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Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$: $$ \mathrm{arg}\max_X \,\mathrm{trace}(X^HA). $$

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    $\begingroup$ If there are no conditions on $X$ then this problem is unbounded, just replace $X$ by $\alpha X$ for $\alpha > 1$. Moreover, your objective need not be real-valued at the moment, e.g., take $X= i A$. $\endgroup$
    – Rammus
    Commented May 10, 2021 at 7:57

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