Abstract
In the last two decades, the emergence of big data mathematical modeling leads to the appearance of large- to huge-scale optimization problems with special structures. Moreover, a wide range of such practical problems does not have Lipschitz (Holder) continuous derivatives. Due to this and the existence of a huge number of data, the classical optimization methods cannot be applied to these types of problems, which increase the demand for new algorithmic developments that are convergent and also computationally reasonable for solving these structured optimization problems. As such, designing, analyzing, and implementing efficient optimization algorithms for nonsmooth and nonconvex problems is the subject of investigation in this proposal. In the other words, we assume that some parts of the objective functions are (high-order) relatively smooth and develop first-, second-, and high-order non-Euclidean methods using generalized Bregman distances to find approximate (high-order) critical points of the objective functions. In addition, we analyze the evaluation complexity of these non-Euclidean methods, which is used as a measure of efficiency. We finally apply our developed algorithms to many applications from signal and image processing, machine learning, data science, and systems biology.
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