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 non-linear in the distribution. Existing methods to optimize nonlinear functions in probability space use the Frechet derivatives, which present both 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 non-linear 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 non-linear DRO problem. Finally, for the minimum variance portfolio selection problem we analyze the regularity conditions and compute the FW-oracle in various settings, and validate the theoretical results numerically.

翻译：本文研究一类分布鲁棒优化（DRO）问题。与现有文献不同，本文所考虑的优化问题中目标函数可能关于概率分布是非线性的。现有方法通过弗雷歇导数优化概率空间中的非线性函数，但该导数在理论和计算层面均存在挑战。受此启发，我们提出基于泛函导数（G-导数）的替代性导数定义及相应的光滑性概念，并将其应用于一般风险度量。通过方差、熵风险及有限支撑集上的风险这三个典型风险度量案例，我们阐释了这些概念。基于此，我们提出一种利用G-导数的Frank-Wolfe（FW）算法，用于求解概率空间中的一般非线性优化问题，并在完全独立于范数的框架下，基于所提出的光滑性概念证明了该算法的收敛性。利用FW算法的框架，我们建立了一种计算非线性DRO问题鞍点的方法。最后，以最小方差投资组合选择问题为例，我们分析了正则性条件，计算了不同情形下的FW预言函数，并通过数值实验验证了理论结果。