A novel generalization of the natural residual function and a neural network approach for the NCP

The natural residual (NR) function is a mapping often used to solve nonlinear complementarity problems (NCPs). Recently, three discrete-type families of complementarity functions with parameter p⩾3 (where p is odd) based on the NR function were proposed. Using a neural network approach based on these families, it was observed from some preliminary numerical experiments that lower values of p provide better convergence rates. Moreover, higher values of p require larger computational time for the test problems considered. Hence, the value p=3 is recommended for numerical simulations, which is rather unfortunate since we cannot exploit the wide range of values for the parameter p of the family of NCP functions. This paper is a follow-up study on the aforementioned results. Motivated by previously reported numerical results, we formulate a continuous-type generalization of the NR function and two corresponding symmetrizations. The new families admit a continuous parameter p>0, giving us a wider range of choices for p and smooth NCP functions when p>1. Moreover, the generalization subsumes the discrete-type generalization initially proposed. The numerical simulations show that in general, increased stability and better numerical performance can be achieved by taking values of p in the interval (1,3). This is indeed a significant improvement of preceding studies.