Stochastic and Robust Optimization Approaches for Financial and Operations Engineering

作为财务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的和易于处理的。然而，它们往往缺乏现实世界中许多重要的方面，如不确定性。例如，行业中使用的最常见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，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，现实世界环境的其他方面在合并时会使此类模型在计算上变得难以处理，例如交易成本或运营设计决策。也许最重要的是，必须在完全了解参数值之前做出决定，因此适当的框架是考虑包含不确定性的模型，以使决策不受模型中重要参数固有的不确定性的影响或稳健。然而，同时考虑不确定性和现实世界的约束导致模型的求解非常具有挑战性。这项建议寻求为重要的金融工程问题和运营管理问题开发模型和找到解决方法，这些问题结合了不确定性和现实世界的实际约束。特别是，拟议的研究方案寻求研究财务指数跟踪和备件管理的随机规划和稳健优化方法。现实世界本质上是嘈杂的，最优鲁棒解的结构不同于认为参数是确定性的模型，该提案试图调查在随机环境中合适的最优或近似策略是什么。算法开发也是拟议研究的一个重要组成部分，将涉及结构分解和稳健凸优化的想法。

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.