On a condition in real linear programming

Question

For me $a,c\in[0,1]$ and $\epsilon>0$ is small (say $0.01$).

Is it possible to set this condition

$$c\leq\max\bigg(\frac{(a-\epsilon)}{(1-\epsilon)}, 0\bigg)$$

in real linear program?

This does not work $$\frac{(a-\epsilon)}{(1-\epsilon)}\leq d$$ $$0\leq d$$ $$c\leq d.$$

Essentially I want $d=0$ if $\frac{(a-\epsilon)}{(1-\epsilon)}<0$ else set $d=\frac{(a-\epsilon)}{(1-\epsilon)}$ and set $c\leq d$ which is equivalent to $c\leq d$ where $d=0$ if $a<\epsilon$ else $d=\frac{(a-\epsilon)}{(1-\epsilon)}$. How do I do that?

See https://www.leandro-coelho.com/how-to-linearize-max-min-and-abs-functions/ — YukiJ, Apr 23 '18 at 08:40
@YukiJ how do you penalize S^- and S^+ in a real linear program? — Turbo, Apr 23 '18 at 08:41
Only way seems to be $bS^- + (1-b)S^+$ where $b\in{0,1}$ which maked it MILP and not a real linear program. — Turbo, Apr 23 '18 at 08:48
@Robut that is indeed the only way because the constraint is not convex in $y$. The link of YukiJ is inappropriate for linear optimization. — LinAlg, Apr 23 '18 at 13:18

LinAlg · Accepted Answer · 2018-04-23T13:19:57.240

The nonlinear constraint $$x\leq\max\bigg(\frac{y-\epsilon}{1-\epsilon}, 0\bigg)$$ where $x,y$ are variables and $\epsilon$ is a parameter, cannot be represented with purely linear constraints since it is not convex. You need an extra binary variable $z$ that indicates which of the two terms in the max is active, and a prespecified upper bound $M$ on the absolute value of the first term of the max expression. Then you can state this problem as: $$\begin{align}x &\leq \frac{y-\epsilon}{1-\epsilon} + zM \\ x &\leq 0 + (1-z)M \\ z &\in \{0,1\}. \end{align}$$

On a condition in real linear programming

1 Answers1