Stochastic and Robust Optimization Approaches for Financial and Operations Engineering

作为金融和运营管理领域建模基础的经典优化框架通常是有洞察力和易于处理的。然而，它们往往缺乏许多重要的真实的世界方面，如不确定性。例如，工业界最常用的金融投资组合模型是均值-方差优化（Markowitz）模型，这是一个确定性问题，并假设金融资产的预期价格（收益）是已知的，这不是一个实际的假设。此外，当结合时，真实的世界环境的其他方面使得这样的模型在计算上难以处理，诸如交易成本或操作设计决策。也许最重要的是，决策必须在参数值完全已知之前做出，因此适当的框架是考虑包含不确定性的模型，以使决策对模型中重要参数固有的不确定性具有免疫力或鲁棒性。然而，考虑到不确定性和真实的世界的约束，在模型中，是非常具有挑战性的解决。该建议旨在为重要的金融工程问题和业务管理问题开发模型并找到解决方法，这些问题将不确定性和真实的世界实际约束结合起来。特别是，拟议的研究计划旨在调查随机规划和稳健的优化方法，财务指标跟踪和备件管理。真实的世界本质上是嘈杂的，最佳的鲁棒解决方案具有不同于模型的结构，该模型认为参数是确定性的，该提案旨在研究在随机环境中适当的最佳或近似策略是什么。数学开发也是拟议研究的一个重要组成部分，将涉及结构化分解和鲁棒凸优化的想法。

The classical optimization frameworks that serve as the basis for modeling in finance and operations management domains are often insightful and tractable. However, they often lack many significant real world aspects such as uncertainty. For example, the most common financial portfolio model used in industry is the mean-variance optimization (Markowitz) model, which is a deterministic problem and assumes that the expected prices(returns) of financial assets are known, which is not a practical assumption. In addition, other facets of real world environments when incorporated renders such models computationally intractable such as transaction costs or operational design decisions . Perhaps most importantly, decisions have to be made before parameter values are fully known and so the proper framework is to consider models that incorporate uncertainty to enable decisions that are immune or robust to uncertainty inherent in important parameters in the models. However, the consideration of both uncertainty and real world constraints results in models that are very challenging to solve. This proposal seeks to develop models and find solution methods for important financial engineering problems and operations management problems that integrate both uncertainty and real world practical constraints. In particular, the proposed research program seeks to investigate stochastic programming and robust optimization approaches for financial index tracking and spare parts management. The real world is inherently noisy and optimal robust solutions have a structure that is different than models that considers parameters to be deterministic and the proposal seeks to investigate what the appropriate optimal or approximate policies are in random environments. Algorithmic development is also an important component of the proposed research and will involve ideas from structured decomposition and robust convex optimization.