A Successive Gap Constraint Linearization Method for Optimal Control Problems with Equilibrium Constraints

TL;DR

This work tackles OCPECs, which couple optimal control with equilibrium constraints modeled via DVIs and VIs, by introducing a gap-constraint-based reformulation that yields a concise, differentiable constraint system and allows constraint regularity through relaxation. It builds on the regularized gap function for VIs, introducing an auxiliary variable to reformulate VI constraints into a pair of equalities and a single gap inequality, then applying a relaxation parameter to obtain a differentiable, well-posed problem. A dedicated SQP-type solver, the successive gap constraint linearization (SGCL) method, is developed to solve the discretized OCPEC; it uses stagewise efficient evaluation of the gap via a skewed projector, a Gauss-Newton Hessian approximation, and a robust QP subproblem solved with FBstab, combined with a filter line search. Numerical experiments on an affine DVI benchmark show that SGCL outperforms MPCC-tailored methods in computational efficiency while delivering competitive or better overall performance, validating the practical viability of the reformulation and solution paradigm. The approach offers a path toward scalable, differentiable OCPEC solvers and may impact trajectory optimization and decision-making in systems with complex equilibrium constraints.

Abstract

In this study, we propose a novel gap-constraint-based reformulation for optimal control problems with equilibrium constraints (OCPECs). We show that the proposed reformulation generates a new constraint system equivalent to the original one but more concise and with favorable differentiability. Moreover, constraint regularity can be recovered by a relaxation strategy. We show that the gap constraint and its gradient can be evaluated efficiently. We then propose a successive gap constraint linearization method to solve the discretized OCPEC. We also provide an intuitive geometric interpretation of the gap constraint. Numerical experiments validate the effectiveness of the proposed reformulation and solution method.

Figures (6)

• Figure 1: Contour of $\varphi^c(\lambda, \eta)$ with $c = 0.5, b_l = -1, b_u = 1$.
• Figure 2: Relaxed feasible set $\mathcal{R}^c = \mathcal{R}^c_1 \bigcup \mathcal{R}^c_2 \bigcup \mathcal{R}^c_3$, with $c = 0.5$, $s = 0.1$, $b_l = -1$, and $b_u = 1$.
• Figure 3: Solution trajectory of the affine DVI example.
• Figure 4: Box plot for relaxation parameter $s^J$ vs. cost.
• Figure 5: Box plot for relaxation parameter $s^J$ vs. the max violation of equilibrium constraints $\log_{10} \Phi$.