Consider an optimization problem $$\min_{\mathbf x}\,\,\,f(\mathbf x)\\s.t.\,\,\,\,\,\,\mathbf x\in\mathcal X$$
Suppose $\mathbf d^t=\mathbf x^{t+1}-\mathbf x^t$ is a descent direction at $\mathbf x=\mathbf x^t$, i.e., $\nabla f(\mathbf x^t)^T(\mathbf x^{t+1}-\mathbf x^t)<0$.
Similarly, suppose $\mathbf d^{t+1}=\mathbf x^{t+2}-\mathbf x^{t+1}$ is a descent direction at $\mathbf x=\mathbf x^{t+1}$, i.e., $\nabla f(\mathbf x^{t+1})^T(\mathbf x^{t+2}-\mathbf x^{t+1})<0$.
Of course, we can assume $f(\mathbf x^{t+1})\leq f(\mathbf x^{t})$. Otherwise, it is a meaningless problem.
Now, I wonder whether $\mathbf x^{t+2}-\mathbf x^t$ is a descent direction at $\mathbf x=\mathbf x^t$, i.e., $\nabla f(\mathbf x^t)^T(\mathbf x^{t+2}-\mathbf x^t)<0$?
Thanks a lot for any help.