my intuition says I'm right, but I couldn't find or get a proof for it:Suppose I have a vector function x → ( t ) ∈ R n . I am interested in finding a r g m a x t | | x → ( t ) | | 2 . For some reasons, this optimization problem is hard to solve, what I can solve easily, however, is a r g m a x t | | x → ( t ) | | 1 .I know that 1-norm and 2-norm are equivalent in R n . Does this also include that the solution of these two problems are equivalent?

Question

my intuition says I&#039;m right, but I couldn&#039;t find or get a proof for it:Suppose I have a vector function             x      →        (  t  )  ∈            R        n  . I am interested in finding             a      r      g      m      a      x        t        |        |              x      →        (  t  )      |              |        2  . For some reasons, this optimization problem is hard to solve, what I can solve easily, however, is             a      r      g      m      a      x        t        |        |              x      →        (  t  )      |              |        1  .I know that 1-norm and 2-norm are equivalent in             R        n  . Does this also include that the solution of these two problems are equivalent?

hopeloothab9m · Accepted Answer

No, the two problems are not equivalent.Take, for example,             x      →        (  t  )  =  (  cos  ⁡  t  ,  sin  ⁡  t  ) to see why:1.   max  ‖            x      →        (  t  )      ‖    1   is attained at   t  =      π    4    +  k      π    2   for   k  ∈      Z  2.   max  ‖            x      →        (  t  )      ‖    2   is attained at all   t  ∈      R  Clearly, the solutions to these two problems are not the same.

my intuition says I'm right, but I couldn't find or get a proof for it: Suppose I have a vector

Answered question

Answer & Explanation

New Questions in High school geometry