Optimizing with Golden Section Method: Choosing Alpha for Maximum Efficiency

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Homework Statement

You are at the point (0,1). Find the minimum of the function in the direction (line) (1, 2)^T using
the Golden-Section line-search algorithm on the step-length interval [0, 1]. Stop when the length of
the interval is less than 0.2. Note: step-length interval could be described by the parameter t, and,
so, all the points along the direction (1, 2)^T
can be expressed as (0, 1) + t · (1, 2).

Relevant Equations

Function is: f(x1,x2)=(x2-x1^2)^2+e^x1^2
Golden Ratio (GR):0.618

My function is f(x₁,x₂)=(x₂-x₁ ²)²+e^x₁^²
I understand the equation for finding a point is x_k+1=x_k+α_kd_k, where d_k=(0,1) and x₀=(0,1). For the first step I can choose α₁=0.618 (GR), but how do we choose α₂? In the solution manual I see they chose 0.382, was this just some arbitrary number? My assumption is that it is arbitrary. I understand afterwards I will have to compare f(x₁) and f(x₂). But I am struggling with how the second α₂ was picked, any help would be appreciated.

 

[scottdave]

I realize this is several months old. What reference or background do you have associated with this problem?

I was not familiar with the Golden Section method, so I looked it up. I found this site, which provides some insight. https://www.geodose.com/2021/06/golden-section-search-python-application-example.html

I need to re-read the article, but I think the 0.382 comes from subtracting 0.618 from 1, though it could come from (0.618)²