Conditional value at risk (CVaR) has been widely studied as a risk measure. In this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with CVaR appearing in the objective and also the constraints. We start by considering a problem in which CVaR is minimized and investigate the way in which it approximates the minimax robust optimization problem as the confidence level is driven to one. We make use of a consistent tail condition which ensures that the CVaR of a random function will converge uniformly to its supremum as the confidence level increases, and establish an error bound for the CVaR optimal solution under second order growth conditions. The results are extended to a minimization problem with a constraint on the CVaR value which in the limit as the confidence level approaches one coincides with a problem having semi-infinite constraints. We study the sample average approximation scheme for the CVaR constraints and establish an exponential rate of convergence for the sample averaged optimal solution. We propose a procedure to explore the possibility of varying the confidence level to a lower value which can give an advantage when there is a need to find good solutions to CVaR-constrained problems out of sample. Our numerical results demonstrate that using the optimal solution to an adjusted problem with lower confidence level can lead to better overall performance.

中文翻译：

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CVaR 风险度量和最小最大限制的不同置信水平

条件风险值 (CVaR) 已被广泛研究作为风险度量。在本文中，我们通过关注置信水平的选择及其对出现在目标和约束中的 CVaR 优化问题的影响来增加这项工作。我们首先考虑一个 CVaR 被最小化的问题，并研究它在置信水平被驱动为 1 时逼近极小极大稳健优化问题的方式。我们利用一致的尾部条件确保随机函数的 CVaR 随着置信水平的增加均匀收敛到其上限值，并在二阶增长条件下为 CVaR 最优解建立误差界限。结果被扩展到一个对 CVaR 值有约束的最小化问题，当置信水平接近 1 时，该限制与具有半无限约束的问题重合。我们研究了 CVaR 约束的样本平均近似方案，并为样本平均最优解建立了指数收敛速度。我们提出了一个程序来探索将置信水平改变为较低值的可能性，当需要在样本外找到 CVaR 约束问题的良好解决方案时，这可以提供优势。我们的数值结果表明，对置信水平较低的调整问题使用最佳解决方案可以带来更好的整体性能。我们研究了 CVaR 约束的样本平均近似方案，并为样本平均最优解建立了指数收敛速度。我们提出了一个程序来探索将置信水平改变为较低值的可能性，当需要在样本外找到 CVaR 约束问题的良好解决方案时，这可以提供优势。我们的数值结果表明，对置信水平较低的调整问题使用最佳解决方案可以带来更好的整体性能。我们研究了 CVaR 约束的样本平均近似方案，并为样本平均最优解建立了指数收敛速度。我们提出了一个程序来探索将置信水平改变为较低值的可能性，当需要在样本外找到 CVaR 约束问题的良好解决方案时，这可以提供优势。我们的数值结果表明，对置信度较低的调整问题使用最佳解决方案可以带来更好的整体性能。