2.1 Recap on Duality
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چکیده
• The dual problem is always convex no matter if the primal problem is convex, i.e., g is always concave. • The primal and dual optimal values, f∗ and g∗, always satisfy weak duality: f∗ ≥ g∗. • Slater’s condition: for convex primal, if there is an x such that h1(x) < 0, · · · , hm(x) < 0 and l1(x) = 0, · · · , lr(x) = 0 (12.5) then strong duality holds: f∗ = g∗. Note that the condition can be further relaxed to strict inequalities over the nonaffine hi, i = 1, · · · ,m.
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