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.^[ref] Slater's condition provides a different constraint qualification for convex optimization, yielding strong duality and dual attainment.^[ref]

At a feasible point for smooth $h=0$, $g\ge 0$, the equality gradients must be independent and there must be a direction $v$ with $Dhv=0$ and $D{g}_{i}v>0$ for every active inequality. It supplies a first-order direction strictly into all active inequality constraints while respecting the equality tangent space. This constraint qualification ensures the usual Karush-Kuhn-Tucker conditions are necessary for local optimality. For a strictly convex quadratic inequality restricted to an affine equality space, it holds at every feasible point exactly when a strictly feasible point exists; moving from a boundary point toward a strictly feasible point supplies the required direction.

For smooth equality constraints $h_j=0$ and inequality constraints $g_i\ge 0$, this condition requires linear independence of all equality gradients and active inequality gradients at the feasible point. It ensures existence of Lagrange multipliers at a local optimum and uniqueness of their values for a fixed objective gradient. It implies the Mangasarian–Fromovitz constraint qualification by solving the independent gradient equations with zero equality derivatives and positive active inequality derivatives.