Optimizing with Golden Section Method: Choosing Alpha for Maximum Efficiency

In summary, the conversation discusses the Golden Section method for finding a point in an equation and choosing the appropriate α value for each step. The initial α value is determined to be 0.618, but it is unclear how the second α value of 0.382 was chosen. The individual seeking help believes it may be arbitrary. They also mention finding a reference or background for this problem, and mention finding an article that explains the method in more detail and potentially provides insight on the choice of α values.
  • #1
ver_mathstats
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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+ex1^2
I understand the equation for finding a point is xk+1=xk+αkdk, where dk=(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.
 
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  • #2
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
 

1. What is the Golden Section Method?

The Golden Section Method is an optimization technique used to find the maximum or minimum value of a function. It involves dividing a search interval into two parts using a specific ratio (known as the golden ratio) and then narrowing down the search interval until the desired value is found.

2. How does the Golden Section Method help with optimizing?

The Golden Section Method helps with optimizing by providing a systematic approach to finding the maximum or minimum value of a function. It eliminates the need for trial and error and can be used for a wide range of functions, making it a versatile tool for optimization.

3. What is Alpha in the context of optimizing with the Golden Section Method?

In the context of optimizing with the Golden Section Method, Alpha refers to the ratio used to divide the search interval. It is typically represented by the Greek letter α and has a value of approximately 0.618. This specific ratio is known as the golden ratio and is believed to be aesthetically pleasing and mathematically significant.

4. How do you choose the best Alpha value for maximum efficiency?

Choosing the best Alpha value for maximum efficiency involves finding the value that results in the most efficient and accurate optimization. This can be achieved by testing different Alpha values and comparing the results, or by using mathematical techniques to determine the optimal value.

5. Are there any limitations to using the Golden Section Method for optimization?

While the Golden Section Method is a useful tool for optimization, it does have some limitations. It may not be suitable for highly complex functions or functions with multiple local maxima or minima. Additionally, it may not always provide the most accurate or efficient results, so it is important to consider other optimization techniques as well.

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