Abstract
We derive an implementable algorithm for solving nonlinear stochastic optimization problems with failure probability constraints using sample average approximations. The paper extends prior results dealing with a failure probability expressed by a single measure to the case of failure probability expressed in terms of multiple performance measures. We also present a new formula for the failure probability gradient. A numerical example addressing the optimal design of a reinforced concrete highway bridge illustrates the algorithm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Royset, J.O., Polak, E.: Implementable algorithm for stochastic programs using sample average approximations. J. Optim. Theory Appl. 122, 157–184 (2004)
Ditlevsen, O., Madsen, H.O.: Structural Reliability Methods. Wiley, New York (1996)
Liu, P.-L., Kuo, C.-Y.: Safety evaluation of the upper structure of bridge based on concrete nondestructive tests. In: Der Kiureghian, A., Madanat, S., Pestana, J.M. (eds.) Applications of Statistics and Probability in Civil Engineering, pp. 1683–1688. Millpress, Rotterdam (2003)
Akgul, F., Frangopol, D.M.: Probabilistic analysis of bridge networks based on system reliability and Monte Carlo simulation. In: Der Kiureghian, A., Madanat, S., Pestana, J.M. (eds.) Applications of Statistics and Probability in Civil Engineering, pp. 1633–1637. Millpress, Rotterdam (2003)
Holicky, M., Markova, J.: Reliability analysis of impacts due to road vehicles. In: Der Kiureghian, A., Madanat, S., Pestana, J.M. (eds.) Applications of Statistics and Probability in Civil Engineering, pp. 1645–1650. Millpress, Rotterdam (2003)
Ermoliev, Y.: Stochastic quasigradient methods. In: Ermoliev, Y., Wets, R.J.-B. (eds.) Numerical Techniques for Stochastic Optimization. Springer, New York (1988)
Benveniste, A., Metivier, M., Priouret, P.: Adaptive Algorithms and Stochastic Approximations. Springer, New York (1990)
Marti, K.: Stochastic Optimization Methods. Springer, Berlin (2005)
Ruszczynski, A., Shapiro, A.: Stochastic Programming. Elsevier, New York (2003)
Shapiro, A., Wardi, Y.: Convergence analysis of stochastic algorithms. Math. Oper. Res. 21, 615–628 (1996)
Sakalauskas, L.L.: Nonlinear stochastic programming by Monte-Carlo estimators. Eur. J. Oper. Res. 137, 558–573 (2002)
Royset, J.O., Der Kiureghian, A., Polak, E.: Optimal design with probabilistic objective and constraints. J. Eng. Mech. 132, 107–118 (2006)
Uryasev, S.: Derivatives of probability functions and some applications. Ann. Oper. Res. 56, 287–311 (1995)
Marti, K.: Differentiation formulas for probability functions: the transformation method. Math. Program. 75, 201–220 (1996)
Tretiakov, G.: Stochastic quasi-gradient algorithms for maximization of the probability function. A new formula for the gradient of the probability function. In: Marti, K. (ed.) Stochastic Optimization Techniques, pp. 117–142. Springer, New York (2002)
Polak, E.: Optimization Algorithms and Consistent Approximations. Springer, New York (1997)
Deak, I.: Three digit accurate multiple normal probabilities. Numer. Math. 35, 369–380 (1980)
Ditlevsen, O., Oleson, R., Mohr, G.: Solution of a class of load combination problems by directional simulation. Struct. Saf. 4, 95–109 (1987)
Bjerager, P.: Probability integration by directional simulation. J. Eng. Mech. 114, 1288–1302 (1988)
Shapiro, A.: Asymptotic properties of statistical estimators in stochastic programming. Ann. Stat. 17, 841–858 (1989)
Rubinstein, R.Y., Shapiro, A.: Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method. Wiley, New York (1993)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York (1998)
Polak, E., Qi, L., Sun, D.: First-order algorithms for generalized semi-infinite min-max problems. Comput. Optim. Appl. 13, 137–161 (1999)
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, New York (1991)
van der Vaart, A., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer, New York (1996)
Matlab, Version 6.5, Mathworks Inc., Natick, MA (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Tseng.
This work was sponsored by the Research Associateship Program, National Research Council. The authors are grateful for the valuable insight from Professors Alexander Shapiro, Evarist Gine, and Jon A. Wellner. The authors also thank Professor Tito Homem-de-Mello for commenting on an early draft of this paper.
Rights and permissions
About this article
Cite this article
Royset, J.O., Polak, E. Extensions of Stochastic Optimization Results to Problems with System Failure Probability Functions. J Optim Theory Appl 133, 1–18 (2007). https://doi.org/10.1007/s10957-007-9178-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-007-9178-0