Probabilistic Constrained Optimization pp 282-307 | Cite as

# Statistical Inference of Stochastic Optimization Problems

## Abstract

We discuss in this paper statistical inference of Monte Carlo simulation based approximations of stochastic optimization problems, where the “true” objective function, and probably some of the constraints, are estimated, typically by averaging a random sample. The classical maximum likelihood estimation can be considered in that framework. Recently statistical analysis of such methods has been motivated by a development of simulation based optimization techniques. We investigate asymptotic properties of the optimal value and an optimal solution of the corresponding Monte Carlo simulation approximations by employing the so-called Delta method, and discuss some examples.

## Keywords

Stochastic Programming Random Element Order Expansion Unique Optimal Solution True Problem## Preview

Unable to display preview. Download preview PDF.

## References

- [1]A. Araujo and E. Giné,
*The Central Limit Theorem for Real and Banach Valued Random Variables*. Wiley, New York, 1980.MATHGoogle Scholar - [2]E. Beale, “On minimizing a convex function subject to linear inequalities”,
*J. Roy. Statist Soc, Ser. B*, 17 (1955), 173–184.MathSciNetMATHGoogle Scholar - [3]P. Billingsley,
*Convergence of Probability Measures*. Wiley, New York, 1968.MATHGoogle Scholar - [4]J.R. Birge and F. Louveaux,
*Introduction to Stochastic Programming*. Springer, New York, 1997.MATHGoogle Scholar - [5]J.F. Bonnans and A. Shapiro,
*Perturbation Analysis of Optimization Problems*. Springer-Verlag, New York, 2000.MATHGoogle Scholar - [6]J.A. Bucklew,
*Large Deviation Techniques in Decision, Simulation, and Estimation*. Wiley, New York, 1990.Google Scholar - [7]H. Chernoff, “On the distribution of the likelihood ratio ”,
*Ann. Math. Statist.*, 25 (1954), 573–578.MathSciNetMATHCrossRefGoogle Scholar - [8]G. Dantzig, “Linear programming under uncertainty ”,
*Management Sci.*,1 (1955), 197–206.MathSciNetMATHCrossRefGoogle Scholar - [9]D. Dentcheva and W. Römisch, “Differential stability of two-stage stochastic programs ”,
*SI AM J. Optimization*, to appear.Google Scholar - [10]J. Dupacovä and R.J.B. Wets, “Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems, ”
*The Annals of Statistics*16 (1988) 1517–1549.MathSciNetCrossRefGoogle Scholar - [11]R.D. Gill, “Non-and-semiparametric maximum likelihood estimators and the von Mises method (Part I)”,
*Scandinavian Journal of Statistics*, 16 (1989), 97–124.MATHGoogle Scholar - [12]R. Griibel, “The length of the short ”,
*Ann. Statist*, 16 (1988), 619–628.MathSciNetCrossRefGoogle Scholar - [13]RJ. Huber, “Robust estimation of a location parameter ”,
*Ann. Math. Statist.*, 35 (1964), 73–101.MathSciNetMATHCrossRefGoogle Scholar - [14]P. J. Huber, “The behavior of maximum likelihood estimates under nonstandard conditions”,
*Proc. Fifth Berkeley Symp.Math. Statist. Probab.*, 1 (1967), 221–233, Univ. California Press.Google Scholar - [15]P.J. Huber,
*Robust Statistics*. Wiley, New York, 1981.MATHCrossRefGoogle Scholar - [16]P. Kall and S.W. Wallace,
*Stochastic Programming*. Wiley, Chichester, 1994.MATHGoogle Scholar - [17]A.J. King, “Asymptotic behavior of solutions in stochastic optimization: Non-smooth analysis and the derivation of non-normal limit distributions”, Ph.D. dissertation, Dept. Applied Mathematics, Univ. Washington, 1986.Google Scholar
