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Is there any incremental (approximate) solution for the following positive least squares problem:

$$\min_x \|Ax-b\|^2\qquad \textrm{s.t.}\qquad x_i> 0,~b_1=1,~b_{i>1}=0$$

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  • $\begingroup$ Your problem comes under the so-called quadratic programming with linear constraints. It is a well studied problem. $\endgroup$ – dineshdileep Jan 18 '15 at 8:27
  • $\begingroup$ @dineshdileep Could you please elaborate more? $\endgroup$ – Elrond Gimli Jan 18 '15 at 8:33
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The problem as written above isn't convex.
If you relax the constraint into $ x \succeq 0 $ then the problem becomes a Convex Problem.

Now you can either solve it using Quadratic Programming Solver or just use the Gradient Descent Method where at each iteration you project the solution onto the non negative orthant:

$$ \begin{align*} {x}^{k + 1} & = {x}^{k} + \alpha {A}^{T} \left( A x - b \right) \\ {x}^{k + 2} & = \max \left\{ {x}^{k + 1}, 0 \right\} \end{align*} $$

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