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In this study, we propose a numerical solution method for the optimal controlproblem with equilibrium constraints (OCPEC). OCPEC can be discretized into aperturbed finite-dimensional nonlinear programming (NLP) problem and furtherrelaxed to satisfy the fundamental constraint qualifications. Therefore, itssolution can be obtained by the continuation method that solves a sequence ofwell-posed NLP problems. However, numerical difficulties that the interior ofthe feasible region shrinks toward the disjunctive and empty set, arise whenthe perturbed parameter is close to zero. Hence, we propose a dedicated NLPsolver that maps the Karush--Kuhn--Tucker conditions into a system ofequations, referred to as the non-interior-point (NIP) method. Compared withactive-set methods, the NIP method considers all inequality constraints at eachiteration, which is similar to the interior-point (IP) method and suitable forlarge-scale problems. However, it does not enforce all iterates to remain inthe feasible interior, leading to a larger stepsize to mitigate the geometricaldifficulties of the feasible region. Moreover, we introduce some numericaltechniques to improve the solver performance. The local convergence andsolution error are analyzed under certain assumptions. Comparisons with anoff-the-shelf IP solver demonstrate that the proposed method is capable offinding a better solution, less sensitive to the ill-posedness of the feasibleregion, and requires significantly fewer iterations.

A Non-Interior-Point Continuation Method for the Optimal Control Problem with Equilibrium Constraints