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I want to solve a generalized form of a quadratic programming problem $$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$, $$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive definite.

As you know, the ordinary quadratic programming only involves one positive definite matrix, i.e. $$\min_x x^TQx+c^Tx$$ $$\textrm{s.t. } Ax\le b$$

Now I want to ask how to solve the first generalized form. Thank you so much.

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    $\begingroup$ Well its still convex, albeit non-smooth. Looks like primal-dual methods, alternating direction method of multipliers or so could be applied… $\endgroup$
    – Dirk
    Feb 17, 2015 at 20:00
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    $\begingroup$ The problem is convex. Moreover, it can be transformed into a second cone problem, which means it can be solved very efficiently . Indeed one can write $x^T P x \le u^2$ and $x^T Q x \le v^2$ (which are both second-order cone constraints) and simply minimize $(u+v)^2+c^T x$, which is quadratic convex (and also second-order cone representable). $\endgroup$
    – F_G
    Feb 21, 2015 at 8:58

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