Inverse Optimization for Imputing Constraints in Mathematical Programs

正向优化方法寻求计算给定模型参数值的决策变量的最优值，而逆优化的目标是推断使给定决策变量值最优的参数，即规定所需的动作或输入以达到最优结果。这项拨款将通过开发一个计算框架来有效地解决一大类逆优化模型，为促进国家健康、繁荣和福利做出贡献。该方法将应用于癌症放疗的系统识别问题，以帮助验证当前的治疗方案。PI将在整个项目中指导博士生进行该研究课题。研究结果将纳入研究生水平的课程和PI正在起草的两本新书，以及针对STEM中代表性不足的学生的优化应用的讲习班和研讨会。目前的逆优化文献几乎全部集中在目标函数参数的输入上。由于这些逆优化模型是非凸的、双线性的，求解困难，因此对约束参数的输入研究很少。该项目将采用两种方法来解决这些模型：(1)如果可能的话，通过变量变换将其转换为等效凸问题；(2)一套定制的近似算法，解决一系列凸问题，如果没有。研究方法将使用几个公开可用的数据集对经典的分支定界算法进行计算评估，并对癌症放疗进行深入的案例研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

While forward optimization methods seek to calculate the optimal values of decision variables for given values of model parameters, the goal of inverse optimization is to infer parameters that render given values of decision variables optimal, i.e., prescribing needed actions or inputs to achieve an optimal result. This grant will contribute to the advancement of national health, prosperity, and welfare by developing a computational framework to efficiently solve a large class of inverse optimization models. The methodology will be applied to system identification problems in cancer radiotherapy to help validate current treatment protocols. The PI will mentor doctoral students on this research topic throughout the project. Results will be incorporated into a graduate-level course and two new books that the PI is drafting, as well as workshops and seminars on applications of optimization for underrepresented students in STEM. The current inverse optimization literature focuses almost entirely on imputing objective function parameters. There has been little work on imputing constraint parameters because these inverse optimization models are nonconvex, bilinear and hence difficult to solve. The project will pursue two approaches to solve these models: (1) conversion into equivalent convex problems via a variable transformation, if possible; and (2) a suite of tailored approximation algorithms that solve a sequence of convex problems, if not. The researched methods will be evaluated computationally against classic branch-and-bound algorithms using several publicly available data sets, together with an in-depth case study in cancer radiotherapy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.