Time Blocks Decomposition of Multistage Stochastic Optimization Problems

Multistage stochastic optimization problems are, by essence, complex because their solutions are indexed both by stages (time) and by uncertainties (scenarios). Their large scale nature makes decomposition methods appealing.The most common approaches are time decomposition --- and state-based resolution methods, like stochastic dynamic programming, in stochastic optimal control --- and scenario decomposition --- like progressive hedging in stochastic programming. We present a method to decompose multistage stochastic optimization problems by time blocks, which covers both stochastic programming and stochastic dynamic programming. Once established a dynamic programming equation with value functions defined on the history space (a history is a sequence of uncertainties and controls), we provide conditions to reduce the history using a compressed "state" variable. This reduction is done by time blocks, that is, at stages that are not necessarily all the original unit stages, and we prove areduced dynamic programming equation. Then, we apply the reduction method by time blocks to \emph{two time-scales} stochastic optimization problems and to a novel class of so-called \emph{decision-hazard-decision} problems, arising in many practical situations, like in stock management. The \emph{time blocks decomposition} scheme is as follows: we use dynamic programming at slow time scale where the slow time scale noises are supposed to be stagewise independent, and we produce slow time scale Bellman functions; then, we use stochastic programming at short time scale, within two consecutive slow time steps, with the final short time scale cost given by the slow time scale Bellman functions, and without assuming stagewise independence for the short time scale noises

The method of endogenous gridpoints for solving dynamic stochastic optimization problems

This paper introduces a method for solving numerical dynamic stochastic optimization problems that avoids rootfinding operations. The idea is applicable to many microeconomic and macroeconomic problems, including life cycle, buffer-stock, and stochastic growth problems. Software is provided. Klassifikation: C6, D9, E2 . July 28, 2005

Stochastic MPC Design for a Two-Component Granulation Process

We address the issue of control of a stochastic two-component granulation process in pharmaceutical applications through using Stochastic Model Predictive Control (SMPC) and model reduction to obtain the desired particle distribution. We first use the method of moments to reduce the governing integro-differential equation down to a nonlinear ordinary differential equation (ODE). This reduced-order model is employed in the SMPC formulation. The probabilistic constraints in this formulation keep the variance of particles' drug concentration in an admissible range. To solve the resulting stochastic optimization problem, we first employ polynomial chaos expansion to obtain the Probability Distribution Function (PDF) of the future state variables using the uncertain variables' distributions. As a result, the original stochastic optimization problem for a particulate system is converted to a deterministic dynamic optimization. This approximation lessens the computation burden of the controller and makes its real time application possible.Comment: American control Conference, May, 201

The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems

This paper introduces a method for solving numerical dynamic stochastic optimization problems that avoids rootfinding operations. The idea is applicable to many microeconomic and macroeconomic problems, including life cycle, buffer-stock, and stochastic growth problems. Software is provided.