  In this paper we are concerned with solving monotone inclusion problems
expressed by the sum of a set-valued maximally monotone operator with a
single-valued maximally monotone one and the normal cone to the nonempty set of
zeros of another set-valued maximally monotone operator. Depending on the
nature of the single-valued operator, we will propose two iterative penalty
schemes, both addressing the set-valued operators via backward steps. The
single-valued operator will be evaluated via a single forward step if it is
cocoercive, and via two forward steps if it is monotone and Lipschitz
continuous. The latter situation represents the starting point for dealing with
complexly structured monotone inclusion problems from algorithmic point of
view.
