  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 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 exponential 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.
