Nonlinear constraints and constraint qualifications

Use equalities h=0 and inequalities g≤0. The Lagrangian adds equality multipliers without sign restrictions and nonnegative inequality multipliers. At a feasible point, only inequalities with zero value are active; inactive ones have zero multiplier under complementarity. A first-order feasible tangent must satisfy zero linearized equality change and nonpositive change for each active inequality. The resulting linearizing cone can be larger than the cone of actual feasible tangent directions when the constraint representation is singular.

The linear independence constraint qualification, LICQ, asks the equality gradients together with all active inequality gradients to be linearly independent at the point. Under continuous differentiability, a local minimum satisfying LICQ admits KKT multipliers, and that multiplier vector is unique. The condition is sufficient rather than necessary. Duplicate active inequalities can break LICQ while still allowing multipliers and valid optimality. A numerical rank test should therefore trigger diagnosis of the formulation, not an unsupported declaration that the optimization problem has no solution.

Mangasarian–Fromovitz constraint qualification, MFCQ, requires independent equality gradients and a direction tangent to every equality that strictly decreases every active inequality. It says that a first-order move can enter the inequality interior while staying tangent to the equalities. If no inequality is active, only the equality-rank requirement remains. LICQ implies MFCQ, but the converse fails because redundant outward normals can still share a strict inward direction. MFCQ suffices for existence of KKT multipliers at a smooth local minimum, without promising uniqueness.

For example, the constraints −x≤0 and −2x≤0 describe the same half-line. At zero their gradients are dependent, so LICQ fails. The direction d=1 strictly decreases both constraint functions, so MFCQ holds. With objective f(x)=x, zero is the optimum and stationarity reads one minus the first multiplier minus twice the second equals zero. Many nonnegative multiplier pairs satisfy it. Redundant descriptions can therefore preserve necessity while destroying uniqueness and worsening the conditioning of numerical systems.

KKT necessity is only a first-order conclusion. Constrained second-order conditions involve the Hessian of the Lagrangian on suitable critical directions, not simply the Hessian of the objective on all coordinates. The constraint curvature contributes through its multipliers. This is why replacing a curved feasible set by its tangent plane can describe possible first-order motions while missing second-order changes in objective value. A full sufficient theorem must specify the feasible point, multipliers, smoothness, and positivity on its critical cone.

Scaling a constraint by a nonzero constant preserves its equality set but changes its multiplier scale. Squaring an equality is more dangerous: its gradient vanishes on every feasible point. That transformation preserves the exact feasible set and destroys the regular first-order representation. Before tuning an optimizer, remove avoidable degeneracy, choose consistent units, and inspect active Jacobian rank. A tiny stationarity residual computed with enormous multipliers may describe numerical cancellation rather than a trustworthy practical certificate.