Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 6 / i / Solution

Use the optimization Lagrangian $L(x,y)=f(x)-y^{T}c(x)$ and let $A=Dc$ be the $m\times n$ constraint Jacobian matrix. Under the standard regularity hypothesis $\mathrm{rank}A(x_{\ast })=m$, a local constrained minimizer satisfies, for some multiplier $y_{\ast }$,The second-order necessary condition isRegularity makes each such tangent vector the velocity of a feasible smooth curve. Twice differentiating the objective along that curve and using stationarity and the differentiated constraints gives the displayed nonnegative second derivative.

The regularity hypothesis is needed for these multiplier conditions to be necessary in this form. For example minimizing $f(x)=x$ subject to $c(x)=x^2=0$ has the unique feasible minimizer zero, but $f^{\prime }(0)-y{c}^{\prime }(0)=1$ for every $y$. The question does not explicitly state the qualification, so it must be supplied when using the usual KKT conditions.