= Constraint qualification
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>.<Nonlinear Optimization, MIT 6.7220 and 15.084, Spring 2025/Lecture 7: Lagrange Multipliers and KKT Conditions>[Lecture 7, §L7.1.3, Theorem L7.3, printed p. 4: independent active gradients]{ref} <Slater's condition> provides a different <constraint qualification> for <convex optimization>, yielding <strong duality> and dual attainment.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§5.2.3, printed pp. 226–227: Slater's condition, strong duality and dual attainment]{ref}
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