bound $\epsilon_{\r}$ for the set can be a constant or a function of $\r$, i.e., $\epsilon_{\r} = f(\r)$. This set enables parameterization of element-wise uncertainties in the nominal steering vector $\mathbf{\hat{a}(\r)}$ at each $\r$.

Besides this consideration, we generalize the problem formulation further by setting a uniform bound (power level) $P$ across the tumor region $\Omega_T$ as an optimization variable to which the power of all the control points in the healthy region $\Omega_S$ are compared. This is in contrast to and the robust formulation in [@14:Shariati2014c], where the power levels of all the healthy grid points $\Omega_S$ are compared with the power of only a single reference point at fictitious tumor center $\r_0$. There is no need to limit our problem to a single point as a reference power level. Rather, the desired tightness of the power level across $\Omega_T$ is specified by the parameter $0 \leq \delta < 1$. This generalization also improves the efficiency when it comes to solving the robust design problem.

With these considerations, the robust beampattern design problem can be formulated as$$\label{eq:robust beampattern problem}
 \begin{aligned}
\underset{\R,t,P}{\textrm{max}} \hspace{.3cm} &t \hspace{.3cm} \textrm{subject to}&\\
&\hspace{-1cm} P - \left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H  \R  \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq t, \forall 