  There are many significant applied contexts that require the solution of
discontinuous optimization problems in finite dimensions. Yet these problems
are very difficult, both computationally and analytically. With the functions
being discontinuous and a minimizer (local or global) of the problems, even if
it exists, being impossible to verifiably compute, a foremost question is what
kind of ''stationary solutions'' one can expect to obtain; these solutions
provide promising candidates for minimizers; i.e., their defining conditions
are necessary for optimality. Motivated by recent results on sparse
optimization, we introduce in this paper such a kind of solution, termed
''pseudo B- (for Bouligand) stationary solution'', for a broad class of
discontinuous piecewise continuous optimization problems with objective and
constraint defined by indicator functions of the positive real axis composite
with functions that are possibly nonsmooth. We present two approaches for
computing such a solution. One approach is based on lifting the problem to a
higher dimension via the epigraphical formulation of the indicator functions;
this requires the addition of some auxiliary variables. The other approach is
based on certain continuous (albeit not necessarily differentiable) piecewise
approximations of the indicator functions and the convergence to a pseudo
B-stationary solution of the original problem is established. The conditions
for convergence are discussed and illustrated by an example.
