= Lagrange sufficiency theorem
{c}
= Lagrangian sufficiency theorem
{c}
{synonym}
For maximizing $f$ subject to $g_i(x)\geq0$ and $h_j(x)=0$, suppose a feasible $x^*$ and multipliers $\lambda_i\geq0,\mu_j$ satisfy complementary slackness and $x^*$ globally maximizes
$$
L(x)=f(x)+\sum_i\lambda_i g_i(x)+\sum_j\mu_jh_j(x).
$$
Then $x^*$ 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.
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