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Hi, I have a cost function of the form

$$F(X) = \operatorname{tr}(X'AX)+\operatorname{tr}(X'B),\quad\textrm{ s.t. }X'X=I.$$

$X$ is a $m\times n$ matrix, ($m>n$), with orthonormal columns. $A$ is symmetric $m\times m$, not necessary positive definite. $B$ is of size $m\times n$.

  1. Is global minimum for this problem guaranteed?
  2. Regardless of the existence of the global minimum, is there any efficient algorithm to solve for a even local minimizer? Or any closed form solutions?

Thanks!

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