Inverse problems, Robust Optimization and Mathematical Programs with Equilibrium Constraints: Algorithms and Applications

本项目涉及建模和求解逆问题、鲁棒优化问题以及平衡约束数学程序的优化方法的研究和发展。反问题的目标是确定观察到的或期望的结果的最可能的原因。这些问题出现在许多重要的军事、成像、信号处理、生物医学、遥感和地质应用中。逆问题是不适定的，因为测量中的小误差会导致估计量的大误差。正则化技术用于求反问题的稳定解，引起一般的非线性规划问题。在特殊情况下，例如成像和信号处理，逆问题可以被重新表述为可处理的凸规划问题。然而，在大多数情况下，例如形状优化，需要解决一个非凸优化问题。鲁棒优化是一种在优化环境中建模和管理不确定性的方法。在这里，不是解决参数识别问题，而是关注约束优化问题，其中参数是数据，并且由于测量或估计误差，只知道位于有界的不确定性集中。在模棱两可的机会约束（鲁棒）优化中，假设参数是具有不确定分布的随机变量，并且只存在于不确定的概率度量集合中。幸运的是，对于某些类型的不确定性集，只要问题的非鲁棒版本本身是凸的，鲁棒优化问题就会产生易于处理的凸规划问题。这些问题出现在工程设计、投资组合选择和服务网络设计等不同领域。mpec源于博弈论和双层优化。它们是优化问题，其中一些约束是参数变分不等式或互补系统，这些约束是由低级优化问题的最优性条件产生的。mpec是许多重要经济问题的良好模型，其中均衡约束来自竞争力量和最优工程设计问题。因此，开发可扩展的mpec解算法具有重要的实际意义。一般来说，解决这三个类中的任何一个问题都相当困难，并且需要大量的计算资源。乍一看，这三种建模范式似乎非常不同。因此，在这三个领域的算法发展之间几乎没有相互作用。虽然所有的范式都有一个独特的结构和一组特殊的问题，但它们有很多共同点。除了它们在形式分析上的相似性之外，根据上下文的不同，这三种范式通常用于为相同的应用程序建模。这个项目的目标是利用这些范例的共同特性来开发新的和更有效的解决方案方法。第二个目标是通过开发新的博士课程和让学生参与理论研究和软件开发，提供反问题、鲁棒优化和mpec方面的跨学科培训。

This project involves the study and development of optimization methods for modeling and solving inverse problems, robust optimization problems, and mathematical programs with equilibrium constraints (MPECs). The goal of inverse problems is to determine the most likely causes for an observed or desired effect. Such problems arise in a number of important military, imaging, signal processing, biomedical, remote-sensing, and geological applications. Inverse problems are ill-posed in the sense that a small error in measurement can lead to large errors in the estimated quantities. The regularization techniques used to obtain stable solutions to inverse problems give rise to general nonlinear programming problems. In special cases, such as imaging and signal processing, the inverse problem can be reformulated as a tractable convex programming problem. In most cases, e.g. shape optimization, however, a non-convex optimization problem needs to be solved. Robust optimization is an approach for modeling and managing uncertainty in an optimization context. Here, rather than solving a parameter identification problem, one is concerned with constrained optimization problems in which the parameters are data and, due to measurement or estimation errors, are only known to lie within bounded uncertainty sets. In ambiguous chance-constrained (robust) optimization it is assumed the parameters are random variables with an uncertain distribution that is only known to lie within an uncertainty set of probability measures. Fortunately, for certain classes of uncertainty sets, robust optimization problems give rise to tractable convex programming problems as long as the non-robust versions of the problems are themselves convex. These problems arise in areas as diverse as engineering design, portfolio selection, and the design of service networks. MPECs have their roots in game theory and bi-level optimization. They are optimization problems in which some of the constraints are parametric variational inequalities or complementarity systems that arise from the optimality conditions for a lower-level optimization problem. MPECs are a good model for a number of important economic problems, where the equilibrium constraints arise from competitive forces, and optimal engineering design problems. Consequently, developing scalable solution algorithms for MPECs is of considerable practical importance. In general, solving problems from any of these three classes is quite difficult and requires significant computational resources. At first glance, these three modeling paradigms appear quite distinct. Consequently, there has been very little interaction between the algorithmic developments that have occurred in these three areas. While all of the paradigms have a unique structure and a special set of issues, they have much in common. Besides their formal analytical similarity, depending on the context all three paradigms are often used to model the same application. A goal of this project is to leverage the common features of these paradigms to develop new and more efficient solution methods. A second goal is to provide interdisciplinary training in inverse problems, robust optimization, and MPECs through the development of new doctoral courses and involvement of students in theoretical research and software development.