Find global minimum of f(x1,x2, xn), with many local minima

In summary, the conversation discusses the problem of finding the global minimum of a highly non-linear function with 1,000,000 arguments. The function takes 1 second to evaluate and has a large number of local minima. The person asks for suggestions on how to approach this problem, mentioning that standard minimum search algorithms struggle with it. The suggestion of using simulated annealing, properly tuned, is offered as a potential solution.
  • #1
mikeph
1,235
18
Hi

I have some function f = f(x1,x2,...xn) over some domain [0,1]^n, and I'd like to find the global minimum. The function is *highly* non-linear and takes about 1 second to evaluate. I know it's positive because it's the sum of squares of about 1,000,000 arguments, each of which pretty much depends (non-linearly) on every single argument x1,x2...xn., where n ≈ 50; As a result I expect there to be a very large number of local minima.

Can anyone suggest a good approach to this problem? Standard minimum search algorithms get lost in a local minima or take forever calculating 50-D Jacobians.

Any suggestions?
Thanks
 
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  • #2
Have you tried simulated annealing? If properly tuned, it can avoid getting stucked into a local minimum.
 

1. What is a global minimum?

The global minimum is the lowest value of a function across its entire domain. It is the absolute lowest point on the graph of the function and represents the overall minimum value of the function.

2. Why is finding the global minimum important?

Finding the global minimum is important because it allows us to determine the optimal solution for a given problem. It helps us understand the behavior of a function and can be used to optimize processes and systems.

3. How do you find the global minimum of a function with many local minima?

To find the global minimum, we need to use optimization techniques such as gradient descent or simulated annealing. These techniques involve iteratively searching for the lowest point on the function until a satisfactory solution is found.

4. What challenges are associated with finding the global minimum of a function?

One of the main challenges is that there may be an infinite number of local minima, making it difficult to determine which one is the global minimum. Another challenge is that the function may be complex and high-dimensional, making it computationally expensive to find the global minimum.

5. Can the global minimum of a function change?

Yes, the global minimum of a function can change if the function is modified or if the domain of the function is changed. Additionally, the global minimum can also change if a different optimization technique is used to find it.

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