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I've got an unconstrained optimization problem, and all function involved can be regarded as differentiable as you like.

The variable is a rectangular matrix $M$. Target Function is $\sum_i f(w^i) +\sum_j g(w_j)$, where $w^i,w_j$ corresponds to the i-th row and j-th column of $M$

The best I can get is to split vars to $f(a^i)$ and $g(b_j)$ with Alternating Direction Method of Multipliers and then solve smaller problems respectively. But the penalty term induces is still expensive.

Is there any preferable methods to solve problems with such structure?

Any thoughts would be helpful. Thanks in advance!

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  • $\begingroup$ Hard to give any suggestion without any clue on $f$ or $g$... $\endgroup$
    – AndreaCassioli
    Commented Mar 5, 2016 at 14:45
  • $\begingroup$ @AC_MOSEK $f$ is a sophisticated function involving SVD decomposition and other things and $g$ is simpler with some L2 norm of a given point after polynomial evaluations of the variable. I just noticed the structure of matrix cols and rows and figuring if there is any approach that would take advantage of this. $\endgroup$
    – Zhongshi Jiang
    Commented Mar 6, 2016 at 7:53
  • $\begingroup$ Well, I would try to write down the problem in formulae. Then there might be some advise to give. ADMM is nowdays very popular, but other unconstrainted minimizers would do anyway. $\endgroup$
    – AndreaCassioli
    Commented Mar 7, 2016 at 8:43

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