Unconstrainted problem -- set descent direction and step length
Consider the following surface z = f(x,y) (viewed from two different angles):
The minimum is achieved at the black dot:
Given an initial guess, as shown by the black dot in the following graph:
Let us select a descent direction, which is shown as the arrow in the following graph:
(A descent direction is a direction such that the directional derivative of f(x,y) on this direction is less than 0.)
Next, following the chosen direction, we project the path on the furface z=f(x,y). This is show by the red curve
in the following graph:
Build a coordinate system as following, and view the 2-D curve in this coordinate system:
