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Theoretical Results

Finite Convergence in Stochastic Programming

A differential inclusion is designed for solving stochastic, finite horizon, convex programs. Under a sharpness condition we demonstrate that the resulting method yields finite convergence.
Sjur D. Flåm

Lattice Rules for Multiple Integration

Numerical methods for stochastic optimization often involve as basic steps the calculation of probabilities and of expected values in a multivariate setting. For this purpose, efficient routines for multidimensional numerical integration are needed. We discuss here a class of quasi-Monte Carlo methods which have been developed recently and which yield powerful tools for multiple integration. For a general background on quasi-Monte Carlo methods we refer to the survey articles of Niederreiter [12], [15] and to the forthcoming book [18].
Harald Niederreiter

Limit theorems on the Robbins-Monro process for different variance behaviors of the stochastic gradient

For finding a solution x* ∈ ℝr of the system of nonlinear equations G(x) = 0 in case that only estimations \( \hat{G}\left( x \right) - G\left( x \right) + Z\left( x \right) \) at each x ∈ ℝr are available, we consider the stochastic approximation procedure
$$ {{X}_{{n + 1}}}: - {{X}_{n}} - {{r}_{n}}{{\hat{G}}_{n}}\left( {{{X}_{n}}} \right), where {{\hat{G}}_{n}}({{X}_{n}}): - G\left( {{{X}_{n}}} \right) + {{Z}_{n}}, n - 1,2, \ldots , $$
with the estimation error Zn: -Z(Xn). In this paper the limiting distribution of the random sequence \( {{\left( {{{\tau }_{n}}\left( {{{X}_{n}} - {{x}^{*}}} \right)} \right)}_{n}} \) is considered for different sequences (rn)n of deterministic step sizes, where (τ n)n is a sequence of positive numbers such that
$$ E{{\left\| {{{X}_{n}} - {{x}^{*}}} \right\|}^{2}} - 0\left( {{{1} \left/ {{\tau _{n}^{2}}} \right.}} \right). $$
Approximations for the limit covariance matrices of (τ n(Xn-x*))n are given for the case where the estimation error Zn in (1) has different variances for indices n from different subsets ℕ(k) of ℕ. Particular attention is paid to the semi-stochastic case where Z -0 for all n contained in an infinite subset N(1) of ℕ.
Ernst Plöchinger

Continuity and Stability in Two-Stage Stochastic Integer Programming

For two-stage stochastic programs where the optimization problem in the second stage is a mixed-integer linear program continuity of the expectation of second-stage costs jointly in the first-stage strategy and the integrating probability measure is derived. Then, regarding the two-stage stochastic program as a parametric program with the underlying probability measure as parameter, continuity of the locally optimal value and upper semicontinuity of the corresponding set of local solutions are established.
Rüdiger Schultz

Applications and Methods

Three Approaches for Solving the Stochastic Multiobjective Programming Problem

In this paper, we consider the multiobjective optimization problem in which each objective function is disturbed by noise. Three approaches using learning automata, random optimization method, and stochastic approximation method are proposed to solve this problem. It is shown that these three approaches are able to find appropriate solutions of this problem. Several computer simulation results also confirm our theoretical study.
Norio Baba, Akira Morimoto

A Stochastic Programming Model for Optimal Power Dispatch: Stability and Numerical Treatment

The economic dispatch of electric power with uncertain demand is modeled as stochastic program with simple recourse. We analyze quantitative stability properties of the power dispatch model with respect to the Li-distance of the marginal distribution functions of the demand vector. These stability results are used to derive asymptotic properties of the model if the (true) marginal distributions are replaced by smooth nonparametric estimates based on the kernel method. Finally, we discuss how smooth estimates can be used efficiently for the numerical treatment of simple recourse models by using nonlinear programming techniques. Numerical results are reported for Dantzig’s Aircraft Allocation Problem.
Nicole Gröwe, Werner Römisch

Computational Techniques for Probabilistic Constrained Optimization Problems

The subject of this paper is to give a survey on algorithms designed for the solution of probabilistic constrained problems with joint probabilistic constraints. A brief summary of the most important convexity results is also included as convexity properties play a central role in developing solution methods for this problem class. For overviews on algorithms aiming the solution of probabilistic constrained problems see also [25], [26] and [50].
János Mayer

Stochastic Optimization in Acid Rain Management with Variable Meteorology

A multi-year linear programming model is developed to examine the impacts of variable meteorology on optimal controls for acid rain abatement in eastern North America. The features impacting the transfer coefficients relating emissions from sources to depositions at receptors, are discussed.
Edward A. McBean

Collapse Load Analysis and Optimal Design by Stochasic Programming with Uncertainties of Loads

The effects of uncertainties of external loads are examined in case of plastic collapse load analysis and optimal design. Stochastic programming model can be written on base of kinematical theorems. In case of collapse load analysis Van de Panne and Popp’s stochastic model of optimal design Kataoka’s model are used. Applications are presented by analysis of large panel structures. The results are compared with the deterministic solutions.
Anna Vásárhelyi


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