Low-Rank KKT Updates and a Parallel Flipping Mechanism for Model-Based Derivative-Free Optimization

Abstract

Model-based derivative-free optimization relies on quadratic interpolation, but maintaining these models typically requires $\mathcal{O}(m^3)$ linear system solves. We show that for the least Frobenius norm updating model, the associated KKT matrix possesses a fixed inner-product structure. Both single-point replacements and a proposed coordinate-axis flipping operation induce exact Rank-2 perturbations to this matrix. Using this structure, we derive an $\mathcal{O}(n^2)$ update formula for the KKT inverse, eliminating costly refactorizations at each iteration. We integrate the update into a parallel trust-region algorithm where workers independently flip interpolation axes, refresh local models, and synchronize the best configuration. Tests on 530 benchmark problems show the method reduces model-maintenance overhead and achieves higher success rates under tight function-evaluation budgets compared to standard solvers.