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

TL;DR

This paper introduces a novel non-interior-point continuation method for solving optimal control problems with equilibrium constraints, effectively addressing numerical difficulties without requiring interior feasible points, and demonstrating improved efficiency over traditional interior-point methods.

Contribution

The paper proposes a two-stage non-interior-point continuation method that regularizes the KKT matrix and tracks solutions efficiently, offering a new approach to solving challenging OCPEC problems.

Findings

The method accurately tracks solution trajectories.

It requires less computation time than interior-point methods.

Convergence properties are theoretically analyzed.

Abstract

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.