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The constrained ridge regression problem is of the form: $\arg\min_{\|\beta\|_2\le t}\|X\beta-y\|_2$. Given a matrix $X$, a vector $y$ and the constrain parameter $t$, how do you solve it numerically?

I'm aware of the equivalent penalized form of ridge regression, which has a simple solution of the form $(X^TX+\lambda I)^{-1}X^Ty$. Unfortunately, my problem is given in the constrained form, and I don't know any analytical relation between $t$ and $\lambda$.

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This is almost a specific optimization problem called a QCQP. The wikipedia article gives solvers that you can use to find the solution. Just use $\lVert\cdot\rVert^2$ instead of $\lVert\cdot\rVert$, this transforms both the objective and the constraint into quadratic forms in $\beta$.

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