiore, G.C., El Ghaoui, L.: On distributionally robust chance-constrained linear programs. J. Optim. Theory Appl. 130(1), 1–22 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Charnes, A., Cooper, W.W.: Deterministic equivalents for optimizing and satisficing under chance constraints. Oper. Res. 11(1), 18–39 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    Article  Google Scholar 

  11. 11.

    Chen, W., Sim, M., Sun, J., Teo, C.P.: From CVaR to uncertainty set: implications in joint chance constrained optimization. Oper. Res. 58(2), 470–485 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, Hoboken (2012)

    MATH  Google Scholar 

  13. 13.

    Danielsson, J.: Stochastic volatility in asset prices estimation with simulated maximum likelihood. J. Econom. 64(1), 375–400 (1994)

    Article  MATH  Google Scholar 

  14. 14.

    Devroye, L., Györfi, L.: Nonparametric Density Estimation: The \(\ell _1\) View. Wiley, Hoboken (1985)

    MATH  Google Scholar 

  15. 15.

    El Ghaoui, L., Oks, M., Oustry, F.: Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51(4), 543–556 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Erdoğan, E., Iyengar, G.: Ambiguous chance constrained problems and robust optimization. Math. Program. 107(1), 37–61 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Gibbs, A.L., Su, F.E.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Article  MATH  Google Scholar 

  18. 18.

    Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41(2), 263–276 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Hu, Z., Hong, L.J.: Kullback–Leibler Divergence Constrained Distributionally Robust Optimization. Technical report, The Hong Kong University of Science and Technology. Available at optimization-online: http://www.optimization-online.org/DB_FILE/2012/11/3677.pdf (2013)

  20. 20.

    Justus, C.G., Hargraves, W.R., Mikhail, A., Graber, D.: Methods for estimating wind speed frequency distributions. J. Appl. Meteorol. 17(3), 350–353 (1978)

    Article  Google Scholar 

  21. 21.

    Kall, P., Wallace, S.: Stochastic Programming. Wiley, Hoboken (1994)

    MATH  Google Scholar 

  22. 22.

    Küçükyavuz, S.: On mixing sets arising in chance-constrained programming. Math. Program. 132(1–2), 31–56 (2010)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Kullback, S.: Information Theory and Statistics. Courier Dover Publications, Mineola (1997)

    MATH  Google Scholar 

  24. 24.

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

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Liao, S., van Delft, C., Vial, J.-P.: Distributionally robust workforce scheduling in call centres with uncertain arrival rates. Optim. Methods Softw. 28(3), 501–522 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Love, D., Bayraksan, G.: Two-stage likelihood robust linear program with application to water allocation under uncertainty. In: Winter Simulation Conference, pp. 77–88 (2013)

  27. 27.

    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). doi: 10.1007/s10107-013-0684-6

  28. 28.

    Luedtke, J., Ahmed, S.: A sample approximation approach for optimization with probabilistic constraints. SIAM J. Optim. 19(2), 674–699 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

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

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Miller, B.L., Wagner, H.M.: Chance constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)

    Article  MATH  Google Scholar 

  31. 31.

    Nemirovski, A., Shapiro, A.: Scenario approximations of chance constraints. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design Under Uncertainty, pp. 3–47. Springer, Berlin (2006)

    Chapter  Google Scholar 

  32. 32.

    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Pagnoncelli, B., Ahmed, S., Shapiro, A.: Sample average approximation method for chance constrained programming: theory and applications. J. Optim. Theory Appl. 142(2), 399–416 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Pardo, L.: Statistical Inference Based on Divergence Measures, vol. 185. CRC Press, Boca Raton (2006)

    MATH  Google Scholar 

  35. 35.

    Parzen, E.: On estimation of a probability density function and mode. Ann. Math. Stat. 33(3), 1065–1076 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Popescu, I.: A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. Math. Oper. Res. 30(3), 632–657 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  37. 37.

    Prékopa, A.: On probabilistic constrained programming. In: Proceedings of the Princeton Symposium on Mathematical Programming, pp. 113–138. Citeseer (1970)

  38. 38.

    Prékopa, A.: Stochastic Programming. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  39. 39.

    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)

    Google Scholar 

  40. 40.

    Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27(3), 832–837 (1956)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Ruszczyński, A.: Probabilistic programming with discrete distributions and precedence constrained knapsack polyhedra. Math. Program. 93(2), 195–215 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

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

  43. 43.

