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I know that if an arbitrary constraint qualification holds in our convex problem, then strong duality holds.

However, does every convex problem which strong duality holds for that should satisfy any CQ? especially LICQ? if no, please provide a counter-example. (A convex minimization problem which strong duality holds for that but LICQ is not)

Phillip A. Hendrick
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  • Take a problem with constraint $0x=0$. LICQ is not satisfied, but strong duality should be satisfied because the constraint is linear. – daw Jul 03 '22 at 18:15
  • $\min { x | x^2 \le 0 }$. – copper.hat Jul 03 '22 at 19:42
  • I highly recommend you to check out the following paper to understand CQs: Wachsmuth, Gerd. "On LICQ and the uniqueness of Lagrange multipliers."(2013). – Kurban Jul 22 '22 at 19:55

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