Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 38 / 1 / a / Solution

For a maximization problem with $g_i(u)\le 0$ and $h_j(u)=0$, use the optimization LagrangianThe Lagrangian sufficiency theorem says: if $u^{\ast }$ is feasible, $u^{\ast }$ globally maximizes $L(\cdot ,{\lambda }^{\ast },{\mu }^{\ast })$ over its original domain, and complementary slackness holds, ${\lambda }_i^{\ast }{g}_i(u^{\ast })=0$, then $u^{\ast }$ globally maximizes $f$ over the feasible set. Equality Lagrange multipliers have no sign restriction. For every feasible $u$,which proves the theorem.

For a minimization problem, reverse the signs in the optimization Lagrangian: take $L=f+{\sum }_i{\lambda }_i{g}_i+{\sum }_j{\mu }_j{h}_j$, with ${\lambda }_i\ge 0$. If a feasible $u^{\ast }$ globally minimizes this optimization Lagrangian and satisfies complementary slackness, thenThe hypothesis is a global extremum of the Lagrangian. Merely solving its stationarity equations is insufficient; no convexity assumption is needed when the global extremum itself has been proved.