Analysis of direct searches for discontinuous functions

作者：Vicente L N^*; Custodio A L

来源：Mathematical Programming, 2012, 133(1-2): 299-325.

DOI：10.1007/s10107-010-0429-8

• Direct-search methods
• Discontinuity
• Directionally Lipschitz
• Lower semicontinuity
• Generalized directional derivatives
• Nonsmooth calculus
• Lipschitz extensions

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 when the number of steps is equal to two). The results of this paper are presented for constrained optimization and illustrated numerically.

• 出版日期2012-6