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The equality-constrained Lagrangian sufficiency theorem says: if $x^{\ast }$ is feasible and, for some fixed multipliers, it globally minimizes the optimization Lagrangian over all $x$, then it globally minimizes the objective over feasible $x$. On the feasible set the constraint terms vanish, so the two minimizations agree. No convexity assumption is needed for this version once global minimization of the optimization Lagrangian has been established.

Let $S_1={\sum }_i{a}_i$, $S_2={\sum }_i{a}_i^2$, and $\Delta =n{S}_2-S_1^2$. Sincethe two constraints are independent. For the squared Euclidean norm objective in the original PDF, takeCompleting the square shows that its unique global minimizer has $x_i=\lambda +\mu a_i$. The constraints become $n\lambda +S_1\mu =1$, $S_1\lambda +S_2\mu =0$, giving $\lambda =S_2/\Delta$, $\mu =-S_1/\Delta$. The Lagrangian sufficiency theorem therefore givesFor the value, use ${\sum }_i(x_i^{\ast }{)}^2=\lambda {\sum }_i{x}_i^{\ast }+\mu {\sum }_i{a}_i{x}_i^{\ast }=\lambda$. Uniqueness also follows directly: for every feasible $x$, the cross term in ${\sum }_i(x_i-x_i^{\ast }{)}^2$ vanishes, so the difference of objectives equals that strictly positive sum unless $x=x^{\ast }$. This is the minimum squared norm under two affine constraints.