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## constrained optimization of dot product

Dear guys,

I have a real matrix $A$ ($1000 \times 20$) which can be written as $[ A_P | A_N ]$ ($A_P$ contains only positive and $A_N$ only negative numbers) and I want a positive vector $x$ of unit length ($x^T x = 1$) for which $x^T A^T A x$ is minimal (closest to 0).

Courat-Fischer-Weyl min-max principle (minimum of $\frac{x^TA^TAx}{x^Tx}$ is lowest eigenvalue of $A^TA$) seems to be close, but not it. Currently, I try to solve the problem rescaling $A^TA$ by a diagonal matrix $D$ to obtain $C = DA^TAD$ which has eigenvector of $n(=20)$ ones corresponding to lowest eigenvalue. Then, diagonal elements of $D^{-1}$ seem to be my vector elements. Can this be a way to solution?

Actually, this was the motivation of my latest post http://mathoverflow.net/questions/93166/bilinear-equation-or-diagonal-matrix-search

Any deterministic algorithm leading to solution (global minimum) in polynomial time is welcome.

Thanks very much

Boris

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 try math.stackexchange.com/… or stackoverflow.com – Will Jagy Apr 7 2012 at 23:33