  We generalize the successive continuation paradigm introduced by Kern\'evez
and Doedel [16] for locating locally optimal solutions of constrained
optimization problems to the case of simultaneous equality and inequality
constraints. The analysis shows that potential optima may be found at the end
of a sequence of easily-initialized separate stages of continuation, without
the need to seed the first stage of continuation with nonzero values for the
corresponding Lagrange multipliers. A key enabler of the proposed
generalization is the use of complementarity functions to define relaxed
complementary conditions, followed by the use of continuation to arrive at the
limit required by the Karush-Kuhn-Tucker theory. As a result, a successful
search for optima is found to be possible also from an infeasible initial
solution guess. The discussion shows that the proposed paradigm is compatible
with the staged construction approach of the COCO software package. This is
evidenced by a modified form of the COCO core used to produce the numerical
results reported here. These illustrate the efficacy of the continuation
approach in locating stationary solutions of an objective function along
families of two-point boundary value problems and in optimal control problems.
