I have two inequalities including trigonometric polynomial: $$ (1) \qquad \sum_{|k| \leq n} u_k e^{i2\pi kt} \leq 1, $$ and $$ (2) \qquad \sum_{|k| \leq n} u_k e^{i2\pi kt} \geq 0. $$ Where the left-hand side is : $$ \sum_{|k| \leq n} u_k e^{i2\pi kt} \\ \qquad = u_{_{-n}}e^{-i2\pi tn} + u_{_{-n+1}}e^{-i2\pi t(n+1)} + \cdots + u_{_{n-1}}e^{i2\pi t(n-1)} + u_{_{n}}e^{i2\pi tn} \\ \qquad = [u_{_{-n}} , u_{_{-n+1}} , \cdots , u_{_{n-1}} , u_{_{n}} ] [e^{-i2\pi tn} , \cdots , e^{i2\pi tn}]^T $$

What are the Semidefinite Programming Representations of them?

In fact, the aforementioned inequalities are the constraints for two optimization problems respectively.

  • $\begingroup$ if your variables are the $u_k$, then this is linear programming in the way it is written. If it is $t$, then your have a (complex) polynomial program in $e^{2i\pi t} $ which you can reformulate as a quadratic problem by adding variables, but without any garanty it will be SDP. $\endgroup$ – Vincent Nov 28 '16 at 10:30

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