On epsilon-Differential Mappings and their Applications in Nondifferentiable Optimization

Nurminski, E.A. ORCID: https://orcid.org/0000-0002-7236-6955 (1978). On epsilon-Differential Mappings and their Applications in Nondifferentiable Optimization. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-78-058

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Abstract

In Section 1 we give some review of the recent developments in nondifferential optimization and discuss the difficulties of the application of subgradient methods. It is shown that the use of epsilon-subgradient methods may bring computational advantages.

Section 2 contains the technical results on continuity of epsilon-subdifferentials. The principal result of this section consists in establishing Lipschitz continuity of epsilon-subdifferential mappings.

Section 3 gives some results on convergence of weighted sums of multifunctions. These results will be used in the study of the convergence of epsilon-subgradient method with sequential averages given in Section 4.

Section 4 gives the convergence theory for several modifications of this method. It is shown that in some cases it is possible to neglect accuracy control for the solution of internal maximum problems in the minmax problems. The results when this accuracy is nonzero and fixed are of great practical importance.

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