    Van Parys, B.P.G., Goulart, P.J., Kuhn, D.: Generalized Gauss Inequalities Via Semidefinite Programming. Technical report, Automatic Control Laboratory, Swiss Federal Institute of Technology Zürich (2014)

  44. 44.

    Vandenberghe, L., Boyd, S., Comanor, K.: Generalized Chebyshev bounds via semidefinite programming. SIAM Rev. 49(1), 52–64 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Wang, Z., Glynn, P.W., Ye, Y.: Likelihood robust optimization for data-driven problems. arXiv:1307.6279v3.pdf (2014)

  46. 46.

    Wasserman, L.: All of Nonparametric Statistics. Springer, Berlin (2006)

    MATH  Google Scholar 

  47. 47.

    Yanıkoglu, I., den Hertog, D., Kleijnen, J.: Adjustable Robust Parameter Design with Unknown Distributions. Available at optimization-online: http://www.optimization-online.org/DB_FILE/2013/03/3806.pdf (2014)

  48. 48.

    Zymler, S., Kuhn, D., Rustem, B.: Distributionally robust joint chance constraints with second-order moment information. Math. Program. 137(1–2), 167–198 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

    Zymler, S., Kuhn, D., Rustem, B.: Worst-case value at risk of nonlinear portfolios. Manage. Sci. 59(1), 172–188 (2013)

    MathSciNet  Article  Google Scholar 

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Acknowledgments

The authors would like to thank the associate editor and referees very much for providing the nice suggestions, which help improve the quality of this paper significantly.

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Correspondence to Yongpei Guan.

Additional information

An early version of this paper is available online at www.optimization-online.org/DB_HTML/2012/07/3525.html. This paper has been presented in the Simulation Optimization Workshop, Viña del Mar, Chile, March 21–23, 2013 and the 13th International Conference on Stochastic Programming, Bergamo, Italy, July 8–12, 2013.

Appendices

Appendix 1: Proof of Lemma 3

Proof

We prove each property as follows:

  1. (i)

    By definition, \(\phi ^*\) is a supremum of linear functions and hence convex.

  2. (ii)

    For any \(x_1, x_2 \in {\mathbb {R}}\) such that \(x_1 < x_2\), we have

    $$\begin{aligned} x_1t - g(t) \ \le \ x_2t - g(t), \ \ \ \forall t \ge 0. \end{aligned}$$

    Also, since \(\phi (t) = +\infty \) for \(t < 0\), we have

    $$\begin{aligned} \phi ^*(x) = \ \sup _{t \in {\mathbb {R}}} \left\{ xt-\phi (t)\right\} = \ \sup _{t \ge 0} \left\{ xt-\phi (t)\right\} , \end{aligned}$$

    and so

    $$\begin{aligned} \phi ^*(x_1) = \ \sup _{t \ge 0} \left\{ x_1t-\phi (t)\right\} \ \le \ \sup _{t \ge 0} \left\{ x_2t-\phi (t)\right\} = \ \phi ^*(x_2). \end{aligned}$$
  3. (iii)

    Since \(\phi (1) = 0\), we have

    $$\begin{aligned} \phi ^*(x) = \ \sup _{t \ge 0} \left\{ xt-\phi (t)\right\} \ \ge \ x. \end{aligned}$$
  4. (iv)

    We prove by contradiction. Suppose that \(\phi ^*(x) = m\) on the interval [ab] and \(\phi ^*(y) = m' \ne m\) for some \(y < a\). First, we observe that \(m' < m\) because \(\phi ^*\) is nondecreasing. Second, there exists some \(\lambda \in [0, 1]\) such that \(a = \lambda y + (1-\lambda )b\). It follows that

    $$\begin{aligned} \phi ^*(a) \ \le \ \lambda \phi ^*(y) + (1-\lambda ) \phi ^*(b) = \ \lambda m' + (1-\lambda ) m < m, \end{aligned}$$

    which gives a desirable contradiction. \(\square \)

