Title: Analysis of direct searches for non-Lipschitzian functions
Authors: Vicente, L. N. 
Custódio, A. L. 
Keywords: Direct-search methods;Directionally Lipschitzian functions;Discontinuities;Lower semicontinuity;Rockafellar directional derivatives;Nonsmooth calculus;Lipschitz extensions
Issue Date: 2009
Publisher: Centro de Matemática da Universidade de Coimbra
Citation: Pré-Publicações DMUC. 09-38 (2009)
Abstract: It is known that the Clarke generalized directional derivative is nonnegative along the limit directions generated by directional direct-search methods at a limit point of certain subsequences of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point. In this paper we generalize this result for non-Lipschitzian functions using Rockafellar generalized directional derivatives (upper subderivatives). We show that Rockafellar derivatives are also nonnegative along the limit directions of those subsequences of unsuccessful iterates when the function values converge to the function value at the limit point. This result is obtained assuming that the function is directionally Lipschitzian with respect to the limit direction. It is also possible under appropriate conditions to establish more insightful results by showing that the sequence of points generated by these methods eventually approaches the limit point along the locally best branch or step function (when the number of steps is equal to two). The results of this paper are presented for constrained optimization and illustrated numerically.
URI: http://hdl.handle.net/10316/13641
Rights: openAccess
Appears in Collections:FCTUC Matemática - Vários

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