Let L be the circular cone in R n which includes a second-order cone as a special case. For any function f from R to R, one can define a corresponding vector-valued function f c (x) on R n by applying f to the spectral values of the spectral decomposition of x R n with respect to L . We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, directional differentiability, Frchet differentiability, continuous differentiability, as well as semismoothness. These results will play a crucial role in designing solution methods for optimization problem associated with the circular cone.

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In this paper, we consider a class of penalized NCP-functions, which includes several existing well-known NCP-functions as special cases. The merit function induced by this class of NCP-functions is shown to have bounded level sets and provide error bounds under mild conditions. A derivative free algorithm is also proposed, its global convergence is proved and numerical performance compared with those based on some existing NCP-functions is reported.

In this article, we extend two classes of merit functions for the second-order complementarity problem (SOCP) to infinite-dimensional SOCP. These two classes of merit functions include several popular merit functions, which are used in nonlinear complementarity problem, (NCP)/(SDCP) semidefinite complementarity problem, and SOCP, as special cases. We give conditions under which the infinite-dimensional SOCP has a unique solution and show that all these merit functions provide an error bound for infinite-dimensional SOCP and have bounded level sets. These results are very useful for designing solution methods for infinite-dimensional SOCP.

Necessary conditions of optimality are derived for multiobjective optimal control problems with pathwise state constraints, in which the dynamics constrain is modeled as a differential inclusion. The obtained result extends results of [2] and [29].