Appendix 2: Proof of Remark 2

Proof

If \(\ell _{\phi } = +\infty \), then \(\ell _{\phi } \ge \overline{m}(\phi ^*)\) and the claim holds. Hence, without loss of generality, we can assume that \(\ell _{\phi } \,{<}\, +\infty \). In the remainder of the proof, we show that \(\ell _{\phi } + \delta \ge \overline{m}(\phi ^*)\) for any \(\delta \,{>}\, 0\), and the claim follows. Because \(\ell _{\phi } = \lim _{x \rightarrow +\infty } \phi (x)/x\), there exists a \(K \,{>}\, 0\) such that \(|\ell _{\phi } - \phi (x)/x| \le \delta /2\), and accordingly \(\ell _{\phi }x - \phi (x) \ge -\delta x /2\) for all \(x \ge K\). If follows that, for any \(M \in {\mathbb {R}}\), there exists an \(N := \max \{K, 2M/\delta \}\) such that

$$\begin{aligned} x \ge N \ \Rightarrow&{\left\{ \begin{array}{ll} \ell _{\phi } x - \phi (x) \ge -\delta x /2, \\ \delta x / 2 \ge M \end{array}\right. } \\ \Rightarrow&(\ell _{\phi } + \delta ) x - \phi (x) = \left( \ell _{\phi } x - \phi (x)\right) + \delta x \ \ge \ \delta x / 2 \ \ge \ M. \end{aligned}$$

It follows that \(\lim _{x \rightarrow +\infty }\{ (\ell _{\phi } + \delta ) x - \phi (x) \} = +\infty \). Hence,

$$\begin{aligned} \phi ^*(\ell _{\phi } + \delta ) = \ \sup _{x \in {\mathbb {R}}} \{ (\ell _{\phi } + \delta ) x - \phi (x) \} \ \ge \ \lim _{x \rightarrow +\infty }\{ (\ell _{\phi } + \delta ) x - \phi (x) \} = \ +\infty . \end{aligned}$$

Therefore, \(\ell _{\phi } + \delta \ge \overline{m}(\phi ^*)\) for any \(\delta > 0\), and the proof is completed. \(\square \)

Appendix 3: Proof of Proposition 2

Proof

First, since \(\phi (x) = (x-1)^2\), we have

$$\begin{aligned} \phi ^*(x) = \left\{ \begin{array}{l@{\quad }l} -1, &{} \hbox {if} \, x \le -2,\\ \frac{1}{4}x^2 + x, &{} \hbox {if} \, x\ge -2. \end{array}\right. \end{aligned}$$

Hence, \(\underline{m}(\phi ^*) = -2\) and \(\overline{m}(\phi ^*) = +\infty \). Second, we solve the problem

$$\begin{aligned} \inf _{\begin{array}{c} z > 0,\\ z_0 + z \ge -2 \end{array}} \frac{\phi ^*(z_0+z) - z_0 - \alpha z + d}{\phi ^*(z_0+z) - \phi ^*(z_0)} = \ \inf _{\begin{array}{c} z > 0,\\ z_0 \ge -2 \end{array}} \frac{\phi ^*(z_0) - z_0 + (1-\alpha ) z + d}{\phi ^*(z_0) - \phi ^*(z_0-z)} \end{aligned}$$

to optimality, where we make a transform by replacing \(z_0\) by \(z_0-z\). We let \(g(z_0, z)\) represent the objective function and discuss the following cases:

  1. 1.

    If \(z_0 - z \le -2\), then \(\phi ^*(z_0-z) = -1\) and \(\phi ^*(z_0) = \frac{1}{4}z_0^2 + z_0\). It follows that

    $$\begin{aligned} g(z_0, z) = \ \frac{\left( \frac{1}{4}z_0^2 + z_0 \right) -z_0+(1-\alpha )z+d}{\left( \frac{1}{4}z_0^2 + z_0 \right) - (-1)} = \ \frac{\frac{1}{4}z_0^2 + (1-\alpha )z + d}{\left( \frac{1}{2}z_0 + 1 \right) ^2}, \end{aligned}$$

    and so

    $$\begin{aligned} \frac{\partial g(z_0, z)}{\partial z_0} = \frac{\frac{1}{2}z_0-(1-\alpha )z-d}{\left( \frac{1}{2}z_0 + 1 \right) ^3}. \end{aligned}$$

