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I'm given a $M\times N$ ($M$ rows, $N$ cols) matrix $X$ and a $1\times M$ column vector $y$ ($M>N$). I need to find such a vector $x$ that minimizes the value of s, where

$s= \sum_{i=1}^{M}\left [(y_i - \hat{y}_i)^2 \right] $

and where $\hat{y} = X \times x$

Basicaly, this is a set of linear constraints. There is also a constraint that for every $x_{i, j} \in X$ , the following is true: $0 \le x_{i,j} \lt 1$

It is known that $X$ is sparse (most of the elements are 0's). The typical number of rows in $X$ is 200 and the typical number of columns is 150

Is Simplex algorithm or gradient-based optimization methods well-suited to perform this job or should I try using global heuristic methods such as Genetic algorithm, taboo search etc? What Python tools would you suggest to attack this problem?

Last, but not least: is there any other SE forum that is better suited for this questoin?

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  • $\begingroup$ As for now it is ok here; proper site is in commitment on Area now. $\endgroup$
    – user88
    Jul 26, 2011 at 8:00
  • $\begingroup$ @David, what you describe is least squares and the solution $x$ is given by $x=(X'X)^{-1}X'y$, so it does not matter whether $X$ is restricted or not. Could you please clarify whether the constraints are on $x$ or on $X$? If there are linear inequality constraints on $x$ then user603 answer applies. $\endgroup$
    – mpiktas
    Jul 26, 2011 at 11:14

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This is a classical problem (quadratic approximation subject ot linear constraints).

Check function $\verb+lsei+$ in $\verb+R+$ package $\verb+lsei+$ or the matlab package cvx or the python package cvxopt.

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