Abstract
In this paper, we study second-order differentiability properties of probability functions. We present conditions under which probability functions involving nonlinear systems and Gaussian (or Student) multi-variate random vectors are twice continuously differentiable. We provide an expression for their Hessian that can be useful in numerical methods for solving probabilistic constrained optimization problems.
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Notes
Since we do not assume complex structure on g, the general case reduces to the normal centered case, as follows. Let \(\tilde{\xi }\sim \mathcal {N}(\mu ,\Sigma )\) be a general multi-variate Gaussian random vector with mean \(\mu \) and covariance matrix \(\Sigma \). We define \(\tilde{g}(x,z) = g(x, Dz + \mu )\) and observe that \(\xi = D^{-1}(\tilde{\xi }- \mu ) \sim \mathcal {N}(0,R)\), where D is the diagonal matrix with \(D_{ii} = {\Sigma _{ii}}^{1/2}\). We see that \(\mathbb {P}[ g(x,\tilde{\xi }) \le 0] = \mathbb {P}[ \tilde{g}(x,\xi ) \le 0 ]\) and that neither convexity in the second argument of g, nor its continuous differentiability properties are perturbed by such a transformation.
References
Arnold, T., Henrion, R., Möller, A., Vigerske, S.: A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pacific J. Optim. 10, 5–20 (2014)
Bremer, I., Henrion, R., Möller, A.: Probabilistic constraints via SQP solver: application to a renewable energy management problem. Comput. Manag. Sci. 12, 435–459 (2015)
Deák, I.: Computing probabilities of rectangles in case of multinormal distribution. J. Stat. Comput. Simul. 26(1–2), 101–114 (1986)
Deák, I.: Subroutines for computing normal probabilities of sets—computer experiences. Ann. Oper. Res. 100, 103–122 (2000)
Genz, A., Bretz, F.: Computation of multivariate normal and t probabilities. Number 195 in Lecture Notes in Statistics. Springer, Dordrecht (2009)
Henrion, R., Möller, A.: Optimization of a continuous distillation process under random inflow rate. Computer Math. Appl. 45, 247–262 (2003)
Henrion, R., Möller, A.: A gradient formula for linear chance constraints under Gaussian distribution. Math. Oper. Res. 37, 475–488 (2012)
Henrion, R., Strugarek, C.: Convexity of chance constraints with independent random variables. Comput. Optim. Appl. 41, 263–276 (2008)
Henrion, R., Strugarek, C.: Convexity of chance constraints with dependent random variables: the use of copulae. In: Bertocchi, M., Consigli, G., Dempster, M.A.H. (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)
Hunter, J.K., Nachtergaele, B.: Applied Analysis. World Scientific Publishing Company, Singapore (2001)
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)
Morgan, D.R., Eheart, J.W., Valocchi, A.J.: Aquifer remediation design under uncertainty using a new chance constraint programming technique. Water Resour. Res. 29, 551–561 (1993)
Naor, A., Romik, D.: Projecting the surface measure of the sphere of \(\ell _p^n\). Ann. I.H. Poincaré 39(2), 241–261 (2003)
Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht (1995)
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)
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)
Uryas’ev, S.: Derivatives of probability and integral functions: general theory and examples, 2nd edn. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, pp. 658–663. Springer, New York (2009)
van Ackooij, W.: Decomposition approaches for block-structured chance-constrained programs with application to hydro-thermal unit commitment. Math. Methods Oper. Res. 80(3), 227–253 (2014)
van Ackooij, W.: Eventual convexity of chance constrained feasible sets. Optimization (J. Math. Program. Oper. Res) 64(5), 1263–1284 (2015)
van Ackooij, W., de Oliveira, W.: Level bundle methods for constrained convex optimization with various oracles. Comput. Optim. Appl. 57(3), 555–597 (2014)
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)
van Ackooij, W., Henrion, R., Möller, A., Zorgati, R.: On probabilistic constraints induced by rectangular sets and multivariate normal distributions. Math. Methods Oper. Res. 71(3), 535–549 (2010)
van Ackooij, W., Henrion, R., Möller, A., Zorgati, R.: Joint chance constrained programming for hydro reservoir management. Optim. Eng. 15, 509–531 (2014)
van Ackooij, W., Minoux, M.: A characterization of the subdifferential of singular Gaussian distribution functions. Set Valued Var. Anal. 23(3), 465–483 (2015)
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)
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van Ackooij, W., Malick, J. Second-order differentiability of probability functions. Optim Lett 11, 179–194 (2017). https://doi.org/10.1007/s11590-016-1015-7
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DOI: https://doi.org/10.1007/s11590-016-1015-7