    Since \(z_0 \le z -2,\,z_0 \ge -2\) and \(\alpha < 1/2\) by assumption, we have \((1/2)z_0-(1-\alpha )z-d\le (\alpha -1/2)z-d-1 < 0\) and \(\frac{1}{2}z_0 + 1 \ge 0\). Hence, \(\partial g(z_0, z)/\partial z_0 < 0\) for any fixed z and it is optimal to choose \(z^*_0 = z-2\). It follows that

    $$\begin{aligned} \inf _{\begin{array}{c} z > 0,\\ z_0 \ge -2 \end{array}} g(z_0, z) = \ \inf _{z > 0} g(z-2, z) = \inf _{z>0} \ 4(d+1)\left( \frac{1}{z}\right) ^2 - 4\alpha \left( \frac{1}{z}\right) + 1. \end{aligned}$$

    Therefore, it is optimal to choose \(z^* = 2(d+1)/\alpha \) and

    $$\begin{aligned} \inf _{\begin{array}{c} z > 0,\\ z_0 \ge -2 \end{array}} g(z_0, z) = \ 1-\frac{\alpha ^2}{d+1}. \end{aligned}$$
  2. 2.

    If \(z_0 -z \ge -2\), then \(\phi ^*(z_0-z) = \frac{1}{4}(z_0-z)^2 + (z_0-z)\) and \(\phi ^*(z_0) = \frac{1}{4}z_0^2 + z_0\). It follows that

    $$\begin{aligned} g(z_0, z) = \ \frac{\left( \frac{1}{4}z_0^2 + z_0 \right) -z_0+(1-\alpha )z+d}{\left( \frac{1}{4}z_0^2 + z_0 \right) - \left( \frac{1}{4}(z_0-z)^2 + (z_0-z) \right) } = \ \frac{\frac{1}{4}z_0^2 + (1-\alpha )z + d}{\frac{1}{2}zz_0 + z - \frac{1}{4}z^2}, \end{aligned}$$

    and so

    $$\begin{aligned} \frac{\partial g(z_0, z)}{\partial z_0} = \frac{z\left( z_0^2 + (4-z)z_0 - 4(1-\alpha )z-4d\right) }{8\left( \frac{1}{2}zz_0 + z - \frac{1}{4}z^2\right) ^2}. \end{aligned}$$

    For fixed z, we set \(\partial g(z_0, z)/\partial z_0 = 0\) and obtain

    $$\begin{aligned} z_0 = \frac{(z-4)\pm \sqrt{z^2+8(1-2\alpha )z+16(d+1)}}{2}. \end{aligned}$$

    Since \(z_0 \ge z-2\), we rule out the negative root and so

    $$\begin{aligned} z^*_0 = \frac{(z-4)+\sqrt{z^2+8(1-2\alpha )z+16(d+1)}}{2} \end{aligned}$$

    is a stationary point of \(g(z_0, z)\) with z fixed and the corresponding objective value

    $$\begin{aligned} g(z_0^*, z) = \ \frac{1}{2}\sqrt{16(d+1)\left( \frac{1}{z}\right) ^2+8(1-2\alpha )\left( \frac{1}{z}\right) +1} \ + \frac{1}{2}\left( 1-4\left( \frac{1}{z}\right) \right) . \end{aligned}$$

    Now we show that \(z_0^*\) is an optimal solution for \(\inf _{z_0\ge z-2}g(z_0, z)\) with z fixed. We compare the value of \(g(z_0^*, z)\) with \(g(+\infty , z)\) and \(g(z-2, z)\) because \(+\infty \) and \(z-2\) are the end points of the feasible region of \(z_0\). We observe that \(g(+\infty , z) = +\infty \). Also, we have

    $$\begin{aligned} g(z-2, z) = \ \frac{\frac{1}{4}(z-2)^2+(1-\alpha )z+d}{\frac{1}{2}z(z-2)+z-\frac{1}{4}z^2} = \ 4(d+1)\left( \frac{1}{z}\right) ^2 - 4\alpha \left( \frac{1}{z}\right) + 1, \end{aligned}$$

    and \(g(z-2, z) \ge g(z_0^*, z)\). To see that, we compare the values of \(g(z-2, z)\) and \(g(z_0^*, z)\) by the following inequalities, where the inequalities below imply those above.

