Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 8 / i / Solution

With $L(x,y)=f(x)-y^{T}c(x)$, the Karush-Kuhn-Tucker conditions areUnder a suitable constraint qualification, such as the linear independence constraint qualification on active constraint gradients, they are necessary for a local minimum. Smoothness alone is insufficient. The sign convention here corresponds to constraints written as $c_i\ge 0$ and a minus sign in the optimization Lagrangian.