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I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$.
$(x = 0 \wedge y = 1) \vee (x \neq 0 \wedge y = 0)$
A simple approach is to write the above constraint as
$(x \leq 0 \wedge x \geq 0 \wedge y \leq 1 \wedge y \geq 1) \vee (x < 0 \wedge y \leq 0 \wedge y \geq 0) \vee (x > 0 \wedge y \leq 0 \wedge y \geq 0)$
To avoid strict inequalities we can reformulate for a sufficiently small $\epsilon > 0$ as follows
$(x \leq 0 \wedge x \geq 0 \wedge y \leq 1 \wedge y \geq 1) \vee (x \leq -\epsilon \wedge y \leq 0 \wedge y \geq 0) \vee (x \geq \epsilon \wedge y \leq 0 \wedge y \geq 0)$
Now for each conjunction we introduce one binary variable, i.e. $b_1, b_2, b_3 \in \lbrace 0, 1 \rbrace$ and we apply big-M. The first conjunction turns into
$x \leq 0 + M (1 - b_1)$
$x \geq 0 - M (1 - b_1)$
$y \leq 1 + M (1 - b_1)$
$y \geq 1 - M (1 - b_1)$
Similarly for the other conjunctions. Adding the constraint $b_1 + b_2 + b_3 \geq 1$ completes the translation to MILP.

Having to introduce $\epsilon$ is not nice as it may cause numerical problems with solvers. Is there a MILP formulation of the above constraint that avoids $\epsilon$ and if not why can't there be such a formulation?
Moreover, I wonder if $\epsilon$ can be avoided if quadratic constraints are permitted. Also if not, why does it not help. Thanks for any insights.

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If you're solving this problem numerically (using floating point numbers), $x > 0$ really means $x \geq \epsilon$ where $\epsilon$ is the next representable floating-point number on the number line for the given precision, in the increasing direction.

If you think about it, floating point numbers are essentially a discrete approximation to real numbers, therefore strict inequalities can only be approximated. Therefore in numerical optimization, one cannot directly formulate constraints with $\gt, \lt$ or $\neq$. There is always an $\epsilon$ involved.

As for $\epsilon$ not being numerically nice, in theory yes. But in practice that rarely happens if one is careful to avoid catastrophic cancellations. Also in many applications $\epsilon$ is chosen to be the numerical tolerance (which is usually many orders of magnitude larger than the machine epsilon).

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