    $$\begin{aligned}&g(z-2, z) \ge g(z_0^*, z) \\&\quad \Leftarrow 8(d+1)\left( \frac{1}{z}\right) ^2 - 8\alpha \left( \frac{1}{z}\right) + 2 \\&\quad \ge \sqrt{16(d+1)\left( \frac{1}{z}\right) ^2+8(1-2\alpha )\left( \frac{1}{z}\right) +1} + \left( 1-4\left( \frac{1}{z}\right) \right) \\&\quad \Leftarrow \left[ 8(d+1)\left( \frac{1}{z}\right) ^2 + 4(1-2\alpha )\left( \frac{1}{z}\right) + 1\right] ^2 \\&\quad \ge 16(d+1)\left( \frac{1}{z}\right) ^2+8(1-2\alpha )\left( \frac{1}{z}\right) +1\\&\quad \Leftarrow 16\left( \frac{1}{z}\right) ^2\left[ 2(d+1)\left( \frac{1}{z}\right) + (1-2\alpha )\right] ^2 \ge 0. \end{aligned}$$

    Hence, \(\inf _{z_0\ge z-2}g(z_0, z) = g(z_0^*, z)\) with z fixed. Therefore, we have

    $$\begin{aligned} \inf _{z > 0, z_0 \ge z-2} g(z_0, z) = \ \inf _{z > 0} \ \frac{1}{2}\sqrt{16(d+1)z^2+8(1-2\alpha )z+1} \ + \frac{1}{2}(1-4z), \end{aligned}$$

    where we have 1 / z replaced by z. Similarly, we set

    $$\begin{aligned} \frac{\partial g(z_0^*, z)}{\partial z} = \ \frac{8(d+1)z+2(1-2\alpha )}{\sqrt{16(d+1)z^2+8(1-2\alpha )z+1}}-2 = \ 0, \end{aligned}$$

    and obtain

    $$\begin{aligned} z^* = \frac{\sqrt{d^2 + 4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{4d(d+1)}. \end{aligned}$$

    Therefore, we have

    $$\begin{aligned} g(z_0^*, z^*) = \ 1-\alpha +\frac{\sqrt{d^2 + 4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{2(d+1)}. \end{aligned}$$

    Again, we shall compare the value of \(g(z_0^*, z^*)\) with \(g(z_0^*, +\infty )\) and \(g(z_0^*, 0)\) since \(+\infty \) and 0 are the end points of the feasible region of z. We observe that \(g(z_0^*, +\infty ) = +\infty \) and \(g(z_0^*, 0) = 1 \ge g(z_0^*, z^*)\), and hence

    $$\begin{aligned} \inf _{z > 0, z_0 \ge z-2} g(z_0, z) = \ 1-\alpha +\frac{\sqrt{d^2 + 4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{2(d+1)}. \end{aligned}$$

Finally, we compare the optimal value of \(g(z_0, z)\) in the two cases. We claim that the optimal value obtained in the latter case is smaller (and hence globally optimal). To see that, we compare the two values by the following inequalities, where the inequalities below imply those above.

$$\begin{aligned}&1 - \frac{\alpha ^2}{d+1} \ge 1-\alpha +\frac{\sqrt{d^2 + 4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{2(d+1)} \\&\quad \Leftarrow d+2\alpha -2\alpha ^2 \ge \sqrt{d^2 + 4d(\alpha -\alpha ^2)} \\&\quad \Leftarrow (d+2\alpha -2\alpha ^2)^2 \ge d^2 + 4d(\alpha -\alpha ^2) \\&\quad \Leftarrow 4\alpha ^2(\alpha -1)^2 \ge 0. \end{aligned}$$

Therefore, the perturbed risk level is

$$\begin{aligned} \alpha ' = \ \alpha - \frac{\sqrt{d^2+4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{2d+2}. \end{aligned}$$

\(\square \)

Appendix 4: Proof of Proposition 3

Proof

First, Since \(\phi (x) = |x-1|\), we have

$$\begin{aligned} \phi ^*(x) = \left\{ \begin{array}{ll} -1, &{}\quad \hbox {if} \, x <-1, \\ x, &{}\quad \hbox {if} \, -1\le x\le 1, \\ +\infty , &{}\quad \hbox {if} \, x > 1. \end{array}\right. \end{aligned}$$

Hence, \(\underline{m}(\phi ^*) = -1\) and \(\overline{m}(\phi ^*) = 1\). Second, we solve the problem

$$\begin{aligned} \inf _{z > 0, -1 \le z_0 + z \le 1} g(z_0, z) := \frac{\phi ^*(z_0+z) - z_0 - \alpha z + d}{\phi ^*(z_0+z) - \phi ^*(z_0)} \end{aligned}$$

to optimality. We discuss the following cases:

  1. 1.

