Gradient Descent

Simply use gradient information.

Let's say you have function $f(x)$ that you want to mimimize.

With gradient descent to just iterate to move $x$ in the negative of the gradient i.e $-f'(x)$. Hence at each step, we take one gradient step as below

$$\mathbf x_{t+1} = \mathbf x_{t} - \alpha f'(\mathbf x)$$

Here $\alpha$is the step size, determines how big do we want to move in the direction of negative of gradient $f'(\mathbf x )$.

The above equation basically represents a eqn of line in vector form. Where the line start passes $\mathbf x_t$ in the direction of vector $f'(\mathbf x)$. For all the values of $\alpha$, you will be able to get all the points that lies on that line.