# Dual of a Feasibility Problem

 0 I am implementing a Benders Decomposition on a large MILP, where the objective is to minimize the cost associated with the integer variables. As a result, my Benders subproblem is only a feasibility problem. This is not an issue since the dual will have an objective and I can use Benders to make feasibility cuts. However, I am not sure whether the subproblem dual should be a minimization or maximization problem. What clue should I look for that will tell me the optimization direction? asked 18 Jan '18, 10:53 leruckle 25●4 accept rate: 0%

 4 If I understand you correctly, you have a problem of the form \begin{align} \min \ & cx + 0f\\ s.t.:\ & Ax + Bf = b\\ \ & x\in \mathcal{X}\cap\{0,1\}\\ \ & f \in\mathcal{F} \end{align} From here you should just continue as usually. If $$\bar{x}$$ is a solution to the integers variables then the subproblem becomes \begin{align} c\bar{x}+\min \ & 0f\\ \ &Bf=b-A\bar{x}\\ \ &f\in\mathcal{F} \end{align} answered 18 Jan '18, 14:02 Sune 958●4●14 accept rate: 20%
 0 Ah, yes, you are exactly correct. It is obvious in retrospect. Thank you. answered 18 Jan '18, 15:17 leruckle 25●4 accept rate: 0% If your question was answered, please accept the answer for future useres comining across your question. (20 Jan '18, 16:08) Sune
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