How to linearize an indicator equality constraints?

 0 Let $$x_i$$ be binary variables and $$a_i$$ given positive integers. I have an equality constraint like the following $x_i\left(\sum_{j=1}^nx_{j}\right)=a_ix_i,$ for all $$i$$. This means that if $$x_i=1$$, then there must exists a set $$S$$ with cardinality $$|S|=a_i-1$$ such that $$x_j=1$$ for all $$j\in S$$. How can I linearize these constraints? If the constraints were of type inequality I could use a large (or small) number and write it as: $\left(\sum_{j=1}^nx_{j}\right)\leq a_ix_i+M(1-x_i),$ for all $$i$$. asked 18 Jan '18, 11:15 zbir 17●1●4 accept rate: 0%

 1 Hint: Think of your equality as two inequalities and apply your big-M approach twice. Also, for numerical reasons be sure to choose $$M$$ as small as possible. answered 18 Jan '18, 11:21 Rob Pratt 1.2k●2●6 accept rate: 28%
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