Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 1 / c

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- Solution c

Solution

The two Lasso problems are

min_{β} ∥ Y - x β ∥_{2}^{2} subject to ∣ β ∣ \leq t

and

min_{β_{1}, β_{2}} ∥ Y - x (β_{1} + β_{2}) ∥_{2}^{2} subject to ∣ β_{1} ∣ + ∣ β_{2} ∣ \leq t .

If

β

solves the first problem, the complete solution set of the second is

{(β_{1}, β_{2}) : β_{1} + β_{2} = β, ∣ β_{1} ∣ + ∣ β_{2} ∣ \leq t} .

Indeed, the feasible totals are exactly

[- t, t]

, the same as in the first problem.

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