Optimization Algorithms For a Class of Engineering Design Problems

这一提议是为了继续长期努力开发专用算法，以解决用标准数学规划或最优控制算法无法处理的重要工程优化问题。这些问题包括设计对中、公差和制造后调整、存在带有角点的障碍物时的机器人和车辆路径规划(用极大-极小型不等式描述)、空中交通管制中出现的最优控制和追逃问题、各种形状优化问题和满足可靠性约束的抗震结构设计问题。这些问题总是具有变量或约束或两者兼有的非光滑性和无穷维性的特点，并且经常涉及到常微分方程组或偏微分方程组。这项研究将利用已开发的几种工具，例如(I)一致逼近理论，(Ii)将看似棘手的问题转化为容易处理的形式的重排技术。一致逼近框架确保逼近问题的局部和全局极小点和固定点收敛于原始问题的局部和全局极小点。此外，它还提供带有离散化调整程序的主算法模型，可与高度精致的数学编程库结合使用。此外，通过从主算法调用超线性收敛的数学规划子例程，可以使这些算法超线性收敛。相比之下，我们设想的问题重构技术必须在特别的基础上开发，就像我们已经对一些路径规划和可靠性约束问题所做的那样。由于我们已经获得了一些可以建立在此基础上的结果，以及主要研究人员和其他人正在进行的互动和合作所提供的多学科环境，提案研究的成功机会将大大增加。

This proposal is for the continuation of long standing efforts in developing special purpose algoritluns for the solution of important engineering optimization problems that are not tractable by standard mathematical programming or optimal control algorithms. The problems include design centering, tolerancing and post manufacture tuning, robot and vehicle path planning in the presence of obstacles with coners, described by max-min type inequalities, optimal control and pursuit-evasion problems arising in air-traffic control, various shape optimization problems, and problems of seismic resistant structural design subject reliability constraints. These problems are invariably characterized by nonsmoothness and infinite dimensionality of either variables or constraints or both, and often involve ordinary or partial differential equations. The proposed research will exploit several tools developed, such as (i) the theory of consistent approximations, (ii) reformulation techniques for converting seemingly intractable problems into tractable forms.The framework of consistent approximations ensures the convergence of local and global minimizers and stationary points of approximating problems to those of the original problem. Also, it provides master algorithm models with discretization adjustment procedures, which can be used in conjunction with highly polished, mathematical programming libraries. Furthermore, these algorithms can be made to converge superlinearly by calling superlinearly converging mathematical programming subroutines from the master algorithms. By contrast, the problem reformulation techniques that we envisage must be developed on an ad hoc basis, as we have already done for some path planning and reliability constrained problems.The chances for success of the proposal research will be considerably enhanced by the fact that we have already obtained some results on which we can build and the multidisciplinary setting provided by the ongoing interactions and collaborations of the principal investigator and others.