Optimality Conditions and Moreau--Yosida Regularization for Almost Sure State Constraints

TL;DR

This paper develops optimality conditions for a risk-averse convex stochastic control problem with state constraints, introducing a Moreau--Yosida regularization approach to handle conical constraints and ensuring consistency as regularization diminishes.

Contribution

It provides strong necessary and sufficient optimality conditions for stochastic problems with state constraints and introduces a novel regularization method with proven consistency.

Findings

Derived strong optimality conditions for constrained stochastic control.

Proposed a Moreau--Yosida regularization for conical constraints.

Showed the regularized solutions converge to the original problem's solutions.

Abstract

We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau--Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity.