New Methodologies for Dynamic Optimization

动态优化是指一类问题参数不确定的优化或决策问题。包括个人、企业甚至政府在内的每个人都需要经常解决这种不确定情况下的决策问题。一种常用的方法是使用概率分布（可能相关）对参数不确定性进行建模，并将问题表述为随机优化问题。虽然它在许多应用程序中是一种合理的方法，但总的来说，它在计算上是难以处理的。此外，我们往往没有足够的历史数据来估计概率分布，以制定随机优化问题。这个项目的主要重点是考虑不同的解决范例，如鲁棒优化和仿射策略的动态优化问题是更容易处理的，并研究这些不同的解决方法之间的关系。这样的研究将为开发更好更快的动态优化问题算法提供重要的见解。提出的工作的宏伟目标是为动态优化问题开发一种可处理的解决方法理论，该理论在许多应用领域具有潜在的重大影响。通过这个项目，PI努力在各个层面上产生积极的影响，包括(i)发展对动态优化的更深层次的理论理解，推进我们知识的前沿，（ii）将该项目所做的研究传播到现实世界，以及（iii）培养研究生的研究和教学，这是创建一个可持续的研究循环及其传播到实践的一个极其重要的环节。PI打算积极追求电力市场中出现的动态优化问题，特别是由于高度可变的可再生能源（如风能和太阳能）和其他因素的广泛整合。在这个项目中开发的工具和见解可以在这个领域产生重大影响，这从经济和气候的角度来看都是非常关键的。

Dynamic optimization refers to the broad class of optimization or decision-making problems where some problem parameters are uncertain.Everyone including individuals, businesses and even Governments need to solve such decision problems under uncertainty at a regular basis. A commonly used approach is to model the parameter uncertainty using probability distributions (possibly correlated) and formulate the problem as a stochastic optimization problem. While it can be a reasonable approach in many applications, it is by and large computationally intractable. Moreover, quite often we do not have sufficient historical data to estimate the probability distributions to formulate the stochastic optimization problem.The primary focus of this project is consider different solution paradigms such as robust optimization and affine policies for dynamic optimization problems that are significantly more tractable, and study the relation between these different solution approaches. Such a study would provide significant insights towards developing better and faster algorithms for dynamic optimization problems. The grand goal of the proposed work is to develop a theory of tractable solution approaches for dynamic optimization problems that has a potential of significant impact in many application areas. Through this project, the PI strives to have a positive impact at various levels that includes (i) developing a deeper theoretical understanding of dynamic optimization that advances the frontier of our knowledge, (ii) the dissemination of research done in this project to the real-world, and (iii) the training of graduate students in research and teaching which is an extremely important link in creating a sustainable cycle of research and its dissemination to practice. The PI intends to actively pursue dynamic optimization problems arising in electricity markets especially due to a broader integration of highly variable renewable sources of power (such as wind and solar) and other factors. The tools and insights developed in this project can have a big impact in this area which is very critical both from an economic as well as climate point of view.