QMC: Operator Splitting Workshop, Open Problems - Heinz Bauschke, Mar 23, 2018

•

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The Statistical and Applied Mathematical Sciences Institute

This talk was presented during the Operator Splitting Workshop

Education

Heinz Bauschke
UBC
Okanagan , Kelowna
Open Problems 3
+ 2
SAMSI March 23,2018
-

UO
X Hilbert
space ZYV convex
,
dosed
, -1-9 ,
# ✓with
projectors B) Pv .
-|FiudxeS÷UnV==
Alternating Projections ( AP)
×n+ ,
:=
pvpuxn
#some
Point in S (
Bregman)
Ttahspgxoif U ,V are linear subspaces
( von Neumann
)
1in ,
,
Hundal'sCountere×am#:
¥X=lz U cone
, Vhyperplane ,
S=ho} .
- V
Xunf 0 but Xu /→ 0 .

Why care ? AP is also ...
2
•
Polyak
's sub
gradient projection method for Max
{ du,d✓ }
•
Projected gradient method
for f= Idf I ⇒
Of = Id -
Pu ⇒ ld -
Pf=pu)
in case
of Hundal 's
example ...
•
Rockafellow 's proximal Point
Algorithm
¥¥¥fg§V
Xnt ,
= PU + PRW
) #Aside-2¥ proximal map BY Moreau
Drfor
•Gradient method
for tyfdjotqtmu,
,o)
Bregman ?
Nguyen;
qz :
www.msn.ng.nrergenaprpwgas.ra.nprdi#fEoeuedfefqyqxna=l2Pv-ld)ol2Pu-ld)tldxn
? ?
Ergod:c-
for2 Hundal ?

- U
3
Back to
ftp.#fI0?
¥ ¥4Uu=PuVn .
, ,Vn=P✓un .
T -
✓
Then
( FGEX ) Vn -
Un →
as
um ,
-
vn→→
}⇒htruu=PBun-un→o
and
EITHER :
uu
-
JEFixPuP✓=Un( KS )
OR :
Hunn → a ← Hull
Vu -
TEFixP✓Pu=VnlUtg)
Hiv ) minimizes
, e.g. , 11×-411 lorllxylpetc ) over UXV
Where
g :=
1%-10) "
gap
vector
"

U
¥9'→,
4
AP for 3 @ more sets ) :
Cyclic Projections
¥gqBouillon -
Combetes -
Cominetk
WJ.%9b.tw
no
function such as 1k -
ylkllytlltllz -
XH =Fl× , y ,z )
exist such that
( J , Jiw ) minimizes F over U×V×W !
cycle
QZ :
What is the
meaning of 1
g ,
,gz,g3 ) ? ? Note
gitgztg 3=0 .
strong conjecture
If Vn -
Un →
g ,
weaker
conjecture ( proved
=
wn -
vn →
gz
} ,
then Un+ ,
-
Un → 0
,
ii. /
Una -
Wu →
93
"
almost
cycles
"
exist .
Write ( umw )=X being a
cycle as
x=
(Yw
)=(bYgYr)=PcRx ,
where C=UxV×W
,
RH ,
,XyX3 ) =
( Xz ,
X , )Xz )
cyclic right shift .

5
×
cycle ←→ x= Pc Rx ⇐ > Rxe ( Nett d) x #
OE Next
Idf
) ×
ld -
R=
flog §)Max mono
, 3*
×
"
almost
cycle
"
←→ . . .
⇐ ) OE rain (
Nctldwhich is TRUE because
Tan ( Nctud .
R )) = ran Nctranlld - R ) 70+0
by
Brezisttarauxthegap
vectors( gngugz ) should be the
unique
W) solution to
the
"
extended
"
Altovd -
Theta dual Nd + (
ld⇐
not hlc .
Max Mon .
. -

Q3 : Is There an extension
of Fenchel -
Rockafellow duality
6
to max non
operators
? ?
Of AxtB× c- 0£ At
y
-
B-
tyg
EXTENDED ! them
Primal dual Variation dam
Rerdski
extended sum
etc
-
Final aside ( time
permitting) .
Theorem ( HB -
Walaa Moursi ,
2018 )
Let IT , .
.
,Tm be
firmly nonexpana
've .
Then 11
dlEAiTi)H< Etilldltilll
11
dLTma .
otzoti )Hf EHDITIIH ,
where , for Tnonexpansire ,
AN :=
Prasad #
b)
is the minimal
displacement vector
of T .

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