Introduce threefour binary variables indicating which region you are in and the function value.

\begin{align} \delta_1 &\rightarrow x_2\geq c, ~x_3\geq d, ~x_1 = a\\ \delta_2 &\rightarrow x_2\geq c, ~0\leq x_3 \leq d, ~x_1 = bx_3\\ \delta_3 &\rightarrow x_2\leq c, ~x_1 = 0 \end{align}\begin{align} \delta_1 &\rightarrow x_2\geq c, ~x_3\geq d, ~x_1 = a\\ \delta_2 &\rightarrow x_2\geq c, ~0\leq x_3 \leq d, ~x_1 = bx_3\\ \delta_3 &\rightarrow x_2\leq c, ~x_1 = 0\\ \delta_4 &\rightarrow x_3\leq 0, ~x_1 = 0 \end{align}

You are in one of the regions so $$\delta_1+\delta_2 +\delta_3=1$$$$\delta_1+\delta_2 +\delta_3+\delta_4=1$$, and all the implications are standard big-M representable such as $$x_2-c\geq -M(1-\delta_1), -M(1-\delta_1) \leq x_1-a\leq M(1-\delta_1)$$ etc. By exloiting structure and linearity in some terms this can be reduced and simplified, but this is the basic model.

Introduce three binary variables indicating which region you are in and the function value.

\begin{align} \delta_1 &\rightarrow x_2\geq c, ~x_3\geq d, ~x_1 = a\\ \delta_2 &\rightarrow x_2\geq c, ~0\leq x_3 \leq d, ~x_1 = bx_3\\ \delta_3 &\rightarrow x_2\leq c, ~x_1 = 0 \end{align}

You are in one of the regions so $$\delta_1+\delta_2 +\delta_3=1$$, and all the implications are standard big-M representable such as $$x_2-c\geq -M(1-\delta_1), -M(1-\delta_1) \leq x_1-a\leq M(1-\delta_1)$$ etc. By exloiting structure and linearity in some terms this can be reduced and simplified, but this is the basic model.

edited body

Introduce three binary variables indicating which region you are in and the function value.

\begin{align} \delta_1 &\rightarrow x_2\geq c, ~x_3\geq d, ~x_1 = a\\ \delta_2 &\rightarrow x_2\geq c, ~0\leq x_3 \leq d, ~x_1 = bx_3\\ \delta_3 &\rightarrow x_2\leq c, ~x_1 = 0 \end{align}

You are in one of the regions so $$\delta_1+\delta_2 +\delta_3=1$$, and all the implications are standard big-M representable such as $$x_2-c\geq -M(1-\delta_1), -M(1-\delta_1) \leq x_1-a\leq M(1-\delta-1)$$$$x_2-c\geq -M(1-\delta_1), -M(1-\delta_1) \leq x_1-a\leq M(1-\delta_1)$$ etc.