Recently this author studied several merit functions systematically for the second-order cone complementarity problem. These merit functions were shown to enjoy some favorable properties, to provide error bounds under the condition of strong monotonicity, and to have bounded level sets under the conditions of monotonicity as well as strict feasibility. In this paper, we weaken the condition of strong monotonicity to the so-called uniform <i>P</i> ^*-property, which is a new concept recently developed for linear and nonlinear transformations on Euclidean Jordan algebra. Moreover, we replace the monotonicity and strict feasibility by the so-called <i>R</i> ₀₁ or <i>R</i> ₀₂-functions to keep the property of bounded level sets.

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.

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 the paper [4], the authors proposed a derivative-free descent algorithm for the nonlinear complementarity problems (NCPs) by the generalized Fischer-Burmeister merit function: p (a, b)= 1

We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order <i>n</i> 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order <i>n</i> 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.