  In this paper we consider the minimization of a continuous function that is
potentially not differentiable or not twice differentiable on the boundary of
the feasible region. By exploiting an interior point technique, we present
first- and second-order optimality conditions for this problem that reduces to
classical ones when the derivative on the boundary is available. For this type
of problems, existing necessary conditions often rely on the notion of
subdifferential or become non-trivially weaker than the KKT condition in the
(twice-)differentiable counterpart problems. In contrast, this paper presents a
new set of first- and second-order necessary conditions that are derived
without the use of subdifferential and reduces to exactly the KKT condition
when (twice-)differentiability holds. As a result, these conditions are
stronger than some existing ones considered for the discussed minimization
problem when only non-negativity constraints are present. To solve for these
optimality conditions in the special but important case of linearly constrained
problems, we present two novel interior trust-region point algorithms and show
that their worst-case computational efficiency in achieving the potentially
stronger optimality conditions match the best known complexity bounds. Since
this work considers a more general problem than the literature, our results
also indicate that best known complexity bounds hold for a wider class of
nonlinear programming problems.
