Gradient descent methods

Question

$f(x,y)=x^2+xy+y^2+yz+z^2-4x-8y-8z-1.$

Can we use gradient methods to determine the minimum of $f(x, y)$? What's the most interesting gradient methods can be used to determine the minimum of $f(x, y)$?

Answer 1

Can we use gradient methods to determine the minimum of $f (x, y, \color{red}{z} )$?

Yes, you can use the gradient.

The function is differentiable and will attain extreme values (minima or maxima) in its stationary points: these are the points for which all partial derivatives are zero, i.e. when the gradient is (equal to the) zero (vector).

So compute $$\nabla f=\left( \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$$ for $$f(x,y,z)=x^2+xy+y^2+yz+z^2-4x-8y-8z-1$$ and then solve $\nabla f = (0,0,0)$.

This will be a system of three linear equations in three unknowns ($x,y,z$).