To "fix" continuous variables in Benders decomposition

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 $$w$$?

• Commented Jun 18 at 19:04
• It can also be related: or.stackexchange.com/questions/9768 The comments might offer some insights. Commented Jun 20 at 10:49