Mathematical Foundations of Distributionally Robust Multistage Optimization

TL;DR

This paper develops a mathematical framework for constructing conditional risk functionals in distributionally robust multistage optimization, highlighting their differences from nested risk measures and enabling robust decision-making under uncertainty.

Contribution

It introduces a method to construct conditional risk functionals that preserve key properties, advancing the theoretical foundation of distributionally robust multistage stochastic programming.

Findings

Conditional risk functionals differ from nested risk measures.

Initial measures and their decompositions can be used in multistage settings.

Framework supports robust optimization under distributional uncertainty.

Abstract

Distributionally robust optimization involves various probability measures in its problem formulation. They can be bundled to constitute a risk functional. For this equivalence, risk functionals constitute a fundamental building block in distributionally robust stochastic programming. Multistage programming requires conditional versions of risk functionals to re-assess future risk after partial realizations and after preceding decisions. This paper discusses a construction of the conditional counterpart of a risk functional by passing its genuine characteristics to its conditional counterparts. The conditional risk functionals turn out to be different from the nested analogues of the original (law invariant) risk measure. It is demonstrated that the initial measure and its nested decomposition can be used in a distributionally robust multistage setting.