On Nonnormal Asymptotic Behavior of Optimal Solutions of Stochastic Programming Problems: The Parametric Case

Under incomplete information about the parameters of the true distribution of the random coefficients, the optimal solutions to stochastic programs can be only approximated. This paper extends the previous results of the author to the case when strict complementarity conditions need not be assumed

Scenario-Based Stochastic Programs: Strategies for Deleting Scenarios

The proposed strategies for deleting scenarios are based on postoptimality analysis of the optimal value function with respect to probabilities of the included scenarios. These strategies can be used to reduce the size of the large scenario based problems or of the problems constructed in the course of specific numerical procedures, such as stochastic decomposition or scenario aggregation. A convex nonsmooth optimization problem is replaced by a sequence of line search problems along recursively updated rays. Convergence of the method is proved and applications indicated

Asymptotic Properties of Restricted L1-Estimates of Regression

Asymptotic properties of L_{1}-estimates in linear regression have been studied by many authors, see e.g. Bassett and Koenker (1978), Bloomfield and Steiger (1983). It is the lack of smoothness which does not allow to we the known results on asymptotic behavior of M-estimates directly (Huber (1967)). The additional lack of a convexity in the nonlinear regression case increases the complexity of the problem even under assumption that the true parameter values belong to the interior of the given parameter set; for a consistency result in this case see e.g. Oberhofer (1982). We shall use the technique developed in Dupacova and Wets (1986, 1987) to get asymptotic properties of the L_{1}-estimates of regression coefficients which are assumed to belong to an a priori given closed convex set given, e.g., by constraints of general equality and inequality form. The method uses, i.a., tools of nondifferentiable calculus and epi-convergence and it can be applied to other classes of L_{1}-estimates as well

Bounds for Stochastic Programs in Particular for Recourse Problems

In this paper, we shall discuss the bounds for the optimal value of recourse problems from the point of view of assumptions and of possible generalizations. We shall concentrate on bounds based on the first order moment conditions and to those based on sample information. We shall indicate when it is possible to remove the convexity assumptions, when there is a hope for extensions to multistage problems and we shall point out reflections of bounds and stability results

Asymptotic Behavior of Statistical Estimators and of Optimal Solutions of Stochastic Optimization Problems, II

This paper supplements the results of a new statistical approach to the problem of incomplete information in stochastic programming. The tools of nondifferentiable optimization used here, help to prove the consistency and asymptotic normality of (approximate) optimal solutions without unnatural smoothness assumptions. This allows the theory to take into account the presence of constraints

Stochastic Programming with Incomplete Information

The possibility of successful applications of stochastic programming decision models has been limited by the assumed complete knowledge of the distribution F of the random parameters as well as by the limited scope of the existing numerical procedures. We shall introduce selected methods which can be used to deal with the incomplete knowledge of the distribution F, to study robustness of the optimal solution and the optimal value of the objective function relative to small changes of the underlying distribution and to get error bounds in approximation schemes. The research was mostly carried out at the Department of Statistics, Charles University, Prague and it was stimulated by a close collaboration of the author with the ADO project of SDS. The present version of the paper was written at IIASA Laxenburg

Stochastic Programming in Water Resources System Planning: A Case Study and a Comparison of Solution Techniques

To analyze the influence of increasing needs upon a given water resources system in Eastern Slovakia and to get a decision on the system development and extension, several stochastic programming models can be used. The two selected models are based on individual probabilistic constraints for the minimum storage and for the freeboard volume supplemented by one joint probabilistic constraint on releases or by a nonseparable penalty term in the objective function. Suitable numerical techniques for their solution are applied to alternative design parameter values. As a result, the paper gives an answer to the case study which is based on multi modeling within the framework of stochastic programming and, at the same time, it gives a comparison of various solution techniques partly included in the SDS/ADO collection of stochastic programming codes

Scenario trees and policy selection for multistage stochastic programming using machine learning

We propose a hybrid algorithmic strategy for complex stochastic optimization problems, which combines the use of scenario trees from multistage stochastic programming with machine learning techniques for learning a policy in the form of a statistical model, in the context of constrained vector-valued decisions. Such a policy allows one to run out-of-sample simulations over a large number of independent scenarios, and obtain a signal on the quality of the approximation scheme used to solve the multistage stochastic program. We propose to apply this fast simulation technique to choose the best tree from a set of scenario trees. A solution scheme is introduced, where several scenario trees with random branching structure are solved in parallel, and where the tree from which the best policy for the true problem could be learned is ultimately retained. Numerical tests show that excellent trade-offs can be achieved between run times and solution quality