  A common theme in mathematics is to define generalized solutions to deal with
problems that potentially do not have solutions. A classical example is the
introduction of least squares solutions via the normal equations associated
with a possibly infeasible system of linear equations. In this paper, we
introduce a "normal problem" associated with finding a zero of the sum of two
maximally monotone operators. If the original problem admits solutions, then
the normal problem returns this same set of solutions. The normal problem may
yield solutions when the original problem does not admit any; furthermore, it
has attractive variational and duality properties. Several examples illustrate
our theory.
