# How to linearise this nonlinear constraint?

I have a constraint in the form

$$\sum_{n=1}^{N}x_{m,n}\omega_{m,n}\ge (t_u-1)\beta_u, \forall u, u=1,2,\cdots, U$$

where $$x_{m,n}$$ is binary variable

$$t_u$$ and $$\beta_u$$ are continuous optimization variables.

How can I linearise this constraint?

• Do you try McCormick linearization technique? Commented Mar 11, 2021 at 8:17
• @A.Omidi, I have not heard about it before!
– KGM
Commented Mar 11, 2021 at 8:31
• Would you see this link? It is a useful method to linearize the product of the two continuous variables. Commented Mar 11, 2021 at 8:48
• @A.Omidi, Thanks, I have seen it already! However, it says 'Values of upper and lower bounds for each term must be estimated if they are not provided explicitly in the problem statement. A very good way to do this is to eliminate the objective function and then maximize or minimize the variable subject to the constraints of the original problem to find the maximum and minimum of the variable. Using these values will give a tighter relaxation on the problem while still containing the optimal solution.' But solving this non convex nonlinear problem is difficult, right?
– KGM
Commented Mar 11, 2021 at 8:54
• As far as I know, by removing the objective function, the remaining problem is solved to find the feasible solution to satisfy the constraints and it does not guarantee to achieve the optimality. In the real situation, it would be possible to estimate the $LB$ and $UB$ of variables based on some predictions or heuristics. Commented Mar 11, 2021 at 9:18

To bound the variables (preparatory to McCormick relaxation), you can maximize (and then minimize) each $$t_u$$ and each $$\beta_u$$ individually, subject to all the linear constraints of the original model (ignoring the nonlinear constraint). Hopefully those problems are all bounded.