Convergence and Termination Criteria

The BFGS algorithm is guaranteed to converge to a local minimum under certain conditions, such as the objective function being smooth and convex. The algorithm terminates when the gradient $\nabla f(\mathbf{x}_k)$ is sufficiently close to zero, or when a maximum number of iterations is reached.

A typical convergence criterion is:

$$\|\nabla f(\mathbf{x}_k)\| \leq \epsilon$$

where $\epsilon$ is a small tolerance value.

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