    If \(z_0 \le -1\), then \(\phi ^*(z_0) = -1\) and \(\phi ^*(z_0+z) = z_0+z\). It follows that

    $$\begin{aligned} g(z_0, z) = \frac{(1-\alpha )z+d}{z_0+z+1}. \end{aligned}$$

    Note here that for any given \(z,\,g(z_0, z)\) is a nonincreasing function of \(z_0\), due to the fact that \(z_0+z+1 \ge 0\). Meanwhile, \(z_0+z \le 1\). Hence, it is optimal to choose \(z_0^* = \min \{1-z, -1\}\) and so

    $$\begin{aligned} g(z_0^*, z) = \left\{ \begin{array}{ll} \frac{(1-\alpha )z+d}{2}, &{}\quad \hbox {if} \, z \ge 2, \\ \frac{(1-\alpha )z+d}{z}, &{}\quad \hbox {if} \, z \le 2. \end{array}\right. \end{aligned}$$

    Therefore, \(g(z_0^*, z)\) is nonincreasing on z on the interval (0, 2] and nondecreasing on z on the interval \([2, +\infty )\), and so \(g(z_0^*, z^*) = 1-\alpha +\frac{d}{2}\).

  2. 2.

    If \(-1 \le z_0 \le 1\), then \(\phi ^*(z_0) = z_0\). Also, we have \(z \le 2\) and \(\phi ^*(z_0+z) = z_0+z\) because \(-1 \le z_0 + z\le 1\). Hence,

    $$\begin{aligned} g(z_0, z) = \frac{(1-\alpha )z+d}{z} = 1 - \alpha + \frac{d}{z} \ge 1-\alpha +\frac{d}{2}, \end{aligned}$$

    and the lower bound is attained at \(z^* = 2\). Therefore, \(g(z_0^*, z^*) = 1-\alpha +\frac{d}{2}\).

To sum up, we have \(1 - \alpha ' = g(z_0^*, z^*) = 1-\alpha +\frac{d}{2}\), or equivalently \(\alpha ' = \alpha -\frac{d}{2}\). \(\square \)

Appendix 5: Proof of Proposition 4

Proof

We divide the proof into two parts. In the first part, we show that the perturbed risk level

$$\begin{aligned} \alpha ' = 1 - \inf _{x\in (0, 1)} \left\{ \frac{e^{-d} x^{1-\alpha } - 1}{x - 1} \right\} . \end{aligned}$$
(15)

In the second part, we show how to compute \(\alpha '\) by using bisection line search.

(Risk level) First, since \(\phi (x) = x\log x -x + 1\), we have \(\phi ^*(x) = e^x-1\). Hence, \(\underline{m}(\phi ^*) = -\infty \) and \(\overline{m}(\phi ^*) = +\infty \). Second, we solve the problem

$$\begin{aligned} \inf _{z > 0} g(z_0, z) := \frac{\phi ^*(z_0+z) - z_0 - \alpha z + d}{\phi ^*(z_0+z) - \phi ^*(z_0)} = \ \inf _{z > 0} \frac{e^z+(d-\alpha z-z_0-1)e^{-z_0}}{e^z-1} \end{aligned}$$

to optimality. Since

$$\begin{aligned} \frac{\partial g}{\partial z_0} = -\frac{d-\alpha z-z_0}{e^{z_0}(e^z-1)}, \end{aligned}$$

we have \(z_0^* = d-\alpha z\) by setting \(\partial g/\partial z_0 = 0\), and so

$$\begin{aligned} \inf _{z > 0} g(z_0, z) =&\ \inf _{z>0} \ g(z_0^*, z) \nonumber \\ =&\ \inf _{z > 0} \ \frac{e^z-e^{\alpha z-d}}{e^z-1} \nonumber \\ =&\ \inf _{z > 0} \ \frac{1-e^{-d}(1/e^z)^{1-\alpha }}{1-(1/e^z)} \nonumber \\ =&\ \inf _{x \in (0, 1)} \ \frac{e^{-d}x^{1-\alpha }-1}{x-1}, \end{aligned}$$
(16)

where Eq. (16) follows by replacing \((1/e^z)\) with x. Therefore, we have proved Eq. (15).

