UBC Theses and Dissertations
Approximate subdifferential of piecewise linear-quadratic functions Bajaj, Anuj
Optimization is a branch of mathematics dealing with the selection of the best element(s) (based on some criteria) from a set of available options. In the study of optimization, we often come across problems involving functions that are not differentiable, thus, one cannot employ the tool of differentiation to solve or analyze such functions. Such an issue can be resolved with the help of subdifferentials. Subgradients generalize derivatives to nondifferentiable functions, which makes them one of the most useful and powerful instruments in nonsmooth optimization. However, they may come with a few limitations. Some limitations may be overcome by using approximate subdifferentials, which are a certain relaxation of true subdifferentials. In this thesis, we present a general algorithm to compute approximate subdifferential of any proper convex function. We then implement the algorithm for the family of convex univariate piecewise linear-quadratic functions - an important model class of functions for nonlinear systems. We provide many numerical examples. Finally, some directions and insights for future work are detailed.
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