Minimum squared norm under two affine constraints ($x_i^{\ast }=(S_2-S_1{a}_i)/(n{S}_2-S_1^2)$)

For real $a_i$ not all equal, let $S_1={\sum }_i{a}_i$, $S_2={\sum }_i{a}_i^2$ and $\Delta =n{S}_2-S_1^2>0$. The unique minimizer of ${\sum }_i{x}_i^2$ under ${\sum }_i{x}_i=1$, ${\sum }_i{a}_i{x}_i=0$ is $x_i^{\ast }=(S_2-S_1{a}_i)/\Delta$, with minimum $S_2/\Delta$. To prove sufficiency, the optimization Lagrangian with multipliers $\lambda =S_2/\Delta$, $\mu =-S_1/\Delta$ is a sum of squares in $x_i-\lambda -\mu a_i$ plus a constant. Equivalently every feasible $x$ satisfies ${\sum }_i{x}_i^2-{\sum }_i(x_i^{\ast }{)}^2={\sum }_i(x_i-x_i^{\ast }{)}^2$.