Nonlinear Optimization: Algorithms, Theory and Software

数值优化在各种科学和工程应用中起着至关重要的作用。医学成像、电力网络模拟、计算金融和大气科学广泛使用优化模型来模拟现实生活中的现象。随着这些模型变得越来越复杂，并且包含了大量的数据，对优化技术的要求已经超出了它们的能力。本提案提出了三个旨在应对这一挑战的项目。第一个项目提出了求解大型约束优化问题的无矩阵方法；我们想到的应用程序中变量和约束的数量在数百万之间。新算法通过将迭代方法应用于内线性方程组来计算非精确牛顿步。在这项研究中要解决的问题包括对不精确性、非凸性和雅可比奇异性的适当管理。第二个项目涉及开发开源软件，用于解决由偏微分方程离散化定义约束的问题。新的优化求解器将与阿贡国家实验室合作创建，并将在无矩阵环境中运行。第三个项目研究检测非线性优化问题是否可行的程序。这一问题在混合整数非线性规划和优化模型的参数化研究中具有重要意义，但尚未引起足够的重视。目标是开发新的优化技术，在优化和可行性之间平稳过渡，反之亦然。所提出的活动的智力优点在于设计能够处理非凸性和非线性的优化方法的复杂性。这项工作产生的更广泛的影响将体现在新算法和软件在电路模拟、计算化学、医学成像、大气科学和机器学习等领域的成功应用中。在这个项目中发展的原则和思想将刺激未来在新的应用领域的研究。

Numerical optimization plays an essential role in a wide variety of scientific and engineering applications. Medical imaging, electrical power network simulations, computational finance, and atmospheric sciences make extensive use of optimization models to simulate real-life phenomena. As these models become increasingly more complex and incorporate a larger amount of data, the demands placed on optimization techniques have surpassed their capabilities. This proposal presents three projects designed to address this challenge.The first project proposes matrix-free methods for very large constrained optimization problems; we have in mind applications where the number of variables and constraints range in the millions. The new algorithms compute inexact Newton steps by applying iterative methods to the inner linear systems of equations. Questions to be addressed in this research include the appropriate management of inexactness, nonconvexity, and Jacobian singularity. The second project concerns the development of open-source software for problems in which the constraints are defined by the discretization of partial differential equations. The new optimization solvers will be created in collaboration with Argonne National Laboratory and will operate in a matrix-free environment. The third project investigates procedures for detecting if a nonlinear optimization problem is feasible. This question has not received sufficient attention in spite of its importance in mixed integer nonlinear programming and in parametric studies of optimization models. The goal is to develop new optimization techniques that transition smoothly between optimization and feasibility, and vice versa. The intellectual merits of the proposed activity lie in the complexity of designing optimization methods that are capable of dealing with nonconvexities and nonlinearities. The broader impacts resulting from this work will be seen in the successful application of the new algorithms and software in areas such as circuit simulation, computational chemistry, medical imaging, atmospheric sciences, and machine learning. The principles and ideas developed in this project will stimulate future research in new areas of application.