Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / B / 5 / i / Solution

Use the optimization Lagrangian $\mathcal{L}(x,y)=f(x)-y^{T}c(x)$, with $A(x)=Dc(x)$ and $H(x,y)={\nabla }_{xx}^2\mathcal{L}(x,y)$. Under the usual linear independence constraint qualification, a local minimizer has multipliers $y_{\ast }$ satisfying the equality-constrained Karush-Kuhn-Tucker conditionsThe second-order necessary condition isOnly feasible tangent directions enter this last test; the Hessian matrix need not be positive semidefinite on the entire ambient space.