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I haven't had much luck searching for this specific problems. Any pointers would be greatly appreciated.

I have an underdetermined system where $ A $ and $ b $ are known. $ x $ is a real vector with $x_i > 0 $.

$$ Ax = b $$

I'm not looking to optimize a function but to explore the solution space.

I'd like to be able to incrementally move around the solution space, using the "current" solution to evaluate a completely different problem (that would be absolutely horrible and infeasible to integrate into this). How would I be able to define this space?

Could anyone give an idea of where to start? Thanks!

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1 Answer

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The solution space is an affine space and can be described as the vector space of the solutions of the corresponding homogeneous system $Ax=0$ shifted by a specific solution $x_0$ of the inhomogeneous system $Ax=b$. Thus, the general solution takes the form $x=x_0+Sv$, where $S$ is a matrix whose columns form a basis of the solution space of the homogeneous equation and $v$ is a column vector of coefficients. You can move around the solution space by varying $v$.

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Great! What method could be used to find $ S $ given the non-negative solution space? – inthecrossfire May 7 at 17:59

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