be addressed using the above approach which are outlined in the following subsection.

Alternative robust formulations {#subsec:alternative}
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Similar robust problems to that of can be formulated in many different ways. For example, by restricting the power level outside the tumor in a weighted fashion. $$\begin{aligned}
\min_{t,R}&& t \hspace{.3cm} \mbox{subject to}\\
&& \hspace{-1.3cm}(\ahat(\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t w(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\
&&\hspace{-1.3cm} (\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\
&& \hspace{-1.3cm}(\ahat (\r)\!\!+\!\!\atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le (1+\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm T}\\
&&\hspace{-1.3cm}R\in \mathcal{R}\end{aligned}$$ where $P,\delta$ are fixed and $w(\r)$ is a weighting function constructed, e.g., so that the energy bound close to the tumor is less restrictive.

One could also construct problems that minimize the sum of the energy in the non-tumor area where $t(\r)$ denotes the energy at $\r$: $$\begin{aligned}
\min_{t(\r),R}&& \sum_{\r\in\Omega_{\rm S}}t(\r)  \hspace{.3cm} \mbox{subject to} \\
&& \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \le t(\r), \forall \atil (\r)\in \mathcal{E}_\r, \r\in \Omega_{\rm S}\\
&& \hspace{-1.3cm} (\ahat(\r) \!\!+\!\! \atil (\r))^H R (\ahat(\r) \!\!+\!\! \atil (\r)) \ge (1-\delta)P, \forall \atil (\r)\in \mathcal{E}_\r, \r\in 