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.

For $\gamma >0$ and $c\ne 0$, minimize $c^{T}u+\gamma z^2$ subject to $\Vert u{\Vert }^2+z=R$. Adding a positive Lagrange multiplier times the constraint gives a positive-definite quadratic optimization Lagrangian, whose minimizer is $u=-c/(2\lambda )$ and $z=-\lambda /(2\gamma )$. Feasibility is the displayed scalar equation. Its left side strictly decreases from positive infinity to negative infinity on $\lambda >0$, so it has a unique root for every real $R$. The Lagrangian sufficiency theorem certifies a unique global optimum on the possibly nonconvex equality surface.