I'm not completely familiar with these sorts of problems, so please bear with me.

Given a homogeneous system Mw=0
where M is a n x n symmetric matrix:

From what I understand, the eigenvectors of M are the set of best fitting solutions to the system above, however, I require a solution that is positive and constrained to a range.

Assuming that you have a vector of eigenvalues d in descending order, and the eigenvectors corresponding to them as the columns of a matrix E, select an index p such that d_p > d_{p+1} = \ldots = d_n

subject to 0 < lb \leq w_i \leq ub
where w_1 is a vector with all elements equal to 1
and w=\sum_{i=p+1}^n\beta_iE_i

The way I'm interpreting the minimisation above is: minimise the square of the Euclidean norm of w - w_1 subject to all the elements of the solution w being in the range [lb, ub]. What's the significance of the subscript \beta here? I don't believe it's constrained in any way.

How can I solve this?


P.S. In case anyone is wondering, these equations are from "NURBS curve and surface fitting for reverse engineering, W. Ma and J. -P. Kruth"