How to find the root of a function within a range in python

Question

I need to find the alpha value in [0,1] of a linear combination alpha*Id+(1-alpha)*M, where Id is the identity matrix, M is a given matrix, such that this linear combination has given mean.

At the moment I am using scipyt.optimize.fsolve but it does not admit the range [0,1] as an input. Any suggestion ?

Answer 1

You can define alpha using a sigmoid function:

alpha = 1/(1+exp(-x))

https://en.wikipedia.org/wiki/Sigmoid_function

Based on this definition, alpha will always be in the range [0, 1]. Then, you can change the target of the optimization in scipy.optimize.fsolve to calibrate the value of x instead of alpha directly.

The variable x is free of constraints, so any optimization method works.

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PS. This technique is very common in machine learning.

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PS2. Adding constraints to an optimizer is only important when they are not being fulfilled. So for example, if your alpha solution is already in the range [0,1], then you can keep the optimizer as it is.