Interior-Point Methods for Nonlinear Optimization and Complementarity

建议#0098040普林斯顿大学范德贝，罗伯特J.优化是确定某些可控参数的值，以便在可行的参数设置的指定区域内实现目标函数的最佳值，即最小值或最大值的过程。随着当前技术解决现实世界困难问题的能力得到越来越广泛的认识，对高性能优化算法和软件的需求迅速增加。现代优化技术的成功在很大程度上源于最近所谓的内点方法的发展。这些方法最初用于线性优化问题，但现在正在积极推广到非线性优化问题。这项研究涉及大规模非线性约束优化的新内点算法和软件的开发。具体目标包括：(1)对当前最先进技术的基本改进，例如用基于过滤器的方法取代步长控制的价值函数，并开发更好的技术来检测无界和不可行的问题；(2)将基本算法扩展到不太适合非线性约束优化的基本范式的新问题类，如二阶锥规划和半定规划；(3)将优化领域进一步扩展到非线性互补问题，这些问题在许多工程问题中都存在；最后(4)开发了一个大型优化模型库，这既是为了展示现代优化技术的力量，也是为未来的算法开发提供一个试验台。

Proposal #0098040Princeton UniversityVanderbei, Robert J.Optimization is the process of determining values of certain controllable parameters so as to achieve the best, i.e. smallest or largest, value of an objective function within a specified domain of feasible parameter settings. The demand for high-performance optimization algorithms and software is increasing rapidly as the power of the current technology to solve difficult real-world problems is becoming more widely recognized.Much of the success of modern optimization techiques stems from the recent development of so-called interior-point methods. These methods were first developed for linear optimization problems but are now actively being extended to nonlinear optimization problems.This research involves the development of new interior-point algorithms and software for large-scale nonlinear constrained optimization. Specific goals include: (1) fundamental enhancements to the current state-of-the-art, such as the replacement of the merit-function for step-length control with afilter-based method and the development of better techniques to detect unbounded and infeasible problems; (2) extension of the basic algorithm to new problem classes, such as second-order cone programming and semidefinite programming, that don't quite fit the basic paradigm of nonlinear constrained optimization; (3) further extension beyond the realm of optimization to nonlinear complementarity problems, which arise in many engineering problems; and finally (4) development of a large repository of optimization models, which serves both to illustrate the power of modern optimization technology and also to provide a test bed for future algorithm development.