# How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression?

How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression?

Given lasso (constrained form): $$\underset{\beta}{\min}{(\frac{1}{2N}||y-x\beta||_2^2)} \space subject \space to \space ||\beta||_1 \leq t$$ The Lagrange form: $$\underset{\beta}{\min}{(\frac{1}{2N}||y-x\beta||_2^2)} + \lambda||\beta||_1$$ I have gone through lots materials and try to understand how these two form are equivalent, but still feel very struggled on how to give a relatively rigorous proof. I guess proof for ridge regression is similar to lasso. So I only post equations for lasso. Any comments that helps would be appreciated.

• This is purely mathematics, it has nothing to do with Lasso or any other statistical estimator. It is the fundamental theory of finding the extreme points of a function under constraints using multipliers. This is the literature you should look up. Mar 11, 2020 at 2:46
• I really can't follow the logic behind when finding the extreme points under constraints, could you please provide some intuitive example? Really appreciated Mar 11, 2020 at 3:14