Linearisation of product of two continuous variable.

Question

I have a quadratic constraint which I want to linearise.

$x_1x_2=0$ where $0\leqslant x_1,x_2\leqslant1$.

What is the best way to do this?

Answer 1

The region described by $0\le x_1,x_2\le 1$ is a square and the condition $x_1x_2=0$ is equivalent to "$x_1=0$ or $x_2=0$" which restricts the solution space to the left and lower edges of the square. The smallest convex set containing these two edges is the triangle described by $x_1+x_2\le 1$.

Depending on your optimization problem (which you didn't tell us) it may be a good or a horrible idea to linearize the condition by $x_1+x_2\le 1$.

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If this doesn't answer your question, please edit your post to clarify what you are asking for and include the full description of the optimization problem you are trying to solve or linearize.