Abstract
Second-order methods play an important role in the theory of optimization. Due to the usage of more information about considered function, they give an opportunity to find the stationary point faster than first-order methods. Well-known and sufficiently studied Newton’s method is widely used to optimize smooth functions. The aim of this work is to obtain a second-order method for unconstrained minimization of nonsmooth functions allowing convex quadratic approximation of the augment. This method is based on the notion of coexhausters—new objects in nonsmooth analysis, introduced by V. F. Demyanov. First, we describe and prove the second-order necessary condition for a minimum. Then, we build an algorithm based on that condition and prove its convergence. At the end of the paper, a numerical example illustrating implementation of the algorithm is given.
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Acknowledgments
The author dedicates the article to the blessed memory of Professor Vladimir Fedorovich Demyanov, whose ideas led to the appearance of this work. The work is supported by the Saint-Petersburg State University under Grant No. 9.38.205.2014.
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Abbasov, M.E. Second-Order Minimization Method for Nonsmooth Functions Allowing Convex Quadratic Approximations of the Augment. J Optim Theory Appl 171, 666–674 (2016). https://doi.org/10.1007/s10957-015-0796-7
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DOI: https://doi.org/10.1007/s10957-015-0796-7