Adaptive Design in Discrete Stochastic Optimization

We present adaptive assignment rules for the design of the necessary simulations when solving discrete stochastic optimization problems. The rules are constructed in such a way, that the expected size of confidence sets for the optimizer is as small as possible

Risk-Reshaping Contracts and Stochastic Optimization

Insurance contracts and lotteries are just the opposite sides of the same coin: These are contracts, which allow to reshape an uncertain financial position by exchanging risks between two contractors. In this paper, we discuss some basic problems of operations research which are connected with such kind of contracts

On the Determination of the Step Size in Stochastic Quasigradient Methods

For algorithms of the Robbins-Monro type, the best choice (from the asymptotic point of view) for the step-size constants a_n is known to be a/n. From the practical point of view, however, adaptive step-size rules seem more likely to produce quick convergence. In this paper a new adaptive rule for controlling the stepsize is presented and its behavior is studied

Metric Entropy and Nonasymptotic Confidence Bands in Stochastic Programming

Talagrand has demonstrated in his key paper, how the metric entropy of a class of functions relates to uniform bounds for the law of large numbers. This paper shows how to calculate the metric entropy of classes of functions which appear in stochastic optimization problems. As a consequence of these results, we derive via variational inequalities confidence bands for the solutions, which are valid for any sample size. In particular, the linear recourse problem is considered

Asymptotic Dominance and Confidence for Solutions of Stochastic Programs

For closed-set valued random processes we introduce a stochastic order relation (dominance) and show that the argmins of a sequence of random processes, which are epi-convergent in distribution satisfy this order relation in an asymptotic sense. The result may be used for the construction of confidence regions for the argmin

Non-standard Limit Theorems for Stochastic Approximation Procedures and Their Applications for Urn Schemes

A limit theorem for the Robbins-Monro stochastic approximation procedure is proved in the case of a non-smooth regression function. Using this result a conditional limit theorem is given for the case when the regression function has several stable roots. The first result shows that the rate of convergence for the stochastic approximation-type procedures (including Monte-Carlo optimization algorithms and adaptive processes of growth being modelled by the generalized urn scheme) decreases as the smoothness increases. The second result demonstrates that in the case of several stable roots, there is no convergence rate for the procedure as whole, but for each of stable roots there exists its specific rate of convergence. The latter allows to derive several conceptual results for applied problems in biology, physical chemistry and economics which can be described by the generalized urn scheme

A Branch and Bound Method for Stochastic Global Optimization

A stochastic version of the branch and bound method is proposed for solving stochastic global optimization problems. The method, instead of deterministic bounds, uses stochastic upper and lower estimates of the optimal value of subproblems, to guide the partitioning process. Almost sure convergence of the method is proved and random accuracy estimates derived. Methods for constructing random bounds for stochastic global optimization problems are discussed. The theoretical considerations are illustrated with an example of a facility location problem

Estimating the Uncertainty in Population Projections by Resampling Methods

This paper proposes a new approach to introducing quantitatively measured uncertainty into population projections. It is to a lesser degree based on past time-series than other approaches, since it uses random walk models for migration, mortality and fertility, for which upper and lower bounds are defined. No parametric distribution is fitted to the observations, but the random walk is resampled from the past data. By putting bounds on the level that fertility can reach in the future, further substantive information is introduced that transcends the information derived from the observed time series. By sampling 10.000 path of the random walks in fertility, mortality and migration, the distributions of population size and structure up to 2050 for Austria, Mauritius and USA are estimated

Coherent risk measures and convex combinations of the conditional value at risk

The conditional-value-at-risk (C V@R) has been widely used as a risk measure. It is well known, that C V@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations of C V@R’s. In this note we show that this conjecture is wrong