TY - JOUR
T1 - Analysis of direct searches for discontinuous functions
AU - Custódio, Ana Luísa da Graça Batista
PY - 2012/1/1
Y1 - 2012/1/1
N2 - 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.
AB - 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.
KW - Directionally Lipschitz
KW - Lower semicontinuity
KW - Discontinuity
KW - Nonsmooth calculus
KW - Lipschitz extensions
KW - Direct-search methods
KW - Generalized directional derivatives
U2 - 10.1007/s10107-010-0429-8
DO - 10.1007/s10107-010-0429-8
M3 - Article
SN - 0025-5610
VL - 133
SP - 299
EP - 325
JO - Mathematical Programming
JF - Mathematical Programming
IS - 1-2
ER -