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Risk-averse optimal control of random elliptic variational inequalities

Amal Alphonse, Caroline Geiersbach, Michael Hintermüller, Thomas M. Surowiec

TL;DR

The paper addresses risk-averse optimal control of a random elliptic variational inequality with obstacle constraints. It introduces a penalised, smoothed surrogate for the VI and derives KKT-type stationarity conditions for the penalised problem, followed by a limit analysis to obtain weak and C-stationarity notions in a stochastic context. A path-following stochastic gradient method with variance reduction is proposed and demonstrated on a modified benchmark, illustrating feasible computation despite the lack of regularity in the stochastic setting. The results reveal fundamental challenges in transferring deterministic stationarity to SMPECs, particularly due to limited regularity with respect to uncertain parameters, and provide a framework for future refinement of theory and numerics in risk-averse PDE-constrained MPECs.

Abstract

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem.

Risk-averse optimal control of random elliptic variational inequalities

TL;DR

The paper addresses risk-averse optimal control of a random elliptic variational inequality with obstacle constraints. It introduces a penalised, smoothed surrogate for the VI and derives KKT-type stationarity conditions for the penalised problem, followed by a limit analysis to obtain weak and C-stationarity notions in a stochastic context. A path-following stochastic gradient method with variance reduction is proposed and demonstrated on a modified benchmark, illustrating feasible computation despite the lack of regularity in the stochastic setting. The results reveal fundamental challenges in transferring deterministic stationarity to SMPECs, particularly due to limited regularity with respect to uncertain parameters, and provide a framework for future refinement of theory and numerics in risk-averse PDE-constrained MPECs.

Abstract

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the algorithm on a modified benchmark problem.
Paper Structure (19 sections, 26 theorems, 132 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 26 theorems, 132 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Notation and background material
  3. Standing assumptions
  4. Example
  5. Analysis of the optimisation problem
  6. Analysis of the VI
  7. Existence of an optimal control
  8. A regularised control problem
  9. A penalisation of the obstacle problem
  10. Stationarity for the regularised control problem
  11. Stationarity conditions
  12. Consistency of the approximation
  13. Passage to the limit
  14. Numerical example
  15. Problem formulation
  16. ...and 4 more sections

Key Result

Lemma 2.1

For almost every $\omega \in \Omega$, there exists a unique solution to eq:VIProblemFormal satisfying the estimate where the constant $C>0$ depends only on $C_b, C_a$ and $C_c$.

Figures (3)

  • Figure 1: Example realisations of random field zero-mean $b$ (left) and lognormal random fields $b$ (middle, right).
  • Figure 2: Mean-zero noise: Control $u_\tau^*$ (left), state $\bar{y}_\tau^*$ (middle), multiplier $\bar{\zeta}_\tau^*$ (right) for $\beta=0.0$.
  • Figure 6: Deterministic solution: Control $u_\tau^*$ (left), state $\bar{y}_\tau^*$ (middle), multiplier $\bar{\zeta}_\tau^*$ (right).

Theorems & Definitions (64)

  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.7
  • proof
  • ...and 54 more