3 research outputs found
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 Capacity of Channels with Feedback
We introduce a general framework for treating channels with memory and
feedback. First, we generalize Massey's concept of directed information and use
it to characterize the feedback capacity of general channels. Second, we
present coding results for Markov channels. This requires determining
appropriate sufficient statistics at the encoder and decoder. Third, a dynamic
programming framework for computing the capacity of Markov channels is
presented. Fourth, it is shown that the average cost optimality equation (ACOE)
can be viewed as an implicit single-letter characterization of the capacity.
Fifth, scenarios with simple sufficient statistics are described