Abstract: In this paper, we propose a new class of penalty functions for solving nonlinear programming problems with inequality constraints, a subclass of which smoothly approximates the $l_1$ penalty function. Based on the new class of penalty functions, we consider a penalty algorithm, the characteristic of which is at each iteration, an exact global optimal solution or an inexact global optimal solution is obtained. Under very weak conditions, the algorithm is always applicable. We present the global convergence without any constraint qualification. Finally, numerical experiments are given.