(Computation) We compute \(\alpha '\) by searching the optimal solution of the minimization problem

$$\begin{aligned} \inf _{x\in (0, 1)} \Bigl \{ \frac{e^{-d} x^{1-\alpha } - 1}{x - 1} \Bigr \}. \end{aligned}$$

First, by denoting \(1 - \alpha ' = \inf _{x\in (0, 1)} h(x)\), we have

$$\begin{aligned} h'(x) = \frac{1 - e^{-d} \alpha x^{1-\alpha } - e^{-d} (1-\alpha ) x^{-\alpha }}{(x-1)^2}, \ \ \forall x \in (0, 1). \end{aligned}$$

It is clear that \((x-1)^2\) decreases as x increases. Meanwhile, since \(x < 1\) and \(x^{-\alpha -1} > x^{-\alpha }\), we have

$$\begin{aligned} (1 - e^{-d} \alpha x^{1-\alpha } - e^{-d} (1-\alpha ) x^{-\alpha })'_x = e^{-d} \alpha (1-\alpha ) (x^{-\alpha -1} - x^{-\alpha }) > 0. \end{aligned}$$

Therefore, \(h'(x)\) increase as x increases in (0, 1), and hence the function h(x) is convex over x in (0, 1). Because \(\displaystyle \lim _{x \rightarrow 0^+} h'(x) = -\infty \) and \(\displaystyle \lim _{x \rightarrow 1^-} h'(x) = +\infty \), we have:

$$\begin{aligned} \hbox {the infimum of } h(x) \hbox { is attained in the interval } (0, 1). \end{aligned}$$
(17)

We can compute the optimal \(x^*\) by forcing

$$\begin{aligned} \frac{1 - e^{-d} \alpha (x^*)^{1-\alpha } - e^{-d} (1-\alpha ) (x^*)^{-\alpha }}{(x^*-1)^2} = 0, \end{aligned}$$

i.e., \((x^*)^{\alpha } = e^{-d}\alpha x^* + e^{-d}(1-\alpha )\). The intersection of functions \(x^{\alpha }\) and \(e^{-d}\alpha x + e^{-d}(1-\alpha )\) can be easily computed by a bisection line search. Finally, to achieve \(\epsilon \) accuracy, i.e., \(|\hat{x} - x^*| \le \epsilon \), of the incumbent probing value \(\hat{x}\), we only have to conduct S steps of bisection, such that \(2^{-S} \le \epsilon \). It follows that \(S \ge \left\lceil \log _2(\frac{1}{\epsilon }) \right\rceil \). \(\square \)

Appendix 6: Proof of Proposition 6

Proof

First, the convergence claim follows from Proposition 5 because \(x = 1\) is the unique minimizer of function \(\phi _{\chi D2}(x) = (x-1)^2,\,x \ge 0\).

Second, we define

$$\begin{aligned} g(d) \ := \ \alpha - \alpha ' = \ \frac{\sqrt{d^2+4d(\alpha -\alpha ^2)}-(1-2\alpha )d}{2d+2} \end{aligned}$$

based on Proposition 2. Then, we have

$$\begin{aligned} g'(d) = \ \frac{1}{2(d+1)^2}\left[ \frac{(2\alpha ^2-2\alpha +1)d+2(\alpha -\alpha ^2)}{\sqrt{d^2+4d(\alpha -\alpha ^2)}} - (1-2\alpha ) \right] . \end{aligned}$$
(18)

Since \(d = d(N)\) by assumption, we have

$$\begin{aligned} \hbox {VoD}_{\alpha }&= \frac{{\hbox {d}}\alpha '}{{\hbox {d}}N} = \left( \frac{{\hbox {d}}\alpha '}{{\hbox {d}}d}\right) \left( \frac{{\hbox {d}}d(N)}{{\hbox {d}}N}\right) \nonumber \\&= {-} g'(d) \ \left( \frac{{\hbox {d}}d(N)}{{\hbox {d}}N}\right) . \end{aligned}$$
(19)

The proof is completed by substituting the definition of \(g'(d)\) in (18) into Eq. (19). \(\square \)

Appendix 7: Proof of Proposition 7

Proof

The convergence claim follows from Proposition 5 because \(x = 1\) is the unique minimizer of function \(\phi _{\hbox {KL}}(x) = x \log (x) - x + 1,\,x \ge 0\). We divide the remainder of the proof into two parts. We develop the relationship between \(\alpha '\) and d in the first part, and compute the value of data in the second part.

