Advances in Global Dynamic Optimization

Paul Barton 0933095智力优点本研究的目的是开发算法的实际规模的动态优化问题的全局解决方案，特别是在化学工程应用的兴趣。在这样的优化问题中，寻找优化指定性能度量的微分方程系统的输入、参数和/或初始条件。 这种类型的优化问题是无处不在的工程学科，由于广泛使用的微分方程模型系统的利益。在化学工程应用中，动态优化问题几乎总是表现出多个次优局部极小点。当在真实的系统上实施时，大多数动态优化算法只能保证收敛到局部最小值，这可能具有严重的经济和环境后果，并且还可能导致不安全的操作条件。到目前为止，大约有三种算法能够解决一般的动态优化问题，以保证全局最优性-所有这些都被限制为少于约五个自由度和五个状态变量的问题，由于计算成本过高，因此不能用于解决许多实际的化学工程问题。因此，这项工作的主要目标是开发更有效的算法来解决动态优化问题globals.Under以前NSF的支持下，PI和同事开发了一个严格的全局动态优化算法，这取决于一个新的理论，为这些问题产生凸松弛。虽然这一理论在许多情况下都能很好地工作，但仍然有大量的问题需要解决，因为这些弛豫太弱了。这项工作涉及的理论发展和有效的数值实现更好的技术产生凸松弛。对于一般的动态优化问题，最近开发了一种新的理论，它可以在许多情况下提供更严格的松弛，但仍然存在许多剩余的挑战，关于实现这一理论，以产生一个完整的算法。这些挑战的主题之一的任务，将在这个项目中进行。所有的技术，放松动态优化问题需要在一个参数范围内的状态变量的边界生成。这些边界的准确性直接反映在所产生的凸松弛的强度。有效地产生更严格的界限比目前可用的方法是第二个具体的研究任务的主题。据发现，重大的改进，可以潜在地作出涉及化学反应系统的优化问题的特定情况下，由于独立已知的物理信息的可用性。这包括一个非常大和重要的化学工程应用类别。然而，充分利用这些信息，提出了一些有趣的理论和计算问题，仍有待解决。最后，我们将研究一种通过简单的一阶泰勒模型以一种新的方式增强我们现有的方法来生成一般动态优化问题的紧界的方法。更广泛的影响解决实际规模的动态优化问题的能力可能会对化学过程工业产生重大影响。有效地模拟大规模动态系统的能力已经引起了许多化学过程的详细动态过程模型的发展。然而，优化这些系统的设计，操作或控制目前只能在本地完成，或通过使用不适当的线性近似。解决涉及这种详细的动态模型的全局动态优化问题的能力，将允许适当的处理重要的问题，如正式的安全验证的化学过程，设计的启动和关闭程序，综合的间歇过程，以及集成的设计和控制。此外，人们相信通用全局动态优化技术将在几乎所有科学和工程学科中得到广泛应用。

Paul Barton0933095Intellectual Merit The objective of this research is to develop algorithms for the global solution of dynamic optimization problems of practical size, with a particular interest in chemical engineering applications. In such optimization problems, inputs, parameters and/or initial conditions for a system of differential equations that optimize a specified performance metric are sought. This type of optimization problem is ubiquitous in engineering disciplines due to the widespread use of differential equations to model systems of interest. For chemical engineering applications, it is widely known that dynamic optimization problems nearly always exhibit multiple suboptimal local minima. Most dynamic optimization algorithms can only guarantee convergence to local minima, which can have serious economic and environmental consequences, and may also result in unsafe operating conditions, when implemented on real systems. To date, there are roughly three algorithms capable of solving general dynamic optimization problems to guaranteed global optimality - all of these are restricted to problems with fewer than about five degrees of freedom and five state variables due to excessive computational cost, and therefore cannot be used to address many practical chemical engineering problems. Thus, the primary objective of this work is to develop more efficient algorithms for solving dynamic optimization problems globally.Under previous NSF support, the PI and coworkers developed a rigorous global dynamic optimization algorithm, which hinges on a novel theory for generating convex relaxations for these problems. Though this theory works well in many cases, there remain large classes of problems for which these relaxations are too weak. This work involves the theoretical development and efficient numerical implementation of better techniques for generating convex relaxations. For general dynamic optimization problems, a new theory has recently been developed which can provide tighter relaxations in many cases, yet there are many remaining challenges regarding implementation of this theory to produce a complete algorithm. These challenges are the topic of one of the tasks that will be undertaken in this project.All techniques to relax dynamic optimization problems require the generation of bounds on the state variables over a range of parameters. The accuracy of these bounds is directly reflected in the strength of the convex relaxations generated. Methods for efficiently generating tighter bounds than are currently available are the subject of the second specific research tasks. It is found that major improvements can potentially be made for the specific case of optimization problems involving chemically reacting systems, due to the availability of independently known physical information. This includes a very large and important class of chemical engineering applications. However, exploiting this information to its fullest extent presents several interesting theoretical and computational questions that remain to be addressed. Finally, a method for generating tight bounds for general dynamic optimization problems by augmenting our existing method with simple first-order Taylor models in a novel manner will be studied.Broader Impact The ability to solve dynamic optimization problems of practical size could have a major impact on the chemical process industry. The ability to simulate large-scale dynamic systems efficiently has caused an increase in the development of detailed dynamic process models for many chemical processes. Yet optimizing these systems for design, operation or control can currently only be done locally, or through the use of inappropriate linear approximations. The ability to solve globally dynamic optimization problems involving such detailed dynamic models would allow appropriate treatment of important problems such as formal safety verification of chemical processes, design of start-up and shut-down procedures, synthesis of integrated batch processes, and the integration of design and control. Furthermore, it is believed that generic global dynamic optimization techniques will find widespread use in nearly all science and engineering disciplines.