 Lagrange sufficiency theorem

For maximizing $f$ subject to $g_i(x)\ge 0$ and $h_j(x)=0$, suppose a feasible $x^{\ast }$ and multipliers ${\lambda }_i\ge 0,{\mu }_j$ satisfy complementary slackness and $x^{\ast }$ globally maximizesThen $x^{\ast }$ globally maximizes $f$ on the feasible set. Convexity or concavity hypotheses are commonly used to prove the required global extremum of the Lagrangian from first-order conditions.