The H-differentiability and calmness of circular cone functions

Let $$\mathcal{L}_{\theta }$$ L θ be the circular cone in $${\mathbb {R}}^n$$ R n which includes second-order cone as a special case. For any function f from $${\mathbb {R}}$$ R to $${\mathbb {R}}$$ R , one can define a corresponding vector-valued function $$f^{\mathcal{L}_{\theta }}$$ f L θ on $${\mathbb {R}}^n$$ R n by applying f to the spectral values of the spectral decomposition of $$x \in {\mathbb {R}}^n$$ x ∈ R n with respect to $$\mathcal{L}_{\theta }$$ L θ . The main results of this paper are regarding the H-differentiability and calmness of circular cone function $$f^{\mathcal{L}_{\theta }}$$ f L θ . Specifically, we investigate the relations of H-differentiability and calmness between f and $$f^{\mathcal{L}_{\theta }}$$ 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. Copyright Springer Science+Business Media New York 2015