How does Slater's condition apply to $\min x_1+x_2$ subject to linear constraints

Question

I have the following problem:

$$\text{min} ~x_1 + x_2$$

subject to

$$x_1 \geq 1 + 0.4 x_1 + 0.4 x_2$$ $$x_2 \geq 3 + 0.56 x_1 + 0.24 x_2$$ $$x_1 -w = 0$$ $$x_2 - w = 0$$

Clearly, the optimum exists and the optimal value is 30. There is no duality gap.

Suppose I penalize the equality constraints and consider the corresponding dual. I am getting a duality gap. Where is the problem with Slater's condition in this example?

Answer 1

By eliminating $w$, this problem is simply $$ \min_w \ 2w \quad \text{s.t.} \quad w \geq 3/1.8. $$ Slater's condition is satisfied and the solution is $w^* = 3/1.8$. Unless you clarify why you want to penalize the equality constraints and what you mean by "the corresponding dual", I can't make any sense of the question.