I have the following expression that I want to maximize: $$ A=\max_{\phi\in [0,2\pi)}\max_{\mathbf{d}\in\{\pm 1\}^{N+1}}\Big\{d_0\operatorname{Re}\{e^{-j\phi}y_0\}+d_1\operatorname{Re}\{e^{-j\phi}y_1\}+...+d_N\operatorname{Re}\{e^{-j\phi}y_N\}+\operatorname{Re}\{e^{-j\phi}z\}\Big\} $$ where $d_i\in\{\pm 1\}$ are binary coefficients and $y_0,y_1,..,y_N,z\in\mathbb{C}$ are all known complex numbers. I have proceeded like this: $$ A=\max_{\phi\in [0,2\pi)}\Big\{\underbrace{\max_{\mathbf{d}\in\{\pm 1\}^{N+1}}\Big\{d_0\operatorname{Re}\{e^{-j\phi}y_0\}+d_1\operatorname{Re}\{e^{-j\phi}y_1\}+...+d_N\operatorname{Re}\{e^{-j\phi}y_N\}\Big\}}_{C}+\underbrace{\operatorname{Re}\{e^{-j\phi}z\}}_B\Big\}\\ $$ and now I am trying to break it down to two maximizations over $\phi$ without using an inequality, otherwise the best I could is to find an upper limit. So $B$ is: \begin{eqnarray*} B&=&\operatorname{Re}\{e^{-j\phi}(\operatorname{Re}\{z\}+j\operatorname{Im}\{z\})\}\\ &=&\operatorname{Re}\{(\cos(-\phi)+j\sin(-\phi))(\operatorname{Re}\{z\}+j\operatorname{Im}\{z\})\}\\ &=&\operatorname{Re}\{\cos(\phi)\operatorname{Re}\{z\}+j\cos(\phi)\operatorname{Im}\{z\}-j\sin(\phi)\operatorname{Re}\{z\}+\sin(\phi)\operatorname{Im}\{z\}\}\\ &=&\operatorname{Re}\{z\}\cos(\phi)+\operatorname{Im}\{z\}\sin(\phi) \end{eqnarray*}
At this point, I have used the identity of linear combination of sine and cosine which is: $$c_1\cos x+c_2\sin x=A\cos(x-\phi)$$ with $A=\sqrt{c_1^2+c_2^2}$ and $\phi=\tan^{-1}(\frac{c_2}{c_1})$
Based on this: \begin{eqnarray*} B&=&\sqrt{\operatorname{Re}^2\{z\}+\operatorname{Im}^2\{z\}}\cos(\phi-\tan^{-1}(\frac{\operatorname{Im}\{z\}}{\operatorname{Re}\{z\}}))\\ &=&|z|\cos(\phi -\angle z) \end{eqnarray*}
So the angle of the cosine of $B$ ranges in $[-\angle z, 2\pi-\angle z)$ and this is the problem which prevents me from taking $B$ out of the maximization. Since this cosine ranges in a whole circle I cannot suppose that it is monotonically increasing. If the range of $(\phi -\angle z)$ was a subset of $[\pi,2\pi)$, we could say: $$A=\max_{\phi\in [0,2\pi)}\{C\}+\max_{\phi\in [0,2\pi)}\{B\}=\max_{\phi\in [0,2\pi)}\{C\}+|z|\cos(\phi_{max} -\angle z)$$ with $\phi_{max}=\mathrm{arg}\max\limits_{\phi\in [0,2\pi)}C$
Am I missing something or any other ideas on how to approach this?