arXiv:2504.04618v2 Announce Type: replace Abstract: In this paper, we develop a distributed solution framework for coordinating traffic light systems and connected automated vehicles (CAVs) in mixed-traffic intersections. We formulate a mixed-integer quadratic program (MIQP) aimed at optimizing the coordination of traffic light systems and CAVs, thereby fully capitalizing on the advantages of CAV integration under conditions of high penetration rates. To effectively solve the challenging multi-agent MIQP approximately, we employ proximal ADMM for solving the convex relaxation of the MIQP while systematically tightening the constraint coefficients using a finite-grid tightening rule to uphold integrality requirements. \hilitediff{The obtained relaxed solutions are corrected using a confidence-guided binary recovery step, which fixes high-confidence binary variables and solves a substantially reduced MIQP over the remaining uncertain binaries to improve the feasibility of the final mixed-integer solution.} The performance of our control framework and the efficacy of the distributed algorithm are rigorously validated through a series of simulations conducted across varying penetration rates and traffic volumes.