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I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ subject~to~&\mathbb{x}^Hx\leq P_1 \\\ & \mathbb{x}^HC\mathbb{x}\leq P_2 \end{align} $P_1$ and $P_2$ are given positive constants. $C$ is a positive-semi-definite matrix. $\mathbb{u}$ is a $N \times 1$ complex vector. As can be seen, constraints are convex. I am not sure if the objective is. Does a closed form exist for this problem? I am familiar with Quadratic Constrained Quadratic Programming and that this problem can be solved using Convex optimization techniques. I was interested in something in the lines of Lagrangian and KKT.

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