Karush-Kuhn-Tucker conditions

For differentiable inequality constraints $g_i(x)\le 0$, a regular local minimum has multipliers ${\lambda }_i\ge 0$ satisfyingFor a convex problem under a suitable constraint qualification, these conditions are sufficient for global optimality.

In a convex resource-allocation problem, tightening an upper bound can change its multiplier from zero to positive. The optimum then moves from a point where only the total-resource constraint is active to the intersection where both constraints are active.