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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?

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Could you provide a bit more detail about what you have done to penalize those constraints? Also, as this is a linear program you know automatically that there can be no duality gap (except for the weird case when both primal and dual are infeasible), so you do not need to bring in a constraint qualification like Slater's condition. – Noah Stein Jun 22 '11 at 16:33
@hari: What's the point of $w$ here really? It's just saying that $x_1 = x_2$. So why don't you simplify your problem into one that has a single variable? I wouldn't dream of penalizing those equality constraints. – Dominique Oct 28 '11 at 20:58

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.

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