  In this article we develop a general theory of exact parametric penalty
functions for constrained optimization problems. The main advantage of the
method of parametric penalty functions is the fact that a parametric penalty
function can be both smooth and exact unlike the standard (i.e. non-parametric)
exact penalty functions that are always nonsmooth. We obtain several necessary
and/or sufficient conditions for the exactness of parametric penalty functions,
and for the zero duality gap property to hold true for these functions. We also
prove some convergence results for the method of parametric penalty functions,
and derive necessary and sufficient conditions for a parametric penalty
function to not have any stationary points outside the set of feasible points
of the constrained optimization problem under consideration. In the second part
of the paper, we apply the general theory of exact parametric penalty functions
to a class of parametric penalty functions introduced by Huyer and Neumaier,
and to smoothing approximations of nonsmooth exact penalty functions. The
general approach adopted in this article allowed us to unify and significantly
sharpen many existing results on parametric penalty functions.
