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Exercise 3.4.7

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We have

\begin{array}{l} \frac{1}{2} ∥ Ax - b ∥_{2}^{2} + λ ∥ x ∥_{1} & = \frac{1}{2} {(4 x_{1} + x_{2} - 1)}^{2} + \frac{1}{2} {(x_{2} - 1)}^{2} + 2 (| x_{1} | + | x_{2} |) \end{array}

Take derivative w.r.t. $x$ , we have

\begin{array}{l} \frac{∂L}{\partial x_{1}} & = 4 (4 x_{1} + x_{2} - 1) + 2 sign (x_{1}) x_{1} = (8 + sign (x_{1})) x_{1} + 2 x_{2} - 2 = 0 \\ \frac{∂L}{\partial x_{2}} & = (4 x_{1} + x_{2} - 1) + (x_{2} - 1) + 2 sign (x_{2}) x_{2} = 2 x_{1} + (1 + sign (x_{2})) x_{2} - 1 = 0 \end{array}

Solve the equations under different cases with $x_{1} \geq 0, x_{2} \geq 0$ , $x_{1} \geq 0, x_{2} 0$ , $x_{1} 0, x_{2} \geq 0$ , $x_{1} 0, x_{2} 0$ , we found two solutions:

$(x_{1}, x_{2}) = (\frac{1}{7}, \frac{5}{14})$ and $(x_{1}, x_{2}) = (\frac{1}{2}, - \frac{5}{4})$

Take these solutions back to the Lagrangian function, we found it achieves minimal when $(x_{1}, x_{2}) = (\frac{1}{7}, \frac{5}{14})$ where the function has a value about $1.2091836734693877$

niuers

2020-03-20 00:00

Exercise 3.4.7

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