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It is evident that the two following optimization problems have the same minimizer: $$ \min_{Ax=b} \| x\|_2 $$ That is, $z=\arg\min_{Ax=b} \| x\|_2$ $$ \min_{Ax=b} \| x\|_2^2 $$ That is, $y=\arg\min_{Ax=b} \| x\|_2^2$

However, what would be the way to show that $y=z$ where $x,y,z \in \mathbb{R}^n$? (Contradiction, straight proof,...)

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    $\begingroup$ You can use the fact that $x \mapsto x^2$ is strictly increasing over the domain $[0, \infty)$. So, since $\|x\|_2 \ge 0$, we have $\|x\|_2 < \|y\|_2 \implies \|x\|^2_2 < \|y\|^2_2$. $\endgroup$ – Theo Bendit Oct 8 '18 at 1:40
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I think a correct statement is: $$\text{the two problems has the same solutions set}. $$ Here is a direct argument. Let $z$ be a solution of the first problem. Our goal is to show that $z$ is a solution of the second problem. Indeed, let $x$ be such that $Ax =b$. Then, $\lVert z\rVert_{2} \leq \lVert x \rVert_{2}$ since $z$ is the solution of the first one and $x$ belongs to the feasible set. Squaring both sides, we get $\lVert z\rVert^2_{2} \leq \lVert x\rVert_{2}^{2}$, which shows that $z$ solves the second one. The converse is similar, just take the square root :)

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