# Objective function with exponential coefficients

I have a linear programming problem, with $$n$$ variables and $$a\leq x_{i} \leq b$$ for each variable $$x_{i}$$, where the objective function is $$\min \sum\limits_{i=1}^{n}{2^{i} x_{i}}$$

Is it true that, given the problem is feasible, there is only one optimal solution of the problem?

EDIT: all the variables have coefficient $$1$$ or $$-1$$ in the constraints.

• If $a\le x_i\le b$ is the only constraint then the minimum is achieved at $x_i=a\forall i$? May 24 at 10:14
• $a\leq x_{i} \leq b$ are not the only constraints, there are other linear constraints. May 24 at 10:51
• No it depends on the other constraints (trivial example if you minimize $c^Tx$ and have a constraint $c^T\geq d$ which becomes active at optimality, then any $x$ satisfying $c^Tx=d$ is optimal. May 24 at 12:06
• @JohanLöfberg Do you mean $c^{T}x \geq d$? May 24 at 12:39
• If each $x_i\in\{0,1\}$, it is true by uniqueness of binary expansion. May 24 at 13:48