Gradient-based optimisation of the conditional-value-at-risk using the multi-level Monte Carlo method

In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework o...

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Bibliographic Details
Published in:Journal of computational physics Vol. 495; p. 112523
Main Authors: Ganesh, Sundar, Nobile, Fabio
Format: Journal Article
Language:English
Published: Elsevier Inc 15.12.2023
Subjects:
CVaR
Gradient descent
Multilevel Monte Carlo methods
Optimisation under uncertainty
Uncertainty quantification
VaR
Uncertainty quantification
Gradient descent
CVaR
Multilevel Monte Carlo methods
VaR
Optimisation under uncertainty
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:In this work, we tackle the problem of minimising the Conditional-Value-at-Risk (CVaR) of output quantities of complex differential models with random input data, using gradient-based approaches in combination with the Multi-Level Monte Carlo (MLMC) method. In particular, we consider the framework of multi-level Monte Carlo for parametric expectations introduced in [24] and propose modifications of the MLMC estimator, error estimation procedure, and adaptive MLMC parameter selection to ensure the estimation of the CVaR and sensitivities for a given design with a prescribed accuracy. We then propose combining the MLMC framework with an alternating inexact minimisation-gradient descent algorithm, for which we prove Q-linear convergence in the optimisation iterations under the assumptions of strong convexity and Lipschitz continuity of the gradient of the objective function. We demonstrate the performance of our approach on two numerical examples of practical relevance, which evidence the same optimal asymptotic cost-tolerance behaviour as standard MLMC methods for fixed design computations of output expectations. •Conditional-Value-at-Risk (CvaR) sensitivities using Parametric Expectations (PE).•Estimating sensitivities using Multi-Level Monte Carlo (MLMC) for PE.•Alternating minimization gradient-descent algorithm using MLMC sensitivities.•Exponential convergence predicted and obtained in the optimization iterations.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2023.112523