A note on the dislocation hyperbolic transformation function

In work of Roman A. Polyak [3], the Modified Chen-Harker- Kanzow-Smale (CHKS) function was studied to relate a multiplier method and a Interior Prox method with the second order distance function. Independently, the dislocated hyperbolic penalty function (DHPF) was proposed by A.E. Xavier (1992). DHPF was rewritten and studied in [1] and [2]. Thus, this function was called the dislocated hyperbolic function (DHF). In this work, we note that DHF is a particular case of CHKS function. Then we will call the DHF function as the dislocation hyperbolic transformation function.