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.

翻译：本文针对具有随机输入数据的复杂微分模型输出量的条件风险价值（CVaR）最小化问题，采用基于梯度的方法与多层级蒙特卡洛（MLMC）方法相结合的策略。具体而言，我们考虑了参数期望的多层级蒙特卡洛框架，并对MLMC估计量、误差估计流程及自适应MLMC参数选择进行改进，以确保在给定设计条件下以预设精度估计CVaR及其灵敏度。随后，我们提出将MLMC框架与交替非精确最小化-梯度下降算法相结合，并在目标函数强凸性及梯度Lipschitz连续假设下，证明了优化迭代的指数收敛性。我们通过两个具有实际意义的数值算例验证了该方法性能，结果表明其在输出期望的固定设计计算中，与标准MLMC方法具有相同的最优渐近成本-容差特性。