# A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align}

$\Re(.)$ denotes the real part. Here $\mathbf{Q}$ is a $N \times N$ positive definite matrix. $\mathbf{b}$ is a $N \times 1$ vector. I am familiar with the technique of putting this problem in the real domain (where it becomes in $2N \times 1$ dimension) and then using the lagrangian technique to solve the resulting problem. I was looking for some analytic technique which would solve it in the complex domain itself. Applying lagrangian technique for complex vectors is also fine.

For problems of this kind it is often convenient to treat the $\mathbf{x}$ and $\mathbf{x}^H$ as independent variables (in the end they are linearly related to real and imaginary part of $x$). So we want to minimize $$E= \mathbf{x}^H Q \mathbf{x} - (\mathbf{x}^H \mathbf{b} + \mathbf{b}^H \mathbf{x}) +1.$$
In this case as $E$ is a real function (as it should otherwise minimization does not make too much sense), the equations $$\nabla_{\mathbf{x}} E =0$$ and $$\nabla_{\mathbf{x}^H} E=0$$ are complex conjugates of each other. So it is enough to solve one of these. We choose $$\nabla_{\mathbf{x}^H} E = Q \mathbf{x} - \mathbf{b} =0$$ with the solution $$\mathbf{x} = Q^{-1} \mathbf{b}.$$
Let $Q^{1/2}$ be a Hermitian square root of $Q$. Then by completing square, we get \begin{align*} x^HQx-2~\Re{(x^Hb)}+1 &=x^HQx-x^Hb-b^Hx+1\\ &=\|Q^{1/2}x - Q^{-1/2}b\|^2 + (1 - b^HQ^{-1}b). \end{align*} Hence the minimum occurs at $x = Q^{-1}b$ and the minimum value is $1 - b^HQ^{-1}b$.
• @user1551 Would you please perform one more step to elaborate, how did you equate the following ? $x^HQx-x^Hb-b^Hx+1=\|Q^{1/2}x - Q^{-1/2}b\|^2 + (1 - b^HQ^{-1}b)$ – kaka Feb 21 '15 at 7:07
• @kaka $\|u\|^2 = u^Hu$. – user1551 Feb 21 '15 at 7:43