# Do Benders cuts exclude current solutions?

I am wondering if optimality cuts in Benders algorithm exclude the possibility to have the same solutions and as a result, have the same optimality cut? I don't know why it is not possible to have the same cuts in different iterations of Benders decomposition algorithm.

It depends on whether you consider the master variable $$\eta$$ to be part of the solution. You can get the same master $$x$$ with a different $$\eta$$ after adding an optimality cut, but then the Benders decomposition algorithm would terminate because the solution is optimal to the original problem.
• Yes, the following can happen in a minimization problem. Solve master problem, yielding solution $(\eta^1, x^1)$. Solve subproblem with fixed $x=x^1$, revealing that the master has underestimated the true objective value, which is $\eta^2 > \eta^1$ for that $x^1$. Add optimality cut to master, and solve again, yielding $(\eta^2, x^1)$. Solving the subproblem would not yield an optimality cut, and you are done. – RobPratt Apr 15 '20 at 23:02
• The current master variables violate the new optimality cut because there would be no optimality cut if they did not. Let $(\hat{x}, \hat{z})$ be the current master solution, where $z$ is the surrogate for the subproblem objective value, and assume you are minimizing. You solve the subproblem with $x=\hat{x}$ and get optimal subproblem value $z*$. If $\hat{z}\ge z*$, there is no new optimality cut. If $\hat{z} < z*$, the new optimality cut $z\ge f(x)$ is generated so that $f(\hat{x})=z^*$, which means $(\hat{x}, \hat{z})$ violates it. – prubin Apr 27 '20 at 22:19