Risk-adverse optimization by Conditional Value-at-Risk and stochastic approximation

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Engineering design is often faced with uncertainties, making it difficult to determine an optimal design. In an unconstrained context, this amounts to choose the desired trade-off between risk and performance. In this paper, an optimization problem with an adaptive risk level is stated using the Conditional Value-at-Risk (\(CVaR_\alpha\)). Under mild conditions on the objective function and taking advantage of the noise, \(CVaR_\alpha\) allows to smooth the problem. Then, a specific algorithm based on a stochastic approximation scheme is developed to solve the problem. This algorithm has two appealing properties. First, it does not use any estimation of quantile to compute the minimum of the \(CVaR_\alpha\) of the noised objective function. Second, it uses only two function evaluations per iteration regardless of the problem dimension. A proof of convergence to a minimum of \(CVaR_\alpha\) of the objective function is established. This proof is based on martingale theory and does not require any information about the differentiability or continuity of the function. Finally, test problems from the literature are combined in a benchmark set to compare our algorithm to a risk-neutral and a worst-case optimization algorithms. These tests prove the ability of the algorithm to be efficient in both cases, especially in large dimension.

, 25 pages

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