I've been given a problem:
min $f(x) = c^Tx$
s.t.
$\sum_{i}x_i = 0$
$\sum_{i}x^2_i$ = 1
I've created the Lagrangian function
$L(x,\lambda) = c^Tx - \lambda_1\sum_{i}x_i - \lambda_2\left(\left(\sum_{i}x^2_i\right)-1\right)$
But I'm not sure how I can arrive at any particular solution given the $\nabla_x$ of the Lagrangian doesn't really help me. Should I augment the function to be an unconstrained minimization by replacing the values of $x_i$ using the 2 equalities? I'm not sure where to proceed.