Escaping from a point in linear programming

Question

Is there a trick for explaining the following constraint as a set of linear (in)equalities?

$$ \sum_{i=1}^n|x_i-a_i|>0, $$ where $a_i$'s are real constants.

Answer 1

Hint
We have by elementary cases
$|x|<a \Leftrightarrow -a < x \wedge x < a$
$|x|>a \Leftrightarrow x < -a \vee a < x$

Answer 2

Suggestion: This constraint actually excludes only the vector $$\vec{x}=\vec{a}$$ Therefore an idea would be to solve the LP without this constraint. If the result is exactly this vector then adjust appropriatelly (for example move $ε$ to some acceptable direction). If the result is not this exact vector, then you obtain at no extra cost that this solution is optimal also for the constrained LP.

The second case is straightforward, while the first case might be more complicated to resolve.

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If you have (as in most LP's) that $x_i \ge 0$ and you have an $a_i<0$ then the constraint is automatically fullfilled in every feasible solution $x$. The point is that you might cirmuvent easily this constraint in many problems, depending on the specific details about the problem, while in full generality it might be more difficult (computationally).