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10 votes
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When is the duality gap of semidefinite programming (SDP) zero?

There is a more complicated theory of duality for SDPs that is exact: there is no 'extra condition' like Slater's condition. This is due to Ramana. (For another take on this involving SOS, see [KS12]...
Ryan O'Donnell's user avatar
9 votes

When is the duality gap of semidefinite programming (SDP) zero?

For the SDP in standard form $$ \min\{ \mathrm{tr}(C^T X): \mathrm{tr}(A_1^T X) = b_1, \ldots, \mathrm{tr}(A_m^T X) = b_m, X \succeq 0\}, $$ Slater's condition reduces to the existence of a positive ...
Sasho Nikolov's user avatar
6 votes
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Is there an approximate version of the strong duality theorem for linear programming?

A candidate for a variant of the strong duality theorem is: the primal LP has a solution $\mathbf{x^*}$ for which: $$ \mathbf{b^T y'} \cdot (1+\epsilon) \geq \mathbf{c^T x^*} \geq b^T \mathbf{y'}/(1+\...
Neal Young's user avatar
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1 vote

When is the duality gap of semidefinite programming (SDP) zero?

There is a nice (I think....) characterization of when strong duality holds, or fails for {\em all} objective functions. We say that the semidefinite {\em system} $(P_{SD}) \,\, \sum_{i=1}^m x_i ...
district9's user avatar
  • 111

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