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### Approximate strictly positive solution to a linear set of equations?

THIS IS AN EXTENDED COMMENT RATHER THAN AN ANSWER. It is inefficient to use Table to generate random numbers. ...

### Nonnegative Least Squares Algorithm (NNLS)

We can compare 6 algorithms based on FindMinimum,NMinimize, ConvexOptimization, ...

### Solution to a linear equation with minimum rank

Your transposed equation looks like Transpose[M].Transpose[X]==Transpose[A] LeastSquares finds the minimal solution ...

### Find min value of a complex Root function

In response to the OP's comment: ...

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### Optimize cost when ordering wood boards (cutting stock problem)

(Solution to original problem further below) MarcoB indicated useful keywords in his comment, in particular this problem is a subclass of the cutting stock problem. For the reference, two links that I ...
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### Find the maximum of a given function

First numerical analysis give a strong hint, that maximum does not depend on m. Test it with different k and m. Leave it to you to proof it exactly. ...
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### How to take the derivative with respect to a function with more than one variable?

I understand that you want to calculate dp/dq. That is the ratio of the change of p to the change of q if x is changed an infinite small amount. To make it simpler I insert the constants and define: <...