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Essentials of numerical nonsmooth optimization

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Abstract

Approximately 60 years ago two seminal findings, the cutting plane and the subgradient methods, radically changed the landscape of mathematical programming. They provided, for the first time, the practical chance to optimize real functions of several variables characterized by kinks, namely by discontinuities in their derivatives. Convex functions, for which a superb body of theoretical research was growing in parallel, naturally became the main application field of choice. The aim of the paper is to give a concise survey of the key ideas underlying successive development of the area, which took the name of numerical nonsmooth optimization. The focus will be, in particular, on the research mainstreams generated under the impulse of the two initial discoveries.

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Notes

  1. Given the features of the adopted machinery, we keep on denoting the current iterate (i.e., the estimate of a minimizer) by \(\overline{{\mathbf {x}}}_k\), although the methods involved in this class are not necessarily of the bundle type.

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  1. DIMES, Università della Calabria, 87036, Rende, CS, Italy

    Manlio Gaudioso, Giovanni Giallombardo & Giovanna Miglionico

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Gaudioso, M., Giallombardo, G. & Miglionico, G. Essentials of numerical nonsmooth optimization. 4OR-Q J Oper Res 18, 1–47 (2020). https://doi.org/10.1007/s10288-019-00425-x

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