Optimization Algorithms for Problems with Stochastic Dominance Constraints

具有随机优势约束问题的优化算法

• 批准号: 1033051
• 负责人: Tito Homem-de-Mello
• 金额: $ 24.84万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-12-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1033051&HistoricalAwards=false
• 关键词: Optimization; Algorithms; Problems; Stochastic; Dominance

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.

这项拟议的研究为优化问题的研究提供了资金，其中使用称为随机优势的概念对问题中约束固有的不确定性进行建模。早期研究重点关注的两个优化问题是二阶线性随机优势概念下的不确定线性和不确定半定规划，它构成了多维随机阶建模的特殊方法。将构建针对此类问题的有效算法。此外，还将开发一种对偶理论，允许显式构造与解决此类问题相关的对偶函数。将分析更一般的随机阶数和相应优化问题的可解性。该研究还将解决随机优势约束中随机实体的支持不是有限的或非常大的情况，因此需要基于采样的方法。最后，可以在随机优势的背景下对具有随机参数的随机实体及其应用进行研究。如果成功，所提出的研究将解决优化某些组件不确定的系统的基本问题，该系统具有应用价值在许多领域，包括运筹学、统计和金融。这项工作将有助于更好地理解在优化问题中使用随机优势概念的好处和缺点——事实证明，随机优势在从经济学到流行病学的许多领域都具有重要意义。这项研究的目标之一是开发解决此类问题的算法，该算法的可用性将有助于更好地对随机模型中的参数不确定性进行建模。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，预计将促进思想的交叉融合，从而有可能导致这两个领域的进一步进步，同时提高应用问题的建模能力。不同领域的结合也将导致新的研究生课程的开发，并通过一系列关于该主题的讲义传播思想。