Let Δ n − 1 := { x ∈ R n : x 1 + x 2 + . . . . x n = 1 , x 1 , x 2 , . . . . x n ≥ 0 }and a ∈ R n Let z := P Δ n − 1 ( a )be the projection of point a onto Δ n − 1 . Show that z satisfies the system of inequalities z − y = a − μ e , z ≥ 0 , y ≥ 0 , z T y = 0where e is the vector of all ones. y , z ∈ R n , μ ∈ R. One can use obtuse angle condition of the projection theorem over the convex set along with Farkas Lemma.I don't know how to approach this problem.

Question

Let       Δ          n      −      1        :=  {  x  ∈      R      n    :      x    1    +      x    2    +  .  .  .  .      x    n    =  1  ,      x    1    ,      x    2    ,  .  .  .  .      x    n    ≥  0  }and  a  ∈      R    n  Let  z  :=      P            Δ              n        −        1              (  a  )be the projection of point a onto       Δ          n      −      1      . Show that   z satisfies the system of inequalities  z  −  y  =  a  −  μ  e  ,  z  ≥  0  ,  y  ≥  0  ,      z    T    y  =  0where   e is the vector of all ones.   y  ,  z  ∈      R    n    ,  μ  ∈  R. One can use obtuse angle condition of the projection theorem over the convex set along with Farkas Lemma.I don&#039;t know how to approach this problem.

iskakanjulc · Accepted Answer

z  =      argmin    {  (  x  −  a      )    T    (  x  −  a  )  :      x    T    e  =  1  ,  x  ≥  0  } use KKT

Callum Dudley · Accepted Answer

Use first order conditions:   z minimizes a smooth convex function   f on a closed convex set   C iff   ⟨  ∇  f  (  z  )  ,  y  −  z  ⟩  ≥  0  ,    ∀  y  ∈  C

Let <mi mathvariant="normal">Δ n −

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