(Relationship between \(\alpha '\) and d) From Proposition 4, we have

$$\begin{aligned} 1 - \alpha ' = \inf _{x\in (0, 1)} \left\{ \frac{e^{-d} x^{1-\alpha } - 1}{x - 1} \right\} . \end{aligned}$$
(20)

Also, the optimal objective value of the embedded optimization problem in equality (20) can be attained by some \(\bar{x} \in (0, 1)\) (based on claim (17) in the proof of Proposition 4), which is the stationary point of the objective function. It follows that

$$\begin{aligned} \left\{ \begin{array}{l} (e^{-d} \bar{x} ^{1-\alpha } - 1)/(\bar{x} - 1) = 1 - \alpha ' \\ \bar{x}^{\alpha } = e^{-d} \alpha \bar{x} + e^{-d} (1 - \alpha ). \end{array} \right. \end{aligned}$$
(21)

Solving this nonlinear equation system, we reformulate the first equation and then substitute the second equation into the first as follows:

$$\begin{aligned}&e^{-d} \bar{x} - (1-\alpha ') \bar{x} \bar{x}^{\alpha } = \alpha ' \bar{x}^{\alpha } \\&\quad \Rightarrow e^{-d} \bar{x} - (1-\alpha ') \bar{x} \Bigl ( e^{-d} \alpha \bar{x} + e^{-d} (1 - \alpha ) \Bigr ) = \alpha ' \Bigl ( e^{-d} \alpha \bar{x} + e^{-d} (1 - \alpha ) \Bigr ) \\&\quad \Rightarrow (\bar{x} - 1) \Bigl ( \alpha (1-\alpha ') \bar{x} - \alpha '(1 - \alpha ) \Bigr ) = 0. \end{aligned}$$

Ruling out the solution \(\bar{x} = 1\), we have \(\bar{x} = \frac{\alpha '(1 - \alpha )}{\alpha (1-\alpha ')} \in (0, 1)\). Finally, we substitute the solution of \(\bar{x}\) back into the second equation in (21) and obtain

$$\begin{aligned} e^{-d} = {\bar{x}^{\alpha }}\big /{(\alpha \bar{x} + 1 - \alpha )} = \bigl ( {\alpha '}\big /{\alpha } \bigr )^{\alpha } \bigl ( {(1 - \alpha ')}\big /{(1 - \alpha )} \bigr )^{1 - \alpha }. \end{aligned}$$
(22)

Finally, by taking the natural logarithm on both sides of Eq. (22), we obtain that

$$\begin{aligned} d = \alpha \log \left( \frac{\alpha }{\alpha '} \right) + (1-\alpha ) \log \left( \frac{1-\alpha }{1-\alpha '} \right) . \end{aligned}$$
(23)

(Value of data) From Eq. (23) we have

$$\begin{aligned} \frac{{\hbox {d}}d}{{\hbox {d}}\alpha '} = -\frac{\alpha }{\alpha '} + \frac{1-\alpha }{1-\alpha '} = \frac{\alpha '-\alpha }{\alpha '(1-\alpha ')}. \end{aligned}$$

It is easy to observe that \({{\hbox {d}}d}\big /{{\hbox {d}}\alpha '}\) is a monotone function of \(\alpha '\) and \({{\hbox {d}}d}\big /{{\hbox {d}}\alpha '} \ne 0\). Hence, we have

$$\begin{aligned} \frac{{\hbox {d}}\alpha '}{{\hbox {d}}d} = 1 \Bigl / \left( \frac{{\hbox {d}}d}{{\hbox {d}}\alpha '}\right) = \frac{\alpha '(1-\alpha ')}{\alpha '-\alpha }. \end{aligned}$$

Therefore,

$$\begin{aligned} \hbox {VoD}_{\alpha } = \left( \frac{{\hbox {d}}\alpha '}{{\hbox {d}}d}\right) \left( \frac{{\hbox {d}}d(N)}{{\hbox {d}}N} \right) = \left[ \frac{\alpha '(1-\alpha ')}{\alpha '-\alpha }\right] d'(N). \end{aligned}$$

\(\square \)

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Jiang, R., Guan, Y. Data-driven chance constrained stochastic program. Math. Program. 158, 291–327 (2016). https://doi.org/10.1007/s10107-015-0929-7

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Keywords

  • Stochastic programming
  • Chance constraints
  • Semi-infinite programming

Mathematics Subject Classification

  • 90C15
  • 90C34