Distributionally robust stochastic optimization (DRSO) is a framework for
decision-making problems under certainty, which finds solutions that perform
well for a chosen set of probability distributions. Many different approaches
for specifying a set of distributions have been proposed. The choice matters,
because it affects the results, and the relative performance of different
choices depend on the characteristics of the problems. In this paper, we
consider problems in which different random variables exhibit some form of
dependence, but the exact values of the parameters that represent the
dependence are not known. We consider various sets of distributions that
incorporate the dependence structure, and we study the corresponding DRSO
problems.
In the first part of the paper, we consider problems with linear dependence
between random variables. We consider sets of distributions that are within a
specified Wasserstein distance of a nominal distribution, and that satisfy a
second-order moment constraint. We obtain a tractable dual reformulation of the
corresponding DRSO problem. This approach is compared with the traditional
moment-based DRSO and Wasserstein-based DRSO with no moment constraints.
Numerical experiments suggest that our new formulation has superior
out-of-sample performance.
In the second part of the paper, we consider problems with various types of
rank dependence between random variables, including rank dependence measured by
Spearman's footrule distance between empirical rankings, comonotonic
distributions, box uncertainty for individual observations, and Wasserstein
distance between copulas associated with continuous distributions. We also
obtain a dual reformulation of the DRSO problem. A desirable byproduct of the
formulation is that it also avoids an issue associated with the one-sided
moment constraints in moment-based DRSO problems.

本文研究了在不确定情况下基于一组概率分布找到表现良好解的分布鲁棒随机优化问题。在两部分中，分别考虑线性依赖和排序依赖的分布情况，提出了包含这些依赖项的新形式的双重优化。实验结果表明，新形式相对于传统的基于单侧矩阵的分布鲁棒随机优化有更好的性能。

具有相关结构的分布鲁棒性随机优化

Distributionally Robust Stochastic Optimization with Dependence  Structure

Distributionally robust stochastic optimization (DRSO) is an approach to
optimization under uncertainty in which, instead of assuming that there is a
known true underlying probability distribution, one hedges against a chosen set
of distributions. In this paper we first point out that the set of
distributions should be chosen to be appropriate for the application at hand,
and that some of the choices that have been popular until recently are, for
many applications, not good choices. We next consider sets of distributions
that are within a chosen Wasserstein distance from a nominal distribution. Such
a choice of sets has two advantages: (1) The resulting distributions hedged
against are more reasonable than those resulting from other popular choices of
sets. (2) The problem of determining the worst-case expectation over the
resulting set of distributions has desirable tractability properties. We derive
a strong duality reformulation of the corresponding DRSO problem and construct
approximate worst-case distributions explicitly via the first-order optimality
conditions of the dual problem.
Our contributions are four-fold. (i) We identify necessary and sufficient
conditions for the existence of a worst-case distribution, which are naturally
related to the growth rate of the objective function. (ii) We show that the
worst-case distributions resulting from an appropriate Wasserstein distance
have a concise structure and a clear interpretation. (iii) Using this
structure, we show that data-driven DRSO problems can be approximated to any
accuracy by robust optimization problems, and thereby many DRSO problems become
tractable by using tools from robust optimization. (iv) Our strong duality
result holds in a very general setting. As examples, we show that it can be
applied to infinite-dimensional process control and intensity estimation for
point processes.

本文对于分布鲁棒随机优化问题提出了一种新方法：使用 Wasserstein 距离来确定最坏情况下的期望，并通过拟合最优对偶解得到最坏分布，这一方法可以用于控制和估计点过程的强对偶结果。