Computer-Aided Nonsmooth Analysis and Applications

计算机辅助非光滑分析和应用研究计划旨在构建新的计算算法，并提高我们对确定性优化算法的理解。新算法将应用于曲面拟合、保形插补等实际问题。随着计算机辅助非光滑分析软件的发布，预期结果将提高研究人员的生产率，提供更快的优化算法，并改进使用非光滑分析变换的领域的计算，例如热力学、机器人导航、计算机视觉和网络通信。 该研究计划将特别关注通过提供更严格的低估和研究二阶信息来改进确定性优化算法。找到全局最优解是一个非常具有挑战性的问题，但最近的进展使这种计算在许多情况下成为可能。本文的研究将进一步促进这一进展。 将创建一个新的工具箱来处理两个变量的函数，并将应用由此产生的新算法，以在确定性全局优化中提供更严格的界限。运算符(也称为集值函数)的操作和可视化将通过另一个工具箱实现，其最具挑战性的任务将是使用颜色和动画的组合来可视化四个维度的对象。现有的计算机辅助凸分析工具箱将通过计算核平均的新算法来增强，核平均是最近在凸分析中引入的一种变换，它概括了众所周知的凸分析变换。最后，将引入一类新的函数来消除分段线性二次函数的固有限制。

The Computer-Aided Nonsmooth Analysis and Applications research program aims at building new computational algorithms, and improving our understanding of deterministic optimization algorithms. The new algorithms will be applied to practical problems such as surface fitting and shape-preserving interpolation. The anticipated results will increase researchers' productivity with the release of a computer-aided nonsmooth analysis software, provide faster optimization algorithms, and improve calculations in fields that use nonsmooth analysis transforms, e.g., thermodynamics, robot navigation, computer vision, and network communication. The research program will specifically focus on improving deterministic optimization algorithms by providing tighter under-estimates and by studying the second-order information. Finding a global optimum is a very challenging problem but recent progress has made that computation possible in a number of cases. The research herein will further contribute to that progress. A new toolbox will be created to manipulate functions of two variables, and the resulting new algorithms will be applied to provide tighter bounds in deterministic global optimization. The manipulation and visualization of operators (also called set-valued functions) will be made possible through another toolbox, whose most challenging task will be to visualize objects in four dimensions using a combination of colours and animations. The existing computer-aided convex analysis toolbox will be enhanced with a new algorithm to compute the kernel average, a transform recently introduced in convex analysis that generalizes well-known convex analysis transforms. Finally, a new class of functions will be introduced to remove an intrinsic limitation of piecewise linear-quadratic functions.