I want to use steepest descent method to minimize the function below and show the first method of iteration analytically with starting point xt[1,1]
f(x) = x^3 + xy + x^2y^2 – 3x
I have gone to the point where I know the next point x1 is found by considering (1,1) + r(3,3) which is (1+3r), (1+3r). However I am not sure how to proceed from there.
I know I need to expand this and find a maximum point but I am not sure.
For $f(\bar{x}) = x^3 + xy + x^2y^2 – 3x$ the gradient is
$$\left\{3x^2+2xy^2+y-3, x(2xy+1)\right\}$$
For starting point $\{1,1\}$ the gradient is $\{3,3\}$. Therefore, the first step strives to minimize $f(\bar{x})$ for $\bar{x} = \{1+3r, 1+3r\}$. Typically, this is done with a one-dimensional search procedure like the golden section search.
WolframAlpha finds the minimum is at $r = -0.138246$.
Provided, the one-dimensional search is successful, you get a new starting point for the next round: Compute the gradient for this point, use the line search to obtain the next point, check if the termination criterion (accuracy or number of steps) for your search is reached, and move on until you either succeeded or failed to reach an optimum.