- [18]A.J. King, “Generalized delta theorems for multivalued mappings and measurable selections ”,
*Mathematics of Operations Research*, 14 (1989), 720–736.MathSciNetMATHCrossRefGoogle Scholar - [19]A.J. King and R.J.-B. Wets, “Epi-consistency of convex stochastic programs, ”
*Stochastics*, 34 (1991), 83–92.MathSciNetMATHGoogle Scholar - [20]A.J. King and R.T. Rockafellar, “Asymptotic theory for solutions in statistical estimation and stochastic programming”,
*Mathematics of Operations Research*, 18 (1993), 148–162.MathSciNetMATHCrossRefGoogle Scholar - [21]O.L. Mangasarian and S. Fromovitz, “The Fritz John necessary optimality conditions in the presence of equality and inequality constraints”,
*Journal of Mathematical Analysis and Applications, 7*(1967), pp. 37–47.MathSciNetCrossRefGoogle Scholar - [22]E.L. Plambeck, B.R. Fu, S.M. Robinson and R. Suri, “Sample-Path Optimization of Convex Stochastic Performance Functions ”,
*Mathematical Programming*, vol. 75 (1996), no. 2, 137–176.MathSciNetMATHCrossRefGoogle Scholar - [23]D. Pollard,
*Convergence of Stochastic Processes*. Springer-Verlag, New York, 1984.MATHCrossRefGoogle Scholar - [24]C.R. Rao,
*Linear Statistical Inference and Its Applications*. Wiley, New York, 1973.MATHCrossRefGoogle Scholar - [25]S.M. Robinson, “Analysis of sample-path optimization, ”
*Math. Oper. Res.*, 21 (1996), 513–528.MathSciNetMATHCrossRefGoogle Scholar - [26]R.T. Rockafellar,
*Convex Analysis*. Princeton University Press, 1970.MATHGoogle Scholar - [27]R.Y. Rubinstein and A. Shapiro,
*Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method*. Wiley, New York, NY, 1993.MATHGoogle Scholar - [28]A. Shapiro, “Asymptotic Properties of Statistical Estimators in Stochastic Programming, ”
*Annals of Statistics*, 17 (1989), 841–858.MathSciNetMATHCrossRefGoogle Scholar - [29]A. Shapiro, “On concepts of directional differentiability, ”
*Journal Optim. Theory and Appl.*, 66 (1990), 477–487.MATHCrossRefGoogle Scholar - [30]A. Shapiro, “Asymptotic analysis of stochastic programs, ”
*Annals of Operations Research*, 30 (1991), 169–186.MathSciNetMATHCrossRefGoogle Scholar - [31]A. Shapiro, “Asymptotic Behavior of Optimal Solutions in Stochastic Programming,”
*Mathematics of Operations Research*, 18 (1993), 829–845.MathSciNetMATHCrossRefGoogle Scholar - [32]A. Shapiro and Y. Wardi, “Nondifferentiability of the steady-state function in Discrete Event Dynamic Systems ”,
*IEEE transactions on Automatic Control*, 39 (1994), 1707–1711.MathSciNetMATHCrossRefGoogle Scholar - [33]A. Shapiro and T. Homem-de-Mello, “A simulation-based approach to two-stage stochastic programming with recourse ”,
*Mathematical Programming*, 81 (1998), 301–325.MathSciNetMATHGoogle Scholar - [34]A. Shapiro and T. Homem-de-Mello, “On rate of convergence of Monte Carlo approximations of stochastic programs ”,
*SIAM J. Optimization*, to appear.Google Scholar - [35]J. Wang, “Distribution sensitivity analysis for stochastic programs with complete recourse ”,
*Mathematical Programming*, 31 (1985), 286–297.MathSciNetMATHCrossRefGoogle Scholar - [36]R.W. Wolff,
*Stochastic Modeling and the Theory of Queues*. Prentice Hall, En-glewood Cliffs, NJ, 1989.MATHGoogle Scholar