Quasidifferential analysis of positively homogeneous functions

Abstract

This paper deals with nonsmooth real continuous positively homogeneous functions and looks for their local first order approximations, suitable for optimization methods.First, the examined functions are supposed directionally differentiable, and necessary and sufficient conditions are stated for the directional derivatives to be continuous. That is proved for f Rⁿ→R in the center of homogeneity and for f:Rⁿ→R everywhere. Therefore these functions may be approximated by quasidifferentials.

Then non directionally differentiable functions are analyzed by showing that their Dini derivatives, whenever continuous, may be expressed in terms of some generalized quasidifferentials. After proving this property within the class of homogeneous functions, having finite Dini derivatives, for these functions as well as for lipschitzian ones, the way to derive generalized quasidifferentials is shown.Finally, for all the examined cases, optimality necessary conditions are formulated and steepest search directions defined

Keywords:

• Nonsmooth optimization
• Positively homogeneous functions
• Quasidifferential
• First order approximation

†Research supported by M.U.R.S.T. and by C.N.R. (P. F. Sistemi informatici e Calcolo Parallelo)

†Research supported by M.U.R.S.T. and by C.N.R. (P. F. Sistemi informatici e Calcolo Parallelo)

Notes

†Research supported by M.U.R.S.T. and by C.N.R. (P. F. Sistemi informatici e Calcolo Parallelo)