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If not all constraints satisfy equalities, does Lagrangian dual method make sense to a convex problem?

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    $\begingroup$ It'll make sense, just make sure that for inequality constraints, their corresponding dual variables must have the correct sign (nonnegative or nonpositive, depending on the inequality and if the problem is maximization or minimization). $\endgroup$
    – dhasson
    Jul 17, 2020 at 16:31
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    $\begingroup$ When you asked at ask.cvxr.com/t/… , you didn't mention anything about "not all constraints satisfy equalities," So I'll provide the same comment here as I did there, which answers the question in the topic title: If the Slater condition is not satisfied, there could be a duality gap, which is contrary to the premise under which primal-dual algorithms operate, and which they rely on for termination. $\endgroup$ Jul 18, 2020 at 4:29

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