Efficient Interior-Point Methods for Mixed-Integer Nonlinear and Conic Programming

最优化问题出现在工程、商业和医学中。计算研究表明，内点方法在解决大规模问题时优于其他方法。这项研究涉及到改进非线性和相关优化问题的内点方法的技术发展。它的智力优势是解决复杂问题及其在不断变化的条件下的重新解决，这是当前必须解决的两个缺陷，以使这些方法保持在优化技术的前沿。测试问题通过鼓励使用适当的建模和求解技术并在必要时开发新的技术，使优化社区和应用领域的研究人员受益。本研究最广泛的影响是通过解决更能反映真实世界的问题和通过更有效地响应动态环境而获得的社会和经济收益。为了改进和扩展内点方法的使用，研究人员研究了(1)原始-对偶惩罚方法，以提高解决数据和问题大小变化的密切相关问题的热启动能力；(2)解决混合整数非线性问题的双层框架，包括改进的热启动和非线性子问题的不可行性识别方案；(3)使用原始-对偶惩罚方法将平衡约束和锥约束融入到非线性优化框架中。这项研究的另一个组成部分是开发和传播具有各种组成部分的应用问题存储库，例如非线性函数、离散变量、平衡约束和锥形约束，以及适用于测试热启动方法的替代数据集和参数设置。所有的工作都被纳入研究生课程和研究中。课程讲稿和测试模型在WWW上分发，软件在NEOS服务器上免费使用。

Optimization problems arise in engineering, business, and medicine.Computational studies show that interior-point methods outperform others for solving large-scale problems. This research involves the development of techniques to improve interior-point methods for nonlinear and related optimization problems. Its intellectual merit is the solution of complex problems and their re-solution under changing conditions, two current deficiencies that must be addressed to keep these methods at the forefront of optimization technology. Test problems benefit the optimization community and researchers in applied fields by encouraging the use of appropriate modeling and solution techniques and the development of new ones as necessary. The broadest impact of this research is the social and financial gains achieved by solving problems that better reflect the real-world and by responding more effectively to a dynamic environment.In order to improve and extend the use of interior-point methods, the investigators study (1) primal-dual penalty methods to improve warm-starting capabilities for solving closely related problems with changing data and problem size, (2) bilevel frameworks for solving mixed-integer nonlinear problems, with improved warm-starting and infeasibility identification schemes for nonlinear subproblems, and (3) incorporation of equilibrium and cone constraints into the nonlinear optimization framework using the primal-dual penalty approach. Another component of this research is the development and dissemination of a repository of applied problems with a variety of components, such as nonlinear functions, discrete variables, equilibrium constraints, and cone constraints, along with alternate data sets and parameter settings suitable for testing warm-start approaches. All work is incorporated into graduate courses and research. Course notes and test models are distributed on the WWW, and software is made available for free use on the NEOS Server.