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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?

  • 2
    @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.2011-10-28

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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.