On minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems

Motivated by the work of Fukushima and Pang (Ref. 1), we study the equivalent relationship between minimizing and stationary sequences of a new class of merit functions for nonlinear complementarity problems (NCP). These merit functions generalize that obtained via the squared Fischer-Burmeister NCP function, which was used in Ref. 1. We show that a stationary sequence {x^k} ⊂ ℜⁿ is a minimizing sequence under the condition that the function value sequence {F(x^k)} is bounded above or the Jacobian matrix sequence {F′(x^k)} is bounded, where F is the function involved in NCP. The latter condition is also assumed by Fukushima and Pang. The converse is true under the assumption of {F′(x^k)} bounded. As an example shows, even for a bounded function F, the boundedness of the sequence {F′(x^k)} is necessary for a minimizing sequence to be a stationary sequence.