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Does a local minimum of a function always satify the Armijo rule?

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Assuming that the function $f$ is continuous (and so $\nabla f$ is defined), then the Armijo rule can't be satisfied as an equality for a global minimum. That is you can rewrite the rule as:
$$f(\mathbf{x}_k+\alpha_k\mathbf{p}_k)\leq f(\mathbf{x}_k)+c_1\alpha_k\mathbf{p}_k^{\mathrm T}\nabla f(\mathbf{x}_k) = f(\mathbf{x}_k)$$
since $\nabla f(\mathbf{x}_k)=0$, but the LHS is larger than the RHS because the point $\mathbf{x}_k$ is a minimum. But if the minimum is only local, then you could meet the condition by arranging for the step to be to a lower minimum.

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