Geodesic
Extremum conditions for a~nonsmooth function in terms of exhausters and coexhausters
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 10-19

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The notions of upper and lower exhausters and coexhausters are discussed and necessary conditions for an unconstrained extremum of a nonsmooth function are derived. The necessary minimum conditions are formulated in terms of an upper exhauster (coexhauster) and the necessary maximum conditions are formulated in terms of a lower exhauster (coexhauster). This involves the problem of transforming an upper exhauster (coexhauster) into a lower exhauster (coexhauster) and vice versa. The transformation is carried out by means of a conversion operation (converter). Second-order approximations obtained with the help of second-order (upper and lower) coexhausters are considered. It is shown how a second-order upper coexhauster can be converted to a lower coexhauster and vice versa. This problem is reduced to using a first-order conversion operator but in a space of a higher dimension. The obtained result allows one to construct second-order methods for the optimization of nonsmooth functions (Newton-type methods).
Keywords: nonsmooth analysis, nondifferentiable optimization, exhauster, coexhauster
Mots-clés : converter. 
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     title = {Extremum conditions for a~nonsmooth function in terms of exhausters and coexhausters},
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M. E. Abbasov; V. F. Demyanov. Extremum conditions for a~nonsmooth function in terms of exhausters and coexhausters. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 10-19. http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a1/