Global Optimization and Parallel Processing of Nonlinear Problems in Engineering Applications

本项目的目标是开发一种有效的非线性优化方法--新颖的：通过外部引导的非线性优化-以及用于解决运筹学、数字信号处理和神经网络学习中的大规模非线性优化问题的辅助硬件/软件。这种方法将允许比现有方法更好、更快地解决这些领域中的许多重要应用问题。该方法首先利用拉格朗日乘子将非线性约束优化问题转化为无约束优化问题。这种转换允许在统一的框架中解决约束和非约束问题。该方法的关键思想是使用用户定义的轨迹将搜索从局部最优(约束优化中的局部鞍点和无约束优化中的局部最小值)中拉出，而不必从新的起点重新开始。该方法在全局搜索和局部搜索之间取得了良好的平衡。在全局搜索中，NOVICE依赖于两种反作用力：将搜索推向局部最优的局部梯度信息和一旦搜索到局部最优就引导搜索走出局部最优的确定性轨迹。轨迹对搜索轨迹的影响用轨迹当前位置与轨迹当前位置之间的距离来表示。然后，使用纯局部搜索更广泛地搜索在全局搜索阶段确定的好的起点。提出的研究包括四个方面：新方法的改进，并行处理，混合非线性优化问题的推广，以及为解决运筹学、数字信号处理和神经网络学习问题而开发的方法的应用。

The objective of this project is to develop an efficient nonlinear optimization method - NOVEL: Nonlinear Optimization Via External Lead - and the supporting hardware/software to solve large-scale nonlinear optimization problems in operations research, digital signal processing, and neural-network learning. Such a method will allow many important application problems in these areas to be solved better as well as faster than existing methods. The method first transforms nonlinear constrained optimization problems using Lagrange multipliers into unconstrained optimization problems. This transformation allows both constrained and unconstrained problems to be solved in a unified framework. The key idea of the method is to use a user-defined trace to pull a search out of a local optimum (a local saddle point in constrained optimization and a local minimum in unconstrained optimization) without having to restart it from a new starting point. The proposed method has a good balance between global search and local search. In global search, NOVEL relies on two counteracting forces: local gradient information that drives the search to a local optimum, and a deterministic trace that leads the search out of a local optimum once it gets there. The effect of the trace on the search trajectory is expressed in terms of the distance between the current position of the trajectory and that of the trace. Good starting points identified in the global-search phase are then searched more extensively using pure local searches. The proposed research consists of four aspects: refinement of the NOVEL method, parallel processing, extension to mixed-nonlinear optimization problems, and applications of the methods developed to solve problems in operations research, digital signal processing, and neural-network learning.