Optimization Algorithms for Problems with Stochastic Dominance Constraints

这项拟议的研究为优化问题的研究提供了资金，在优化问题中，约束的内在不确定性是使用一个称为随机优势的概念来建模的。在二阶线性随机优势概念下的不确定线性规划和不确定半定规划是最早受到关注的两个优化问题，它构成了对多维随机订单进行建模的一种特殊方法。对于这类问题，将构建有效的算法。此外，还将发展一种对偶理论，它允许显式地构造与解决这类问题相关的对偶函数。更一般的随机序和相应的优化问题的可解性将被分析。研究还将解决随机实体在随机优势约束中的支持度不是有限的或非常大的情况，因此需要基于抽样的方法。最后，在随机支配的背景下研究具有随机参数的随机实体及其应用。如果成功，所提出的研究将解决一些组件未知的系统的优化的基本问题，这在许多领域都有应用，包括运筹学、统计学和金融学。这项工作将有助于更好地理解在优化问题中使用随机支配概念的好处和缺点-这一概念已被证明在从经济学到流行病学的许多领域具有极其重要的意义。这项研究的目标之一是开发解决此类问题的算法，该算法的可用性将导致对随机模型中的参数不确定性进行更好的建模。拟议的研究建立在两个不相关的领域(优化和随机优势)之上，预计它将促进思想的交叉，这可能会导致这两个领域的进一步进展，同时允许提高应用问题的建模能力。这种不同领域的结合还将导致开发新的研究生课程，并通过一套关于这一主题的课堂讲稿来传播想法。

This proposed research provides funding for the study of optimization problems where the uncertainty intrinsic to the constraints in the problem is modeled using a concept known as stochastic dominance. Two optimization problems which will receive an early focus in the research are the uncertain linear and uncertain semidefinite programs under a second-order linear stochastic dominance concept, which constitutes a particular way to model multi-dimensional stochastic orders. Efficient algorithms for such problems will be constructed. In addition, a duality theory that allows explicit construction of dual functions associated with the solution of such problems will be developed. More general stochastic orders and the solvability of corresponding optimization problems will be analyzed. The research will also address the situation where the support of the random entities in the stochastic dominance constraints is not finite or is very large, so that sampling based approaches are required. Finally, a study of stochastic entities with random parameters and their applications may be conducted within the context of stochastic dominance.If successful, the proposed research will address the fundamental problem of optimizing a system where some components are not known with certainty, which has applications in many areas, including operations research, statistics and finance. The work will help to develop a better understanding of the benefits and drawbacks of using the concept of stochastic dominance --- which has proven to be of capital importance in many areas, ranging from economics to epidemiology --- in an optimization problem. One goal of this research is to develop algorithms for such problems, the availability of which will result in better modeling of parameter uncertainty in stochastic models. The proposed research builds upon two unrelated areas (optimization and stochastic dominance) and it is expected to promote a cross-fertilization of ideas that can potentially lead to further advances in both areas, while allowing for improved modeling abilities of application problems. This combination of different areas will also lead to the development of new graduate courses and the dissemination of ideas through a set of lecture notes on the topic.