  In this study, we focus on the numerical solution method for the optimal
control problem with equilibrium constraints (OCPEC).It is extremely
challenging to solve OCPEC owing to the absence of constraint regularity and
strictly feasible interior points. To solve OCPEC efficiently, we first relax
the discretized OCPEC to recover the constraint regularity and then map its
Karush--Kuhn--Tucker (KKT) conditions into a perturbed system of equations.
Subsequently, we propose a novel two-stage solution method, called the
non-interior-point continuation method, to solve the perturbed system. In the
first stage, a non-interior-point method, which solves the perturbed system
using the Newton method and globalizes convergence using a dedicated merit
function, is employed. In the second stage, a predictor-corrector continuation
method is utilized to track the solution trajectory as a function of the
perturbed parameter, starting at the solution obtained in the first stage. The
proposed method regularizes the KKT matrix and does not enforce iterates to
remain in the feasible interior, which mitigates the numerical difficulties of
solving OCPEC. Convergence properties are analyzed under certain assumptions.
Numerical experiments demonstrate that the proposed method can accurately track
the solution trajectory while demanding significantly less computation time
compared to the interior-point method.
