Convex Underestimators for Dynamic Optimization Problems

Barton研究所：MITProposal number：0120441本研究的目标是为非凸动态嵌入优化问题、混合整数动态优化问题和非凸变分问题发展确定性全局优化算法。研究：在第一个研究任务中，PI将为具有非常一般的积分目标函数和嵌入线性时变动态系统的最优化问题发展凸性理论和凸低估估计器。有了这些理论基础，就有可能将现有的确定性全局优化算法应用于非凸非线性规划和混合整数非线性规划，从而也可以解决混合整数动态优化问题。接下来，PI将探索如何将这种复合函数方法扩展到具有嵌入的非线性动态系统的问题。这一扩展将在很大程度上借鉴为线性情况开发的理论。在最后的研究任务中，他将探索变分和最优控制问题的凸低估估计的构造，即函数的线性空间上的凸低估估计。在这些变分凸低估算子的应用中存在着许多主要的研究挑战，包括从凸低估问题生成严格下界的数值生成，以及如何在例如分支定界过程中划分函数的线性空间。影响：该研究将通过发展动态嵌入优化问题的凸性理论，以及发展动态嵌入优化问题和变分问题的凸低估估计构造方法，做出基础性贡献。这一理论将为这些问题的求解提供一系列实用的确定性全局优化算法。解决此类问题以保证全球最优的能力将具有广泛的实际意义。例如，在工艺操作领域，使用详细的动态模型，有望解决诸如正式安全验证、综合批处理工艺以及设计主要工艺瞬变过程(如启动和关闭程序)等问题。大多数严重的工业事故都发生在这种短暂的事件中。此外，该方法还将适用于其他工程学科以及应用数学。

ABSTRACTPI: Paul I. Barton Institution: MITProposal Number: 0120441The objective of this research is to develop deterministic global optimization algorithms for nonconvex dynamic embedded optimization problems, mixed-integer dynamic optimization problems and nonconvex variational problems. The approaches need to be practically implementable and they should provide theoretical guarantees of locating the global optimum.Research: In the first research task, the PI will develop a convexity theory and convex underestimators for optimization problems with a very general integral objective function and linear time varying dynamic system embedded. With these theoretical foundation, it will be possible to adapt existing deterministic global optimization algorithms for nonconvex nonlinear programs and mixed-integer nonlinear program, hence also addressing mixed-integer dynamic optimization problems. Next, the PI will explore how this composite function approach can be extended to problems with nonlinear dynamic systems embedded. This extension will draw heavily on the theory developed for the linear case. In the final research task, he will explore the construction of convex underestimators for variational and optimal control problems, i.e., convex underestimators on linear spaces of functions. A number of major research challenges exist in the application of these variational convex underestimators, including the numerical generation of rigorous lower bounds from the convex underestimating problems, and how to partition a linear space of function in, for example, a branch-and-bound procedure.Impact: The research will make fundamental contributions via developing a convexity theory for dynamic embedded optimization problems, and developing methods for the construction of convex underestimators for dynamic embedded optimization problems and variational problems. This theory will lead to a series of practical deterministic global optimization algorithms for the solution of these problems. The capability to solve such problems to guaranteed global optimality will have broad practical implications. For example, in the area of process operations there is hope for solving problems such as formal safety verification, 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. Most serious industrial accidents occur during such transient events. Furthermore, the method will have applicability in other engineering disciplines as well as applied mathematics.