  Study about theory and algorithms for constrained optimization usually
assumes that the feasible region of the optimization problem is nonempty.
However, there are many important practical optimization problems whose
feasible regions are not known to be nonempty or not, and optimizers of the
objective function with the least constraint violation prefer to be found. A
natural way for dealing with these problems is to extend the constrained
optimization problem as the one optimizing the objective function over the set
of points with the least constraint violation. Firstly, the minimization
problem with least constraint violation is proved to be an Lipschitz equality
constrained optimization problem when the original problem is a convex
optimization problem with possible inconsistent conic constraints, and it can
be reformulated as an MPEC problem. Secondly, for nonlinear programming
problems with possible inconsistent constraints, various types of stationary
points are presented for the MPCC problem which is equivalent to the
minimization problem with least constraint violation, and an elegant necessary
optimality condition, named as L-stationary condition, is established from the
classical optimality theory of Lipschitz continuous optimization. Finally, the
smoothing Fischer-Burmeister function method for nonlinear programming case is
constructed for solving the problem minimizing the objective function with the
least constraint violation. It is demonstrated that, when the positive
smoothing parameter approaches to zero, any point in the outer limit of the
KKT-point mapping is an L-stationary point of the equivalent MPCC problem.
