Dual solutions in convex stochastic optimization

TL;DR

This paper explores duality and optimality in convex stochastic optimization, providing conditions for dual solutions and no duality gap, with implications across operations research, control, and finance.

Contribution

It introduces new sufficient and necessary conditions for duality and optimality in convex stochastic problems within locally convex spaces of random variables.

Findings

Established conditions for absence of duality gap.

Derived optimality conditions for specific problem classes.

Extended previous models in control, portfolio optimization, and programming.

Abstract

This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control and financial mathematics. Our analysis builds on the theory of Fr\'echet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models e.g.\ on stochastic optimal control, portfolio optimization and mathematical programming.