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arXiv · 2608.16009

On the Complexity of BFGS Method for Smooth Convex Optimization

Lijun Ding·Jinwen Yang·Baoyu Zhou
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Abstract

We study the BFGS method with an Armijo-Wolfe line search for minimizing convex functions with Lipschitz-continuous gradients, without assuming strong convexity. We establish a global iteration complexity bound of $\mathcal{O}(k^{-1/2})$ for the smallest gradient norm among the first $k$ iterates. Moreover, when the initial sublevel set is bounded, we show that the function value gap converges at a rate of $\mathcal{O}(k^{-1})$. Our analysis leverages the classical trace-log-determinant potential function and reveals that a key inequality underlying this potential function remains valid without strong convexity.

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Lijun Ding, Jinwen Yang, Baoyu Zhou. 2026-08-17. On the Complexity of BFGS Method for Smooth Convex Optimization. https://arxiv.org/abs/2608.16009

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