Mathematical Sciences: Continuation and Bifurcations Investigations In Constrained Optimization

数学科学：约束优化中的延拓和分岔研究

• 批准号: 8704679
• 负责人: Aubrey Poore
• 金额: $ 1.39万
• 依托单位: Colorado State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-08-01 至 1989-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8704679&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Continuation; Bifurcations; Investigations

Constrained optimization problems permeate much of applied mathematics, engineering and the sciences ranging from linear and nonlinear programming, calculus of variations and optimal control to variational formulations of parameter identification and inverse problems, ill-posed problems, and continuum mechanics. The primary objective of this research program is the development of a new class of robust and very fast algorithms for the solution of these problems. These algorithms are based on the use of fast numerical continuation techniques to follow smooth penalty paths to optimality. The key to the development of these algorithms is an expanded Lagrangian system of equations which contains the penalty path as a solution, is a perturbation of the first order necessary conditions, and sometimes, but not always, provides a basis for finding multiple optima. The problem formulation and continuation methodology provide a natural framework for the development and application of Tikhonov regularization techniques to the solution of sensitive, ill- conditioned, or ill-constrained optimization problems arising from model error or uncertainty and for the investigation of the parametric optimization problem. Also, in this setting the traditional ill-conditioning associated with a sequential optimization algorithm is removed for three smooth penalty functions: the quadratic penalty function for equality constraints and the logarithmic barrier and quadratic loss function for inequality constraints. In linear programming one of the problem formulations and correspondence algorithms is very similar to the Karmarkar algorithm with similar expectations in speed; however, these fast continuation-penalty path algorithms are also applicable to the aforementioned constrained optimization problems with similar expectations in speed. Research of this type aims at developing new computer software for optimization of functions of many variables with constraints in the most efficient and robust way. Software of this type has a large number of potential applications, ranging from management of resources to airline scheduling and machine design.

约束优化问题渗透到许多应用领域， 数学，工程和科学，从线性和 非线性规划、变分法与最优控制 参数识别的变分公式， 逆问题、不适定问题和连续介质力学。 该研究项目的主要目标是开发 一类新的鲁棒和非常快速的算法， 解决这些问题。 这些算法是基于 使用快速数值延拓技术， 最优的惩罚路径 这些发展的关键 算法是一个扩展的拉格朗日方程系统， 包含惩罚路径作为解，是 一阶必要条件，有时，但不总是， 为寻找多个最优值提供了基础。 问题 制定和延续方法提供了一个自然的 吉洪诺夫的开发和应用框架 正则化技术的解决方案的敏感，病态， 条件或病态约束优化问题 从模型误差或不确定性和调查的 参数优化问题 此外，在这种情况下， 传统的病态与连续的 优化算法去除了三个光滑惩罚 函数：等式的二次罚函数 约束和对数障碍和二次损失 不等式约束的函数。 线性规划一 的问题公式和对应算法是非常 类似于Karmarkar算法，具有类似的期望， 速度;然而，这些快速连续惩罚路径算法 也适用于上述受约束的 在速度上具有相似期望的优化问题。 这种类型的研究旨在开发新的计算机 多变量函数优化软件 以最有效和最强大的方式进行约束。 软件 这种类型具有大量的潜在应用， 从资源管理到航空公司调度和机器 设计