nidantasnu

Answered

2022-07-10

Let

${\mathrm{\Delta}}_{n-1}:=\{x\in {R}^{n}:{x}_{1}+{x}_{2}+....{x}_{n}=1,{x}_{1},{x}_{2},....{x}_{n}\ge 0\}$

and

$a\in {R}^{n}$

Let

$z:={P}_{{\mathrm{\Delta}}_{n-1}}(a)$

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.

${\mathrm{\Delta}}_{n-1}:=\{x\in {R}^{n}:{x}_{1}+{x}_{2}+....{x}_{n}=1,{x}_{1},{x}_{2},....{x}_{n}\ge 0\}$

and

$a\in {R}^{n}$

Let

$z:={P}_{{\mathrm{\Delta}}_{n-1}}(a)$

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

$z={\textstyle \text{argmin}}\{(x-a{)}^{T}(x-a):{x}^{T}e=1,x\ge 0\}$ 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 $\u27e8\mathrm{\nabla}f(z),y-z\u27e9\ge 0,\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall}y\in C$

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