This article focuses on a class of distributionally robust optimization (DRO) problems where, unlike the growing body of the literature, the objective function is potentially nonlinear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present theoretical and computational challenges. Motivated by this, we propose an alternative notion for the derivative and corresponding smoothness based on Gateaux (G)-derivative for generic risk measures. These concepts are explained via three running risk measure examples of variance, entropic risk, and risk on finite support sets. We then propose a G-derivative-based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in probability spaces and establish its convergence under the proposed notion of smoothness in a completely norm-independent manner. We use the set-up of the FW algorithm to devise a methodology to compute a saddle point of the nonlinear DRO problem. Finally, we validate our theoretical results on two cases of the $entropic$ and $variance$ risk measures in the context of portfolio selection problems. In particular, we analyze their regularity conditions and "sufficient statistic", compute the respective FW-oracle in various settings, and confirm the theoretical outcomes through numerical validation.

翻译：本文聚焦于一类分布鲁棒优化问题，与现有文献不同之处在于其目标函数在分布上可能是非线性的。概率空间中非线性函数优化的现有方法使用弗雷歇导数，这带来了理论和计算上的挑战。受此启发，我们基于一般风险度量的加托导数提出了一种替代的导数概念及相应的光滑性定义。这些概念通过方差、熵风险及有限支撑集风险这三个典型风险度量示例进行阐释。随后，我们针对概率空间中的一般非线性优化问题，提出了一种基于加托导数的弗兰克-沃尔夫算法，并在完全独立于范数的条件下，基于所提出的光滑性概念建立了算法的收敛性。利用弗兰克-沃尔夫算法的框架，我们设计了一种计算非线性分布鲁棒优化问题鞍点的方法。最后，我们在投资组合选择问题的背景下，通过熵风险和方差风险两种情形验证了理论结果。具体而言，我们分析了它们的正则性条件和"充分统计量"，计算了不同设置下的弗兰克-沃尔夫预言机，并通过数值实验验证了理论结论。