waveforms used to excite the array have the carrier frequency of $500$ kHz. The acoustic wave speed for the breast tissue is considered $1500$ m/s.

![A schematic 2-D breast model with an $16$-mm embedded tumor at $(0,34)$ as a reference geometry. A curvilinear ultrasonic array with $51$ transducers is located near to the organ at risk. The ultrasonic array is used for hyperthermia treatment.[]{data-label="fig:2D_model"}](schematic_2D_model_col "fig:"){width="\columnwidth" height="5.5cm"}\

To characterize (discretize) the healthy tissue region $\Omega_S$ and the tumor region $\Omega_T$, two grid sets with the spacing $4$mm are considered. For optimization, a rectangular surface of the dimension $ 64 \times 42$ in mm is assumed symmetric around the tumor to model the healthy region $\Omega_S$, while the grid points belonging to the circular tumor region are excluded from this surface and they model $\Omega_T$. Overall, $174$ and $13$ number of control points are considered to characterize $\Omega_S$ and $\Omega_T$ in order to optimize the array beampattern.

The total transmitted power is constrained to $\gamma = 1$. For simplicity, the uncertainty set $\mathcal{E}_{\r}$ is modeled with ${\mathbf{W}} = {\mathbf{I}}_M$ with $\epsilon_{\r} \equiv \epsilon$ for all $\r$ where $\epsilon = 0.25$. Furthermore, the tightness of the desired power level in the across tumor 