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In nearly all applications I have seen, the master problem variables $x$ that define the subproblem are binary.

(Logic-based) Benders decomposition can applied to a problem of the form: $$\min_{x,y} f(x)+cy \\ \text{s.t.} \quad g(x)+Ay\geqslant b, \\x,y \in \{0,1\}^n $$

However, in my application, the problem also includes continuous variables $w$: $$\min_{x,y,w} f(x)+cy \\ \text{s.t.} \quad g(x)+Ay\geqslant b+h(w) \\ Cy\geqslant Dw \\ x,y \in \{0,1\}^n, w\geqslant0 $$ where $w$ are in the constraints of the subproblem. For given $\overline{x}$ and $\overline{w}$, the subproblem is: $$\min_{y} f(\overline{x})+cy \\ \text{s.t.} \quad g(\overline{x})+Ay\geqslant b+h(\overline{w}) \\ Cy\geqslant D\overline{w} \\ y \in \{0,1\}^n $$

My question is, during the iteration, are there any cuts to deal with the continuous variables $w$?

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