- #1
NaturePaper
- 70
- 0
Let
[tex]
F=z_1\cos\theta_1\cos\theta_2+z_2(e^{i\alpha_1}\sin\theta_1\cos\theta_2+e^{i\alpha_2}\cos\theta_1\sin\theta_2)+z_3e^{i(\alpha_1+\alpha_2)}\sin\theta_1\sin\theta_2
[/tex]
where [tex] 0\le\alpha_i\le\pi,~0\le\theta_i\le\pi/2 [/tex] and [tex] z_i[/tex]
are some fixed complex numbers.
Then how to find
[tex]\max_{\theta_i, \alpha_i}|F|[/tex]
We note that [tex]|F|\le1[/tex].
Particularly, I want to know if there is any set of constraints like the case of optimization over real variables. [I know definitely that the conditions are [tex] \theta_1=\theta_2; \alpha_1=\alpha_2[/tex]. But I have to establish it. So, how to prove it?]
[tex]
F=z_1\cos\theta_1\cos\theta_2+z_2(e^{i\alpha_1}\sin\theta_1\cos\theta_2+e^{i\alpha_2}\cos\theta_1\sin\theta_2)+z_3e^{i(\alpha_1+\alpha_2)}\sin\theta_1\sin\theta_2
[/tex]
where [tex] 0\le\alpha_i\le\pi,~0\le\theta_i\le\pi/2 [/tex] and [tex] z_i[/tex]
are some fixed complex numbers.
Then how to find
[tex]\max_{\theta_i, \alpha_i}|F|[/tex]
We note that [tex]|F|\le1[/tex].
Particularly, I want to know if there is any set of constraints like the case of optimization over real variables. [I know definitely that the conditions are [tex] \theta_1=\theta_2; \alpha_1=\alpha_2[/tex]. But I have to establish it. So, how to prove it?]