Quantitative stability of optimal values and solution sets to stochastic programming problems is studied when the underlying probability distribution varies in some metric space of probability measures. We give conditions that imply that a stochastic program behaves stable with respect to a minimal information (m.i.) probability metric that is naturally associated with the data of the program. Canonical metrics bounding the m.i. metric are derived for specic models, namely for linear two-stage, mixed-integer two-stage and chance constrained models. The corresponding quantitative stability results as well as some consequences for asymptotic properties of empirical approximations extend earlier results in this direction. In particular, rates of convergence in probability are derived under metric entropy conditions. Finally, we study stability properties of stable investment portfolios having minimal risk with respect to the spectral measure and stability index of the underly...

Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions, it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case.

: Epi-convergence in distribution is a useful tool in establishing limiting distributions of "argmin" estimators; however, it is not always easy to find the epi-limit of a given sequence of objective functions. In this paper, we define the notion of stochastic equi-lower-semicontinuity of a sequence of random objective functions. It is shown that epi-convergence in distribution and finite dimensional convergence in distribution (to a given limit) of a sequence of random objective functions are equivalent under this condition. Key words and phrases: argmin estimators, convergence in distribution, epi-convergence, equi-semicontinuity AMS 1991 subject classifications: Primary 62F12, 60F05; Secondary 62E20, 60F17. Running head: Stochastic equi-semicontinuity 1 Introduction Many statistical estimators are defined as the minimizer (or maximizer) of some objective function; common examples include maximum likelihood estimation and M-estimation. Since any maximization problem can be re-exp...

An analysis of convex stochastic programs is provided if the underlying probability distribution is subjected to (small) perturbations. It is shown, in particular, that ε-approximate solution sets of convex stochastic programs behave Lipschitz continuous with respect to certain distances of probability distributions that are generated by the relevant integrands. It is shown that these results apply to linear two-stage stochastic programs with random recourse. Consequences are discussed on associating Fortet-Mourier metrics to two-stage models and on the asymptotic behavior of empirical estimates of such models, respectively.

informs doi 10.1287/moor.1060.0222

. To justify the use of sampling to solve stochastic programming problems one usually relies on a law of large numbers for random lsc (lower semicontinuous) functions when the samples come from independent, identical experiments. If the samples come from a stationary process, one can appeal to the ergodic theorem proved here. The proof relies on the `scalarization' of random lsc functions. 1 Introduction Stochastic programming models can be viewed as extensions of linear and nonlinear programming models to accommodate situations in which only information of a probabilistic nature is available about some of the parameters of the problem. The following formulation includes both the stochastic programming with recourse models and the stochastic programming with chance constraints models : min Eff 0 (¸¸ ¸; x)g (1) so that Eff i (¸¸ ¸; x)g 0; i = 1; : : : ; m; x 2 IR n where - ¸¸ ¸ is a random vector with support \Xi ae IR N , - P is a probability distribution function on IR N , - f ...

This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainties and nonsmoothness. The well known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the generalized gradients and metric regularity of the set-valued mappings which characterize the first order equilibrium conditions, it is shown that with probability approaching one exponentially fast with the increase of sample size, the statistical estimator converge to its true counterparts. Finally, the model is applied to an equilibrium problem in electricity market. Key words. Stochastic Nash equilibrium, exponential convergence, H-calmness, Clarke generalized gradients, metric regularity. 1

Abstract We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained problems. Numerical experiments are performed to correctly tune the parameters involved in the SAA. In addition, we present a method for constructing statistical lower bounds for the optimal value of the considered problem and discuss how one should tune the underlying parameters. We apply the SAA to two chance constrained problems. The first is a linear portfolio selection problem with returns following a multivariate lognormal distribution. The second is a joint chance constrained version of a simple blending problem.

We study sample approximations of chance constrained problems. In particular, we consider the sample average approximation (SAA) approach and discuss the convergence properties of the resulting problem. We discuss how one can use the SAA method to obtain good candidate solutions for chance constrained ()Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de