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 solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the <i>P</i>-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan <i>P</i>-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the

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.