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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.

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    $\begingroup$ 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. $\endgroup$ Mar 11, 2020 at 2:46
  • $\begingroup$ I really can't follow the logic behind when finding the extreme points under constraints, could you please provide some intuitive example? Really appreciated $\endgroup$ Mar 11, 2020 at 3:14

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