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

Caroline Geiersbach, René Henrion

TL;DR

This paper derives optimality conditions for optimization problems subject to random state constraints and compares them to a model based on robust constraints with respect to the (compact) support of the given distribution.

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.

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

TL;DR

This paper derives optimality conditions for optimization problems subject to random state constraints and compares them to a model based on robust constraints with respect to the (compact) support of the given distribution.

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.
Paper Structure (12 sections, 25 theorems, 169 equations)

This paper contains 12 sections, 25 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. A Model PDE
  4. Probabilistic constraints in optimal control
  5. General framework
  6. Probabilistic state constraints in the concrete PDE
  7. Explicit formulae for the subdifferential of the radial probability function
  8. Subdifferential of the probability function and optimality conditions in the concrete PDE setting
  9. Almost sure and robust constraints
  10. Almost sure state constraints
  11. Robust state constraints
  12. Connection to semi-infinite programming.

Key Result

Lemma 1.2

Suppose Assumption ass:PDE-standing holds. Then for any $u, f_0, \phi_i\in L^2(D)$ ($i=1, \dots, m$) and every $z \in \Xi$, there exists a unique solution $\hat{y}(\cdot,z) \in H_0^{1}(D)\cap \mathcal{C}(\bar{D})$ of eq:robust-PDE, where $\bar{D}$ denotes the closure of $D$. Moreover, there exists a

Theorems & Definitions (55)

  • Lemma 1.2
  • proof
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Theorem 2.2: Hantoute2019, Theorem 5, Corollary 2 and Proposition 6
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 45 more