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I have been trying to solve a system of linear equations $$\mathbf{Ax}=\mathbf{b}$$ Where $\mathbf{A}$ is a $m\times n$ matrix, (such that $m>n$) and $\mathbf{x}=\{x_1 ,x_2, x_3...x_n\}$ and $\mathbf{b}=\{b_1 ,b_2, b_3...b_n\}$ are vectors of length $n$. In particular, I aim to minimize the norm $||\mathbf{Ax-b}||$.

I am doing this by performing the singular value decomposition of matrix $\mathbf{A}$ as $\mathbf{U w^{-1}} V^{T}$. Components of the vector $\mathbf{x}$ corresponding to the minimum norm are then given simply by $\mathbf{x=VwU^{T}b}$.

This method gives me a set of $x_i$ for minimum possible value of $||\mathbf{Ax-b}||$.

My question is as follows, is there any method to seek the minimum of $||\mathbf{Ax-b}||$ while forcing one of the components of $\mathbf{x}$, say $x_n$ to have non negative values?

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For the first part either you wrote $A$ by accident or something but if

$$ A = U \Sigma V^{T} \tag{1} $$

and we have

$$ Ax=b \\ U \Sigma V^{T} x = b \\ x = V\Sigma^{\dagger}U^{T}b \tag{2} $$

for the second part, you need to use linear programming to enforce that $x_{n}$ be non-negative. I.e

$$ \textrm{ minimize } c^{T}x \\ \textrm{ subject to } Ax \leq b \\ \textrm{ and } x \geq 0 \tag{3}$$

Typically we have

$$ lb \leq x \leq ub \tag{4} $$

Which is modified afterward

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  • $\begingroup$ Thanks, let me try that $\endgroup$ – Abhijit Oct 5 '18 at 6:43

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