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suppose we have a primal semidefinite programming. for finding its dual we use Lagrange multiplier $w_i$ for each semidefinite constraint. If the Lagrange multiplier be zero for one constraint what we could say for its related complementary constraint in dual?

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For all you ever wanted to know about this, check out Boyd and Vanderberghe's Convex Optimization, especially the part on KKT. In fact, that page advertises a MOOC on Convex Optimization...

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