# How to model the following Constraint

I would like to model the following:

$$B \le \alpha \implies \sum_i W(i) \ge \beta$$, where $$B$$ a continuous variable, $$W(i)$$ binary variables, $$\alpha$$ a real constant number, $$\beta$$ an integer constant number

I could do this by introducing a new binary variable $$Z$$ and requiring

$$B \le \alpha \implies Z = 1$$ and $$Z =1 \implies \sum_i W(i) \ge \beta$$

My question is, is there a possibility of avoiding the introduction of $$Z$$?

• Jan 31 at 0:33
• Here the constraint $\sum_i W(i) \ge \beta$ is an all integer constraint. Could this somehow avoid the introduction of $Z$? Jan 31 at 8:44
• Dear @RobPratt, in a case, where one of $w_i$ is equal to one, then $B \leq \alpha \implies w = 1$. That can be linearized without any more binary variables. Would you say, does not any general procedure exist to do that? Jan 31 at 9:39
• I should state that $1 \le \beta \le \lvert I \rvert, i \in I$ Jan 31 at 15:24
• Do you also have bounds on $B$ and $\alpha$? Jan 31 at 17:15

As in How to enforce logical implication $\sum_j a_j x_j \le b \implies \sum_j c_j x_j \le d$, (temporarily) introduce constant $$\epsilon>0$$ and binary variable $$Z$$, and impose linear big-M constraints \begin{align} \alpha + \epsilon - B &\le (\alpha + \epsilon - 0) Z \tag1\label1 \\ \beta - \sum_i W_i &\le (\beta - 0) (1-Z) \tag2\label2 \end{align} Now eliminate $$Z$$ by multiplying \eqref{1} by $$\beta$$ and \eqref{2} by $$\alpha + \epsilon$$ and adding them up to obtain the valid inequality $$\beta(\alpha + \epsilon - B) + (\alpha + \epsilon)\left(\beta - \sum_i W_i\right) \le \beta(\alpha + \epsilon) Z + (\alpha + \epsilon) \beta (1-Z),$$ which simplifies to $$B + \frac{\alpha + \epsilon}{\beta}\sum_i W_i \ge \alpha + \epsilon. \tag3\label3$$
But this is too weak unless $$B=0$$ or $$\sum_i W_i = 0$$. The original disjunction $$B \ge \alpha + \epsilon \lor \sum_i W_i \ge \beta$$ is nonconvex, and \eqref{3} is a convex relaxation.