Regularized Decomposition of Stochastic Programs: Algorithmic Techniques and Numerical Results

A finitely convergent non-simplex method for large scale structured linear programming problems arising in stochastic programming is presented. The method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. Algorithmic techniques taking advantage of properties of stochastic programs are described and numerical results for large real world problems reported

A Partial Regularization Method for Saddle Point Seeking

This article generalizes the Nash equilibrium approach to linear programming to the saddle point problem. The problem is shown to be equivalent to a non-zero sum game in which objectives of the players are obtained by partial regularization of the original function. Based on that, a solution method is developed in which the players improve their decisions while anticipating the steps of their opponents. Strong convergence of the method is proved and application to convex optimization is discussed

Computing Normalized Equilibria in Convex-Concave Games

Abstract. This paper considers a fairly large class of noncooperative games in which strategies are jointly constrained. When what is called the Ky Fan or Nikaidô-Isoda function is convex-concave, selected Nash equilibria correspond to diagonal saddle points of that function. This feature is exploited to design computational algorithms for finding such equilibria. To comply with some freedom of individual choice the algorithms developed here are fairly decentralized. However, since coupling constraints must be enforced, repeated coordination is needed while underway towards equilibrium. Particular instances include zero-sum, two-person games - or minimax problems - that are convex-concave and involve convex coupling constraints.Noncooperative games; Nash equilibrium; joint constraints; quasivariational inequalities; exact penalty; subgradient projection; proximal point algorithm; partial regularization; saddle points; Ky Fan or Nikaidô-Isoda functions.

Parallel Decomposition of Multistage Stochastic Programming Problems

A new decomposition method for multistage stochastic linear programming problems is proposed by the author. The method combines the ideas of the regularized decomposition method for two-stage programs and dynamic programming. With each node of the decision tree of the multistage stochastic problem a certain regularized subproblem is associated which generates decisions for its successors and some backward information for its predecessor. The subproblems are solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. This method is especially convenient for implementation on a parallel computer

On the Regularized Decomposition Method for Two Stage Stochastic Linear Problems

A new approach to the regularized decomposition (RD) algorithm for two stage stochastic problems is presented. The RD method combines the ideas of the Dantzig-Wolfe decomposition principle and modern nonsmooth optimization methods. A new subproblem solution method using the primal simplex algorithm for linear programming is proposed and then tested on a number of large scale problems. The new approach makes it possible to use a more general problem formulation and thus allows considerably more freedom when creating the model. The computational results are highly encouraging

Noncooperative Convex Games: Computing Equilibrium By Partial Regularization

A class of non-cooperative constrained games is analyzed for which the Ky Fan function is convex-concave. Nash equilibria of such games correspond to diagonal saddle points of the said function. This feature is exploited in designing computational algorithms for finding such equilibria

Cost-effective Sulphur Emission Reduction under Uncertainty

The problem of reducing SO2 emissions in Europe is considered. The costs of reduction are assumed to be uncertain and are modeled by a set of possible scenarios. A mean-variance model of the problem is formulated and a specialized computational procedure developed. The approach is applied to the transboundary air pollution model with real-world data

Constraint Aggregation in Infinite-Dimensional Spaces and Applications [Updated January 1998]

An aggregation technique for constraints with values in Hilbert spaces is suggested. The technique allows to replace the original optimization problem by a sequence of subproblems having scalar or finite-dimensional constraints. Application to optimal control, games and stochastic programming are discussed in detail

Managing Water Quality under Uncertainty: Application of a New Stochastic Branch and Bound Method

The problem of water quality management under uncertain emission levels, reaction rates and pollutant transport is considered. Various performance measures: reliability, resiliency and vulnerability are taken into account. A general methodology for finding a cost-effective water quality management program is developed. The approach employs a new idea of the stochastic branch and bound method, which combines random estimates of the performance for subsets of decisions with iterative refinement of the most promising subsets