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Find the minimum value of ||x||1 subject to ||x||2 = 1 in R2. Which x achieves such minimum?

Find the min value of the 1 norm x subject to 2 norm x

Screen Shot 2018-05-01 at 1.57.23 PM.png

Screen Shot 2018-05-01 at 1.58.05 PM.png

Screen Shot 2018-05-01 at 1.57.23 PM.png

Find the minimum value of ||x||1 subject to ||x||2 = 1 in R2. Which x achieves such minimum?

Screen Shot 2018-05-01 at 1.58.05 PM.png

Top Answer

For any  ( x , y ) ∈ R ​ 2 ​ ​  , we have  ∣ ∣ ( x , y ) ∣ ∣ ​ 2 ​ ​ = √ ​ x ​ 2 ​ ​... View the full answer

2 comments
  • hi.. im wondering if $||x||_{infty} = max (|x|, |y|) since according to theorem that ||x||_{1} = |x|+|y| so can we => ||x||_{1} = max (|x|, |y|) ? thank you
    • danie0723
    • May 02, 2018 at 2:27am
  • is it ||x||_1=|x|+|y| in your definition? Then i would have to redo it. In that case answer would be minimum value 0 , which is attained at 3pi/4 and -pi/4
    • Mathwarrior
    • May 02, 2018 at 5:31am

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