why constrained regression and Regularized regression equivalent

problem 1:

　　$min_{eta} ~f_alpha(eta):=frac{1}{2}Vert y-XetaVert^2 +alphaVert etaVert$

problem 2:

　　$min_{eta} ~frac{1}{2}Vert y-XetaVert^2 \ s.t.~Vert etaVert-cleq 0$

problem 2 Lagrangian:

      $mathcal{L}(eta,lambda)=frac{1}{2}Vert y-XetaVert^2+lambda (Vert etaVert-c)$

kkt shows:

dual-inner optimal:$eta^*=min_{eta}~mathcal{L}(eta,lambda):=frac{1}{2}Vert y-XetaVert^2+lambda (Vert etaVert-c)$

primal-inner optimal:$lambda^*(Vert etaVert-c)=0$

 

for problem 1:

$eta^*=min_{eta} ~f_alpha(eta):=frac{1}{2}Vert y-XetaVert^2 +alphaVert etaVert$

set $lambda = alpha$ and $c=Vert etaVert$

can see both kkt conditions meet