Optimization Algorithms For a Class of Engineering Design Problems

这一建议是为继续长期的努力，在发展特殊用途algoritluns的解决方案的重要工程优化问题，是不听话的标准数学规划或最优控制算法。 这些问题包括设计定心、公差和制造后调整、机器人和车辆在存在具有拐角的障碍物的情况下的路径规划（由最大-最小型不等式描述）、空中交通管制中出现的最优控制和追踪-规避问题、各种形状优化问题以及抗震结构设计受可靠性约束的问题。 这些问题的特点是不光滑和无穷维的变量或约束或两者，往往涉及常微分方程或偏微分方程。 本研究将利用几种工具，如（i）一致逼近理论，（ii）将看似棘手的问题转化为易于处理的形式的重构技术，一致逼近的框架确保了局部和全局极小值的收敛性以及逼近问题的稳定点到原问题的稳定点。 此外，它还提供了具有离散化调整程序的主算法模型，可以与高度完善的数学编程库结合使用。 此外，这些算法可以通过调用超线性收敛的数学规划子程序从主算法超线性收敛。 相比之下，我们设想的问题重新表述技术必须在特别的基础上发展，正如我们已经为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我们可以建立的结果，以及校长正在进行的互动和合作所提供的多学科环境，调查员和其他人。

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.