Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 42 / 1 / a

For a minimization problem on a domain $D$, write the equality constraints as $h(x)=0$ and the inequality constraints as $g(x)\le 0$. Define the optimization LagrangianThe Lagrangian sufficiency theorem states that a feasible $x^{\ast }$ is globally optimal if it globally minimizes $L(\cdot ,{\lambda }^{\ast },{\mu }^{\ast })$ on $D$ for some multipliers with ${\mu }^{\ast }\ge 0$ and satisfies complementary slackness, ${\mu }_j^{\ast }{g}_j(x^{\ast })=0$ for every $j$. Equality multipliers have unrestricted signs.

For any feasible $x$, the multiplier sign gives $L(x,{\lambda }^{\ast },{\mu }^{\ast })\le f(x)$. Feasibility and complementary slackness at $x^{\ast }$ give $L(x^{\ast },{\lambda }^{\ast },{\mu }^{\ast })=f(x^{\ast })$. HenceThis proves sufficiency. Global Lagrangian minimization, rather than stationarity alone, is the certificate. No convexity or constraint qualification is needed for this implication; convexity is one way to verify the required global minimum.