# Linearize a logical constraint

 0 Does anyone have any idea of how I can linearize the following logical constraints set? $$(\sum_{t'=1}^t x_{it'}<1 \quad \rightarrow \quad \lambda_{it}=0), \quad \forall i,t$$ $$(\sum_{t'=1}^t x_{it'}=1 \quad \rightarrow \quad \lambda_{it}=1), \quad \forall i,t$$ $$\lambda_{it} \in \textbf{boolean}$$ $$x_{it} \in [0,1]$$ asked 27 Dec '17, 03:06 monash 37●2●4●11 accept rate: 0%

 1 You cannot. Strict inequalities are incompatible with mathematical programming. You can linearize, but only one of two relaxations. You either have to convert $$\ldots < 1$$ to $$\ldots \le 1$$ (which means $$\lambda_{it}$$ can be either 0 or 1 if the sum equals 1), or $$\ldots < 1$$ to $$\ldots \le 1 - \epsilon$$ for some $$\epsilon > 0$$ (in which case any value between $$1-\epsilon$$ and 1 for the sum becomes infeasible). Linearizing implications is a FAQ here, so details for whichever one you choose shouldn't be too hard to find using the search functionality. answered 27 Dec '17, 15:18 Paul Rubin ♦♦ 14.6k●4●13 accept rate: 19%
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Asked: 27 Dec '17, 03:06

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Last updated: 27 Dec '17, 18:07

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