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 discontinuous 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 Lipschitz 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.
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Support for L. N. Vicente’s research was provided by FCT under the grants PTDC/MAT/64838/2006 and PTDC/MAT/098214/2008. Part of this work was developed while this author was visiting the Courant Institute of Mathematical Sciences of New York University under a FLAD scholarship.
Support for A. L. Custódio was also provided by Centro de Matemática e Aplicações da Universidade Nova de Lisboa.
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Vicente, L.N., Custódio, A.L. Analysis of direct searches for discontinuous functions. Math. Program. 133, 299–325 (2012). https://doi.org/10.1007/s10107-010-0429-8
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DOI: https://doi.org/10.1007/s10107-010-0429-8
Keywords
- Direct-search methods
- Discontinuity
- Directionally Lipschitz
- Lower semicontinuity
- Generalized directional derivatives
- Nonsmooth calculus
- Lipschitz extensions