This study presents a numerical method for the optimal control problem with equilibrium constraints (OCPEC). It is extremely difficult 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 parameterized system of equations. Subsequently, we solve the parameterized system using a novel two-stage solution method called the non-interior-point continuation method. In the first stage, a non-interior-point method is employed to find an initial solution, which solves the parameterized system using Newton’s method and globalizes convergence using a dedicated merit function. In the second stage, a predictor–corrector continuation method is utilized to track the solution trajectory as a function of the parameter, starting at the initial solution. The proposed method regularizes the KKT matrix and does not enforce iterates to remain in the feasible interior, which mitigates the numerical difficulties in solving OCPEC. Convergence properties are analyzed under certain assumptions. Numerical experiments demonstrate that the proposed method can solve OCPEC while demanding remarkably less computation time than the interior-point method.

中文翻译：

---

一种用于平衡约束的最优控制问题的非内点连续法


本研究提出了一种平衡约束最优控制问题 （OCPEC） 的数值方法。由于缺乏约束规则性和严格可行的内部点，求解 OCPEC 非常困难。为了有效地求解 OCPEC，我们首先放宽离散化的 OCPEC 以恢复约束规则性，然后将其 Karush-Kuhn-Tucker （KKT） 条件映射到参数化方程组。随后，我们使用一种称为非内点连续法的新型两阶段求解方法求解参数化系统。在第一阶段，采用非内点方法来寻找初始解，该解使用牛顿方法求解参数化系统，并使用专用评价函数全球化收敛。在第二阶段，使用预测器-校正器延拓法从初始解开始跟踪解轨迹作为参数的函数。所提出的方法对 KKT 矩阵进行正则化，并且不强制迭代以保持在可行的内部，这减轻了求解 OCPEC 的数值困难。收敛属性是在某些假设下分析的。数值实验表明，所提方法可以求解 OCPEC，同时比内点方法需要的计算时间明显减少。