Exercise 3.4.3

We have Lagrangian as $L(x,y)=\lambda \parallel x{\text{∥}}_1+y(x^{T}w-1)$ where $w=(2,3)$, assume $x=(x_1,x_2)$, we have $L(x,y)=\lambda (|x_1|+|x_2|)+y(2x_1+3x_2-1)$. Take derivatives of $L(x,y)$ w.r.t. $x_1,x_2,y$, we have

• $\frac{\mathit{\partial L}}{\partial x_1}=\lambda x_1+2y=0$ when $x_1>0$
• $\frac{\mathit{\partial L}}{\partial x_2}=\lambda x_2+3y=0$ when $x_2>0$
• $\frac{\mathit{\partial L}}{\mathit{\partial y}}=2x_1+3x_2-1=0$

Solve the equations we find $x_1=\frac{2}{13},x_2=\frac{3}{13}$