Let Δ n − 1 := { x ∈ R n : x 1 +...

nidantasnu

nidantasnu

Answered

2022-07-10

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 = 0
where 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.

Answer & Explanation

iskakanjulc

iskakanjulc

Expert

2022-07-11Added 18 answers

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

Callum Dudley

Expert

2022-07-12Added 4 answers

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

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