We generalize the successive continuation paradigm introduced by Kernévez and Doedel [1] 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 optimal solutions of an objective function along families of two-point boundary value problems and in optimal control problems.