The H-differentiability and calmness of circular cone functions

Abstract

Let L θ be the circular cone in R n which includes second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f L θ on R n by applying f to the spectral values of the spectral decomposition of x ∈ R n with respect to L θ . The main results of this paper are regarding the H -differentiability and calmness of circular cone function f L θ . Specifically, we investigate the relations of H -differentiability and calmness between f and f L θ . In addition, we propose a merit function approach for solving the circular cone complementarity problems under H -differentiability. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.

On the calm b-differentiability of projector onto circular cone and its applications

“…Recently, generalizations of second-order cones and second-order cone complementarity sets have been examined by many authors 5,[14][15][16][17][18][19][20][21][22] . For example, authors in 14,[19][20][21][22] considered circular cones, which are generalizations of second-order cones and are, in general, nonsymmetric and non-self-dual cones. The generalized differentiability of the projection operator onto the circular cone was provided in 14,22 .…”

“…The generalized differentiability of the projection operator onto the circular cone was provided in 14,22 . Moreover, the differentiability and calmness of vectorvalued functions associated with the circular cone were also studied in 19,23 . In particular, authors in 21 showed that the results of the projection operator onto a circular cone could not be shown by simply resorting to the results of the projection operator onto the second-order cone, and hence, it is necessary to study the results of circular cone directly.…”