How can I use Lagrange multipliers (multiple constraints)/(with a vector constraint λ)

Question

To show that the pseudoinverse solution based on this form H^+ = H^T(HH^T)^−1 . can be derived by finding the shortest image vector f that satisfies the imaging equation. In other words, show that the solution to the problem f(hat)=argmin1/2f^Tf
s.t g = Hf is the pseudoinverse solution f(hat) = H^T(HH^T)^−1g