Global Dynamic Optimization

摘要：保罗一世巴顿研究所：马萨诸塞州理工学院提案编号：0521962标题：全局动态优化动态优化确定动态系统的输入/控制简档、真实的值参数、初始条件和/或边界条件的值，这些值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。 化工动态优化问题中经常出现多个次优局部极小点。传统的动态优化问题的解决方案只能保证定位局部最小值，这可能是次优的。 如果在真实的系统上实施次优解决方案，则次优性会产生直接的经济、安全和环境影响。 该项目将开发理论、数值方法和软件，以保证在有限次迭代内找到动态优化问题的全局解。知识产权PI在NSF支持下的先前研究已经为嵌入线性时变常微分方程（ODE）和非线性常微分方程的优化问题创建了全局优化理论和算法。 该方法已被证明为相对较小的动态系统和最多约十个自由度的优化问题。 本研究的目的是进一步发展理论和算法，以便解决涉及大型动态系统（可能有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。 这将使化学工程中的大多数动态优化问题在全局优化方法的范围内。 一个关键的理论和实际问题，在这个扩展是紧估计的图像下的非准单调微分方程（大多数化学工程应用是非准单调）的解决方案的参数集的计算。 基于微分不等式经典结果的扩展的理论和算法将被用来解决这个问题。 此外，化学工程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（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一个构造性的证明，证明了动态系统是安全的，或者保证了反例的位置。然而，形式化验证总是针对模型，并没有考虑到模型的预测与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。以往的研究没有考虑模型不确定性的问题，在正式的安全验证，但这是一个潜在的关键问题，在保证一个物理系统的安全。提出了一种基于嵌入微分方程的半无限程序的方法来解决安全验证中的模型不确定性。更广泛的影响：解决动态优化问题以保证全局最优的能力不断增长可能具有广泛的实际意义。 例如，在工艺操作领域，有希望解决的问题，如正式的安全验证下的不确定性，综合批处理过程的合成，和设计的主要过程瞬态，如启动和关闭程序，使用详细的动态模型。这项工作的结果将通过期刊文章、公开分发的软件、课程和PI目前正在编写的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网络免费分发给学术研究人员。 该项目将为学生提供多学科教育和研究的许多机会。该系有大量的女本科生和研究生，这将有助于吸引她们中的一些人参与这一项目。

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.