Scenario-decomposition Solution Framework for Nonseparable Stochastic Control Problems

TL;DR

This paper introduces a novel scenario-decomposition framework for nonseparable, non-monotonic stochastic control problems, extending traditional methods and enabling solutions for complex, real-world applications like portfolio optimization.

Contribution

It develops a new scenario-decomposition approach based on the progressive hedging algorithm, addressing challenges in nonseparable and non-monotonic stochastic control problems.

Findings

Framework successfully handles nonseparable problems

Applicable to online quadratic programming

Effective in dynamic portfolio selection

Abstract

When stochastic control problems do not possess separability and/or monotonicity, the dynamic programming pioneered by Bellman in 1950s fails to work as a time-decomposition solution method. Such cases have posted a great challenge to the control society in both theoretical foundation and solution methodologies for many years. With the help of the progressive hedging algorithm proposed by Rockafellar and Wets in 1991, we develop a novel scenario-decomposition solution framework for stochastic control problems which could be nonseparable and/or non-monotonic, thus extending the reach of stochastic optimal control. We discuss then some of its promising applications, including online quadratic programming problems and dynamic portfolio selection problems with smoothing properties.