Discover How to Find a Function's Maximum with scipy.minimize

Learn how to optimize functions in Python by maximizing them with `scipy.minimize` using the BFGS method. We break down the steps and provide code examples!
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This video is based on the question https://stackoverflow.com/q/70308748/ asked by the user 'Andrés Ortega' ( https://stackoverflow.com/u/17647054/ ) and on the answer https://stackoverflow.com/a/70308891/ provided by the user 'AbbeGijly' ( https://stackoverflow.com/u/13418705/ ) at 'Stack Overflow' website. Thanks to these great users and Stackexchange community for their contributions.

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Maximizing Functions in Python Using scipy.minimize

When it comes to optimization in Python, you're likely to encounter situations where you need to find the maximum of a given function. In this guide, we'll explore how to achieve that using the scipy.optimize.minimize function, particularly applying the BFGS method. If this sounds challenging to you, worry not! We'll break down the solution step-by-step.

The Challenge: Finding a Maximum

Suppose you have a function defined as follows:

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Your task is to find the maximum value of functions A and B by leveraging scipy.minimize. However, you are restricted from modifying the functions themselves. Instead, you can only adjust how they are called during the minimization.

The Approach: Minimizing the Negative

The fundamental insight is that to find the maximum of a function, you can minimize its negative. This means that instead of maximizing f(x), you will minimize -f(x). Here’s how you can implement this:

Step 1: Set Up the Environment

First, ensure you have the necessary library imported:

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Step 2: Create a Function for Minimization

For function A, you create a lambda function that calls -A(x):

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This code effectively finds the maximum of A starting from an initial guess of 0. As a result, it will print:

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Step 3: Repeat for Function B

Apply the same logic to function B:

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You should get an output close to -2, which in this case will be printed as:

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Step 4: Compact One-Liner

If you prefer a more concise approach, you can encapsulate the minimization in a one-liner:

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Step 5: Minimizing Both Functions Simultaneously

If you want to find the maximum for both functions simultaneously, you can create a function that takes a vector as input. This would look like:

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This will give you results like this:

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Conclusion

Finding the maximum of a function using scipy.minimize might initially seem daunting, but once you recognize that maximizing a function is akin to minimizing its negative, the process becomes much simpler. By following these structured steps, you can effectively tackle various optimization problems in Python with ease.

Feel free to play around with the initial guesses in the minimization routine and explore how they impact the results! Happy coding!