How can I solve this nonlinear projection problem? Can you give me any idea?
Constraints are as shown in the image. The norm can be anyone.
Variables : $z_1,\;z_2,\;\underline{x}\in\mathbb{R}^d,\;\overline{x}\in\mathbb{R}^d,\;\underline{y},\;\overline{y}$
$\hat{x}\in \mathbb{R}^d, \; \hat{y}$ are data, and $y,\;\hat{y}$ are 1 or -1.
$z_1 \geq 0,\; z_2 \geq 0, \;z_1+z_2 \geq 0$
$z_1 (\lVert \underline{x} - \hat{x} \rVert + \lvert {\underline{y}-\hat{y}} \rvert )
+z_2 (\lVert \overline{x} - \hat{x} \rVert + \lvert {\overline{y}-\hat{y}} \rvert) \leq \epsilon $
Thank you very much.