# What does the objective of the primal tell us about the dual?

So lets say the primal of the problem is written like this:

$$\max \sum^n_{j=1}c_jx_j$$ $$\text{subject to } \sum^n_{j=1}a_{ij}x_j\le b_i\qquad i=1,2,\ldots,m$$

$$\qquad x_j\ge 0\qquad j=1,2,\ldots,n$$

If the objective function (c) is negative, what does that mean for the dual?

• can you show us your attempt? Oct 23, 2022 at 16:30

The dual constraints are $$\sum_i a_{ij} y_i \ge c_j,$$ so $$c_j < 0$$ implies that the constant zero solution $$y\equiv 0$$ is dual feasible.
• $\sum_i a_{ij} \cdot 0 = 0 \ge c_{ij}$, so $0$ is dual feasible. The primal cannot be unbounded here because the primal objective function to be maximized is $\sum_j c_j x_j \le 0$ since $c_j < 0$ and $x_{ij} \ge 0$. Oct 23, 2022 at 18:35
• I meant $x_j \ge 0$. Oct 23, 2022 at 18:41