Optimality conditions in control problems with random state constraints in probabilistic or almost-sure form

TL;DR

This paper derives optimality conditions for control problems with random state constraints, addressing probabilistic and almost-sure cases, and applies them to PDEs with random inputs, highlighting differences in approaches.

Contribution

It introduces distinct optimality conditions for probabilistic and almost-sure constraints, utilizing spherical-radial decomposition and robust constraint modeling.

Findings

Explicit optimality conditions for probabilistic constraints using spherical integrals.

Comparison of optimality conditions between probabilistic and almost-sure cases.

Application to PDEs with random inputs demonstrating the methods.

Abstract

In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. While the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation (PDE) with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them to a model based on robust constraints with respect to the (compact) support of the given distribution.