Approximately sixty 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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