Global Dynamic Optimization

保罗·I·巴顿研究所：麻省理工学院建议号：0521962标题：全局动态优化动态优化确定动态系统的输入/控制配置文件、实值参数、初始条件和/或边界条件的值，该动态系统根据指定的度量在一段时间内优化其性能。动态优化问题几乎出现在现代化学工程的各个方面。化工中遇到的动态优化问题往往会出现多个次最优局部极小值。传统的求解动态优化问题的方法只能保证找到局部极小值，而局部极小值可能是次优的。如果在实际系统上实施次优解决方案，次优可能会对经济、安全和环境产生直接影响。该项目将开发理论、数值方法和软件，以确保在有限的迭代次数内找到动态优化问题的全局解。智能优点NSF支持的PI先前的研究为嵌入线性时变常微分方程组(ODE)和非线性ODE的优化问题创造了全局优化理论和算法。对于相对较小的动态系统和最优化问题中的大约10个自由度，这种方法已经被证明是可行的。本研究的目的是进一步发展理论和算法，以便能够解决涉及大系统(可能是1,000-10,000个状态变量)和数十个自由度的优化问题，以保证全局最优性。这将把化学工程中的大部分动态优化问题纳入全局优化方法的范围。在这一推广中的一个关键的理论和实践问题是非拟单调微分方程解的参数集象的紧估计的计算(大多数化学工程应用都是非拟单调的)。基于微分不等式中经典结果的推广的理论和算法将被用来解决这个问题。此外，化工中的许多动态优化问题也嵌入了微分-代数方程(DAE)和/或偏微分方程组(PDE)。设计了将全局优化方法扩展到DAE和PDE嵌入式系统的理论和算法，全局动态优化的一个应用是形式化的安全验证；确定性全局优化提供了动态系统安全的构造性证明，或保证了反例的定位。然而，形式验证总是与模型有关，没有考虑到模型的预测和相应物理系统的行为之间总是存在差异的事实。这通常被称为模型不确定性。以前的研究没有考虑形式安全验证中的模型不确定性问题，但这是保证物理系统安全的一个潜在的关键问题。提出了一种基于嵌入微分方程的半无限规划的方法来解决安全验证中的模型不确定性。广泛的影响：解决动态优化问题以保证全局最优性的能力的增强可能具有广泛的实际意义。例如，在工艺操作领域，有希望解决不确定情况下的正式安全验证、集成批处理工艺的综合以及使用详细的动态模型设计主要工艺瞬变过程(如启动和关闭程序)等问题。这项工作的结果将通过期刊文章、公开分发的软件、课程大纲和国际和平研究所目前正在编写的一本关于全球优化的教科书广泛传播。此外，通过该项目开发的软件将通过Web免费分发给学术研究人员。该项目应该为学生提供许多多学科教育和研究的机会。该系大量的女本科生和研究生应该有助于吸引她们中的一些人参与这个项目。

ABSTRACTPI: Paul I. Barton Institution: Massachusetts Institute of TechnologyProposal Number: 0521962Title: Global Dynamic OptimizationDynamic optimization determines values for input/control profiles, real valued parameters, initial conditions and/or boundary conditions of a dynamic system that optimize its performance over some period of time according to a specified metric. Dynamic optimization problems appear in almost every aspect of modern chemical engineering. The dynamic optimization problems encountered in chemical engineering often exhibit multiple suboptimal local minima. Conventional approaches for the solution of dynamic optimization problems can only guarantee locating local minima, which may be suboptimal. Suboptimality can have direct economic, safety and environmental impacts if a suboptimal solution is implemented on a real system. This project will develop theory, numerical methods and software that can guarantee locating a global solution of a dynamic optimization problem within a finite number of iterations.Intellectual Merit Previous research by the PI supported by the NSF has created global optimization theory and algorithms for optimization problems embedding linear time varying ordinary differential equations (ODEs) and nonlinear ODEs. The approach has been demonstrated for relatively small dynamic systems and up to about ten degrees of freedom in the optimization problem. The purpose of this research is to develop the theory and algorithms further so that optimization problems involving large-scale dynamic systems (possibly 1,000s-10,000s of state variables) and tens of degrees of freedom can be solved to guaranteed global optimality. This would bring a majority of the dynamic optimization problems in chemical engineering within the scope of global optimization methods. A key theoretical and practical issue in this extension is the computation of tight estimates of the image of a parameter set under the solution of nonquasi-monotone differential equations (most chemical engineering applications are nonquasi-monotone). Theory and algorithms based on extensions of classical results in differential inequalities will be used to address this issue. Moreover, many dynamic optimization problems in chemical engineering also have differential-algebraic equations (DAEs) and/or partial differential equations (PDEs) embedded. Theory and algorithms extending the global optimization approach to DAE and PDE embedded systems are planned.An application of global dynamic optimization is formal safety verification; deterministic global optimization provides a constructive proof that a dynamic system is safe, or guarantees location of a counterexample. However, formal verification is always with respect to a model and does not take into account the fact that there is always a discrepancy between the predictions of a model and the behavior of the corresponding physical system. This is commonly referred to as model uncertainty. Previous research has not considered the issue of model uncertainty in formal safety verification, but this is a potentially critical issue in guaranteeing the safety of a physical system. An approach based on a semi-infinite program with differential equations embedded is proposed to address model uncertainty in safety verification.Broader Impacts: The growing capability to solve dynamic optimization problems to guaranteed global optimality could have broad practical implications. For example, in the area of process operations there is hope for solving problems such as formal safety verification under uncertainty, the synthesis of integrated batch processes, and the design of major process transients such as start-up and shut-down procedures, using detailed dynamic models. The results of this work will be broadly disseminated through journal articles, publicly distributed software, course curricula and a textbook on global optimization currently being prepared by the PI. Moreover, the software developed through this project will be freely distributed via the Web to academic researchers. The project should provide students many opportunities for multidisciplinary education and research. The large number of female undergraduate and graduate students in the department should help in attracting some of them t o work on this project.