A constraint qualification is a regularity hypothesis on optimization constraints that guarantees a specified optimality or duality conclusion. For the necessity of Karush-Kuhn-Tucker conditions, an appropriate constraint qualification ensures that a local minimum admits suitable Lagrange multipliers; linear independence of the gradients of equality optimization constraints and active constraints is one sufficient hypothesis in KKT regularity at a local minimum.[ref]
Lecture 7: Lagrange Multipliers and KKT Conditions. Lecture 7, §L7.1.3, Theorem L7.3, printed p. 4: independent active gradients
Slater's condition provides a different constraint qualification for convex optimization, yielding strong duality and dual attainment.[ref]Convex Optimization by Stephen Boyd and Lieven Vandenberghe. §5.2.3, printed pp. 226–227: Slater's condition, strong duality and dual attainment
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