Computing Bounds for the Solution of the Stochastic Optimization Problem with Incomplete Information on Distribution of Random Parameters

The paper deals with the solution of a stochastic optimization problem under incomplete information. It is assumed that the distribution of probabilistic parameters is unknown and the only available information comes with observations. In addition the set to which the probabilistic parameters belong is also known. Numerical techniques are proposed which allow to compute upper and lower bounds for the solution of the stochastic optimization problem under these assumptions. These bounds are updated successively after the arrival of new observations

Numerical Techniques for Finding Estimates which Minimize the Upper Bound of the Absolute Deviation

The paper deals with the numerical techniques for finding the special type of parameter estimates based on the minimization of L_{1}-norm of error. More specifically, these estimates are derived by minimization of the upper bound of the error, which is evaluated similarly to the upper bounds on the solution of stochastic optimization problem in WP-86-72. The research reported in this paper was performed in the Adaptation and Optimization Project of the System and Decision Sciences Program

Optimization of Functionals Which Depend on Distribution Functions: 1. Nonlinear Functional and Linear Constraints

The main purpose of this paper is to discuss numerical optimization procedures for problems in which both the objective function and the constraints depend on distribution functions. The objective function is assumed to be nonlinear and to have directional derivatives, while the constraints are taken as linear. The proposed algorithm involves linearization of the objective function at the current point and solution of an auxiliary linear subproblem. This last problem is solved using duality relations and cutting-plane techniques

Combining Generalized Programming and Sampling Techniques for Stochastic Programs with Recourse

This paper deals with an application of generalized linear programming techniques for stochastic programming problems, particularly to stochastic programming problems with recourse. The major points which needed a clarification here were the possibility to use the estimates of the objective function instead of the exact values and to use the approximate solutions of the dual subproblem instead of the exact ones. In this paper conditions are presented which allow to use estimates and approximate solutions and still maintain convergence. The paper is a part of the effort on the development of stochastic optimization techniques at the Adaptation and Optimization Project of the System and Decision Sciences Program

A Stochastic Algorithm for Minimax Problems

This paper deals with minimax problems in which the "inner" problem of maximization is not concave. A procedure based on the approximation of the inner problem by a stochastic set of elements which can contain only two elements at each iteration is shown to converge with probability 1

Duality Relations and Numerical Methods for Optimization Problems on the Space of Probability Measures with Constraints on Probability Densities

In this paper, the authors look at some quite general optimization problems on the space of probabilistic measures. These problems originated in mathematical statistics but have applications in several other areas of mathematical analysis. The authors extend previous work by considering a more general form of the constraints, and develop numerical methods (based on stochastic quasigradient techniques) and some duality relations for problems of this type. This paper is a contribution to research on stochastic optimization currently underway within the Adaptation and Optimization Project

Simultaneous Nonstationary Optimization, Estimation and Approximation Procedures

The main aim of this paper is to investigate those algorithmic procedures which solve optimization problems whilst either estimating the unknown parameters of these problems or approximating them by more simple problems. The problem of nonstationary optimization with time-varying functions and a set of optimal solutions (set of equilibria) is considered. The proposed solution technique is based on the application of nonmonotonic optimization procedures. We derive the convergence of such procedures by studying the Hausdorf distance between a current approximate solution and the set of E-optimal solutions. The Lipschitz continuity of the Hausdorf distance between sets of E-optimal solutions upon the parameters of the problem is also discussed

Modeling of Competition and Collaboration Networks under Uncertainty: Stochastic Programs with Resource and Bilevel

We analyze stochastic programming problems with recourse characterized by a bilevel structure. Part of the uncertainty in such problems is due to actions of other actors such that the considered decision maker needs to develop a model to estimate their response to his decisions. Often, the resulting model exhibits connecting constraints in the leaders (upper-level) subproblem. It is shown that this problem can be formulated as a new class of stochastic programming problems with equilibrium constraints (SMPEC). Sufficient optimality conditions are stated. A solution algorithm utilizing a stochastic quasi-gradient method is proposed, and its applicability extensively explained by practical numerical examples

Stochastic Quasigradient Methods and their Implications

A number of stochastic quasigradient methods are discussed from the point of view of implementation. The discussion revolves around the interactive package of stochastic optimization routines (STO) recently developed by the Adaptation and Optimization group at IIASA. (This package is based on the stochastic and nondifferentiable optimization package (NDO) developed at the V. Glushkov Institute of Cybernetics in Kiev.) The IIASA implementation is described and its use illustrated by application to three problems which have arisen in various IIASA projects