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

Kangyu Lin, Toshiyuki Ohtsuka

TL;DR

OCPEC combines optimal control with equilibrium constraints modeled as a differential variational inequality, leading to challenging NLPs with constraint regularity issues. The authors propose a two-stage non-interior-point continuation method that relaxes the equilibrium constraints into a perturbed KKT system solved first by a non-interior-point Newton method and then by a predictor-corrector continuation to $s\to 0$, leveraging a smoothed Fisher–Burmeister function to handle inequalities. They establish nonsingularity of the KKT matrix, global convergence, and local quadratic convergence, along with an error bound in the relaxation and smoothing parameters. Numerical results on Cart Pole with Friction show accurate trajectory tracking with significantly reduced computation time compared to interior-point solvers, and demonstrate the necessity of continuation to preserve correct sensitivity. The approach offers a scalable framework for OCPEC with potential for real-time applications and further real-time embedding improvements.

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.

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

TL;DR

OCPEC combines optimal control with equilibrium constraints modeled as a differential variational inequality, leading to challenging NLPs with constraint regularity issues. The authors propose a two-stage non-interior-point continuation method that relaxes the equilibrium constraints into a perturbed KKT system solved first by a non-interior-point Newton method and then by a predictor-corrector continuation to , leveraging a smoothed Fisher–Burmeister function to handle inequalities. They establish nonsingularity of the KKT matrix, global convergence, and local quadratic convergence, along with an error bound in the relaxation and smoothing parameters. Numerical results on Cart Pole with Friction show accurate trajectory tracking with significantly reduced computation time compared to interior-point solvers, and demonstrate the necessity of continuation to preserve correct sensitivity. The approach offers a scalable framework for OCPEC with potential for real-time applications and further real-time embedding improvements.

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.
Paper Structure (25 sections, 5 theorems, 42 equations, 7 figures, 3 algorithms)

This paper contains 25 sections, 5 theorems, 42 equations, 7 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Contributions
  4. Outline
  5. Notation
  6. Problem Formulation
  7. Variational inequalities
  8. Differential variational inequalities
  9. Optimal control problems with equilibrium constraints
  10. The Proposed Solution Method
  11. Optimality conditions
  12. Treatments of inequality constraints
  13. Stage 1: Non-interior-point solution method
  14. Stage 2: Predictor-corrector continuation method
  15. Convergence Analysis
  16. ...and 10 more sections

Key Result

Theorem 1

Let Assumption assumption: relaxed NLP problem hold and pick $\sigma > 0$. Then, for any given $s \geq 0$, the KKT matrix $\mathcal{K}(\boldsymbol{Y}, p)$ with $p = [s, \sigma]^T$ is nonsingular for any $\boldsymbol{Y} \in \mathbb{R}^{n_Y}$ in the neighborhood of a KKT point $\boldsymbol{Y}^*(s)$.

Figures (7)

  • Figure 1: Relaxed feasible set formed by (\ref{['equation: discretized OCPEC set K']}) -- (\ref{['equation: discretized OCPEC relax bilinear 2']}), with $s = 0.02$, $b_l = -1$, and $b_u = 1$.
  • Figure 2: Parameter sequence $\{ p^j = [s^j, \sigma^j]^T\}^J_{j = 0}$
  • Figure 3: Solution trajectory
  • Figure 4: Comparison of different time steps and methods
  • Figure 5: Comparison of VI natural residual
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 1: Stationarity Properties
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4