Lecture WS 16/17 Advanced Topics in Numerical Methods in Science and Technology

Lecturers

Dr. Seyedehsomayeh Hosseini and Prof. André Uschmajew

Nonsmooth optimization is devoted to the general problem of minimizing functions that are typically not differentiable at their minimizers. In order to optimize nonsmooth functions, the classical theory of optimization cannot be directly used due to lacking certain differentiability and strong regularity conditions. However, because of the complexity of the real world, functions used in practical applications are often nonsmooth. Nonsmooth problems appear in many fields of applications, for example in image denoising, optimal control, neural network training, data mining, economics and computational chemistry and physics, therefore it is of eminent importance to be able to optimize a nonsmooth function. This course aims to provide a basic working understanding of nonsmooth analysis and its applications in optimization. Intended topics include:

• Nonsmooth differential theory
• Tangent and Normal cones
• Nonsmooth optimization theory
• Bundle methods
• Quasi-newton methods

Recommended literature:

• Marko. M. Mäkelä, Pekka. Neittaanmäki, Nonsmooth Optimization; Analysis and Algorithms with Applications to Optimal Control, World Scientific, 1992.
• Frank H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial and Applied Mathematics, 1990.
• Boris. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, Grundlehren Series, 331, Springer, New York (2006).
• F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics 178, Springer, New York (1998).

Prerequisites: analysis and linear algebra (Bachelor level), smooth optimization methods (Steepest descent methods, Newton methods etc.).