&\Omega_S\!\!:\!\!\left[ \begin{array}{cc}
\beta_i \mathbf{W} - \R   &    -\R \ahat(\r_i) \\
-\ahat(\r_i)^H \R         &     P - t - \ahat(\r_i)^H \R \ahat(\r_i) - \beta_i \epsilon_{\r_i}
\end{array} \right] \succeq {\mathbf{0}}, &\\
&\Omega_T\!\!:\!\!\left[ \begin{array}{cc}
\beta_{j,1} \mathbf{W} + \R   &    \R \ahat(\r_j) \\
\ahat(\r_j)^H \R              &    \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! (1-\delta)P \!\!-\!\! \beta_{j,1} \epsilon_{\r_j}
\end{array} \right] \succeq {\mathbf{0}}, &\\
&\Omega_T\!\!:\!\!\left[ \begin{array}{cc}
\beta_{j,2} \mathbf{W} - \R   &    -\R \ahat(\r_j) \\
-\ahat(\r_j)^H \R             &     (1+\delta)P \!\!-\!\! \ahat(\r_j)^H \R \ahat(\r_j) \!\!-\!\! \beta_{j,2} \epsilon_{\r_j}
\end{array} \right] \succeq {\mathbf{0}}, &\\
&\R \in \mathcal{R},\beta_i,\beta_{j,1},\beta_{j,2} \geq 0,  i=1,\ldots,N_S, j=1,\ldots,N_T.&
\end{aligned}$$

*Proof:* See Appendix \[app A\].

Observe that the notations $\Omega_S$ and $\Omega_T$ indicate that the corresponding linear matrix inequalities (LMIs) should be satisfied for the points $\r_i \in \Omega_S$ and $\r_j \in \Omega_T$, respectively. Note that the robust SDP problem in this paper, which is stated in Theorem \[theo:robustSDP\], can be solved more efficiently than the SDP problem in [@14:Shariati2014c] since the matrices $\R$ and $\mathbf{W}$ in the current formulation have half of the size of the matrices involved in the latter problem. This occurs due to the generalization of the robust problem by using the uniform power level as a benchmark.

Note that other robust problems with similar objectives can also 