Computer-Aided Nonsmooth Analysis and Applications

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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.