# how to linerize this product of three variables? (binary * binary * continuous)

 1 2 Hi all, Take the non-linear inequality: z >= C + Axy where z is continuous, C contains a bunch of linear stuff, A is a real variable, and x and y are binary variables. I've been having headaches over how to linearize this for a few days, or at least to prove it is not possible. Any input is appreciated. asked 30 Nov '13, 19:25 LC Coelho 88●1●6 accept rate: 100%

 3 First, replace the $$xy$$ term with a new variable $$w:=xy$$ using this idea. Then you can replace the term $$Aw$$, e.g., using this idea, assuming that $$A$$ is bounded. answered 30 Nov '13, 20:39 Austin Buchanan 1.3k●4●13 accept rate: 42% This does not work because one assumes that the original constraint is an equality. In my case, it is a >=. (30 Nov '13, 20:42) LC Coelho 2 It should work. Let your constraint be z>=C+v where you enforce v=Aw as the second link suggests. (30 Nov '13, 20:46) Austin Buchanan I'll give it another try tomorrow morning and I'll come back here later. Thanks Austin. (30 Nov '13, 21:43) LC Coelho 1 @LC Coelho: If you follow your own advice, things should go just fine ;-) [...depending on how you define $$z$$ – if it's a continuous variable, then an additional $$z \geq 0$$ – as in the formulation suggested by @Austin – is required.] (01 Dec '13, 05:23) fbahr ♦ Thanks Austin and Fbahr, you were both right. I was trying to rewrite everything at once. By adding several steps and intermediate variables those tips really work. (01 Dec '13, 22:57) LC Coelho
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