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?