nidantasnu

Answered

2022-07-10

Let
${\mathrm{\Delta }}_{n-1}:=\left\{x\in {R}^{n}:{x}_{1}+{x}_{2}+....{x}_{n}=1,{x}_{1},{x}_{2},....{x}_{n}\ge 0\right\}$
and
$a\in {R}^{n}$
Let
$z:={P}_{{\mathrm{\Delta }}_{n-1}}\left(a\right)$
be the projection of point a onto ${\mathrm{\Delta }}_{n-1}$. Show that $z$ satisfies the system of inequalities
$z-y=a-\mu \mathbf{\text{e}},z\ge 0,y\ge 0,{z}^{T}y=0$
where $\mathbf{\text{e}}$ is the vector of all ones. $y,z\in {R}^{n},\mu \in R$. 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

Expert

2022-07-11Added 18 answers

$z=\text{argmin}\left\{\left(x-a{\right)}^{T}\left(x-a\right):{x}^{T}e=1,x\ge 0\right\}$ use KKT

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 $⟨\mathrm{\nabla }f\left(z\right),y-z⟩\ge 0,\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }y\in C$

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