Note that $$N_{(t-1)}$$ can be viewed as the nodes in a scenario tree at time step $$t-1$$, whilst $$N_{(t-1)m}$$ are the nodes at time $$t$$ which are descendant of node $$m$$ at time $$t-1$$. The $$\delta_{tmn}$$ term is the probability of reaching node $$n$$ at time $$t$$ from node m at time $$t-1$$.
\begin{aligned} \min _{\beta_{t n}} & \sum_{m \in \mathcal{N}_{t-1}} \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n} \cdot\left(P_{t n}-C\right) \cdot \mathbf{x}_{t n} \\ \text { s.t. } & \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n} \cdot f_{t n}^i=f_{(t-1) m}^i \quad i \in \mathcal{I}, m \in \mathcal{N}_{t-1} \\ & \sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n}^2 \leq A_{(t-1) m}^2 \quad m \in \mathcal{N}_{t-1} \\ & \beta_{t n} \geq 0 \quad n \in \mathcal{N}_t \end{aligned}
• If I am not mistaken, your only nonlinear constraint is $$\sum_{n \in \mathcal{N}_{(t-1) m}} \delta_{t m n} \cdot \beta_{t n}^2 \leq A_{(t-1) m}^2 \quad m \in \mathcal{N}_{t-1}$$ Define the set $K$ of all nonnegative vectors $\beta_t$ satisfying this constraint. This will be a cone and section 3.2.3 of the Mosek modeling cookbook linked above covers how to express this cone. Then your feasible region is in the form $M\beta_t = f, \, \beta_t \in K$ and you can apply the theory of conic duality described there. Feb 29 at 19:34