Mathematical Sciences: Dynamical Systems, Complexity and Optimization

将分析动力系统在优化问题中的作用。 将特别注意不等式约束的优化问题。 其中要考虑的问题是：半定规划，非凸问题的多面体，一般非线性凸数学规划问题，线性二次控制问题与二次约束。 相应的动力系统的哈密顿结构及其与约束力学问题的关系将被研究。还将考虑动力系统的定性性质对障碍函数的选择的依赖性。 各种方法的基础上的动力系统的算法建设进行了比较。 半定问题，措施空间中的优化问题，和无限维二次问题的新方法将被建议。 作为该项目的结果，预计将取得实质性的进展，对建设的边界点算法的上述类别的问题。更重要的是，相邻点技术和序列二次规划方法的结合将导致有效的算法来解决广泛的最优控制问题的状态空间和控制不等式约束。 控制理论在从导弹制导到激光CD调谐的实际问题中有着广泛的应用。 控制策略的选择是由可用资源和性能指标决定的，这是很自然的。 这就是所谓的最优控制理论的主要范式。直到最近，只有很少的最优控制问题可以在真实的时间解决。所谓的邻域点算法的快速发展完全改变了这种情况。现在很有可能解决最优控制和其他涉及不等式约束的无限维问题。机器人技术（包括运动规划、灵巧抓取力优化）和量子过程控制（对新材料的开发很重要）是可以使用这种技术的问题的很好例子。相邻点算法的令人兴奋的方面是，它们的效率随着问题的大小而增长。 在这一建议中，一个程序的发展边界点算法的建议，这将无法将这种方法应用到广泛的一类最优控制问题。 方法概括的走廊点算法，同时保持其效率（复杂性估计测量）。 动力系统理论为工程的实现提供了重要的理论工具。其结果是，它是预期的，有效的邻点算法适合于最优控制的应用将被开发。 将获得这些算法的复杂性估计。

The role of dynamical systems arising in connection with optimization problems will be analyzed. Special attention will be paid to optimization problems with inequality constraints. Among problems to be considered are: semidefinite programming, nonconvex problems on polytopes, general nonlinear convex mathematical programming problems, and linear-quadratic control problems with quadratic constraints. The Hamiltonian structure of corresponding dynamical systems and its relation to the problems of constraint mechanics will be studied. The dependence of qualitative properties of dynamical systems on the choice of barrier functions will also be considered. Various approaches to the construction of algorithms based on dynamical systems are compared. New approaches to semidefinite problems, optimization problems in spaces of measures, and infinite-dimensional quadratic problems will be suggested. As a result of the project, it is expected that substantial progress will be made toward construction of interior-point algorithms for the above mentioned classes of problems. More im portantly, the combination of interior-point techniques and sequential quadratic programming approaches will lead to efficient algorithms for solving broad classes of optimal control problems with both state space and control inequality constraints. Control theory has numerous applications to practical problems ranging from missile guidance to laser CD tuning. It is quite natural to expect that the choice of control strategy is determined by the available resources and the performance index. This is the main paradigm of so-called optimal control theory. Until very recently only few optimal control problems could be solved in real time. The very fast development of so-called interior-point algorithms completely changes the situation. It is now quite possible to address optimal control and other infinite-dimensional problems involving inequality constraints. Robotics (including motion planning, dexterous grasping force optimization) and control of quantum processes (important for the development of new materials) are good examples of problems where such techniques can be used. The exciting aspect of interior-point algorithms is that their efficiency grows with the size of the problem. In this proposal, a program of development interior-point algorithms is suggested which would unable one to apply this methodology to a broad class of optimal control problems. Ways are outlined to generalize interior-point algorithms while keeping their efficiency (measured by complexity estimates). Dynamical system theory provides an important theoretical tool for the realization of the project. As a result, it is expected that the efficient interior-point algorithms suitable for optimal control applications will be developed. Complexity estimates will be obtained fot these algorithms.