What's the meaning of extra variables present in constraints, but not in the objective?

Question

What's the meaning of extra variables present in constraints, but not in the objective?

E.g.

$$\min (x_1,x_2)$$ $$s.t. x_1+x_2-x_3 \geq 1$$

Does this mean that $x_3$ is redundant, since it's not present in the objective?

Answer 1

They impose constraints, they can influence the values of the other variables that are present in the objective function.

For example $$\max x_1$$

subject to $$x_1+x_2 =1 .$$ $$x_2\ge 0.5$$

Hence even though $x_2$ doesn't appear in the objective function, they impose the constraint that $x_1 \le \frac12$.