Affirmative Action Programs
30
pages
English
Documents
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
30
pages
English
Documents
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
- fiche de synthèse - matière potentielle : job announcements for distribution
- program progress on specific action items
- occupational data
- available labor pools of the communities
- wsdot
- affirmative plan executive summary
- percent of the wsdot workforce to the percent of the available workforce
- su
- affirmative action plan
- career development programs
- workforce
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methods
12.Cutting-plane
•
•
•
•
EE236C(Spring2010-11)
1-21
)ecitcarpdnayroehtni(tneicffieeromhcumebnactubnahtpetsrepnoitatupmocdnayromemesmelborp)xevnocisauqdna(xevnocelbaitnereffidnoneldnahpetshcaetasnoitcnuftniartsnocroevitcejbofotneidargbusenoeriuqerpetshcaetarellamssemocebhcihw,tesemosnitniopderised’ezilacol‘
Localizationmethods
•
•
typicallyrequiremor
subgradientmethod
•
•
•
-212
Cutting-planemethods
Cutting-planeoracle
n
goal:
findapointinconvexset
C
⊆
R
,ordeterminethat
C
isempty
n
cutting-planeoracle:
when
queried
at
x
∈
R
,oracleeither
•
assertsthat
x
∈
•
returnsasepraaCti,ngrohyperplanebewtee
a
T
z
≤
b
for
z
∈
C,
oracleprovidesblack-boxdescriptionof
C
Cutting-planemethods
nxandC:a
Tb≥xa
6=,0
13-2
Neutralanddeepcuts
neutralcut:
a
T
x
=
b
(querypointisonboundaryofhalfspacethatiscut)
deepcut:
a
T
x>b
(querypointisininteriorofhalfspacethatiscut)
Cutting-planemethods
C
x
C
x
214-
Unconstrainedminimization
takesetofminimizersof
f
as
C
cutting-planeoracle
(forconvex
f
):if
0
6
=
g
∈
∂f
(
x
)
,then
g
T
(
z
−
x
)
≤
0
definesa(neutral)cut
(
a,b
)=(
g,g
T
x
)
at
x
proof:
g
T
(
z
−
x
)
>
0
implies
z
6∈
C
because
Cutting-planemethods
f
(
z
)
≥
f
(
x
)+
g
T
(
z
−
x
)
)x(f>
5-21
interpretation
levelcurvesof
f
•
byevaluating
g
∈
∂f(x)ewruleo
x
ut
g
g
T
(
z
−
x
)
≥
0
halfspaceinserach
•
getone‘bit’ofinfo(onlocationof
x
⋆
)byevaluating
g
Cutting-planemethods
frooptimum
6-21
Deepcutforunconstrainedminimization
supposeweknowanumber
f
¯
with
f
(
x
)
>f
¯
≥
f
⋆
(
e.g.
,thesmallestvalueof
f
foundsofarinanalgorithm)
deepcut:
if
g
∈
∂f(x),thenadeepcutisgivenb
g
T
(
z
−
x
)+
f
(
x
)
−
f
¯
≤
0
y
proof:
f
(
x
)+
g
T
(
z
−
x
)
>f
¯
implies
f
(
z
)
>f
⋆
,so
z
6∈
C
Cutting-planemethods
7-21
Feasibilityproblem
C
issolutionsetofconvexinequalities
cutting-planeoracle
f
i
(
x
)
≤
0
,i
=1
,...,m
if
x
notfeasible,find
j
with
f
j
(
x
)
>
0
andevaluate
g
j
∈
∂f
j
(
x
)
;
isadeepcut
f
j
(
x
)+
g
jT
(
z
−
x
)
≤
0
proof:
f
j
(
x
)+
g
jT
(
z
−
x
)
>
0
implies
z
6∈
C
because
Cutting-planemethods
f
j
(
z
)
≥
f
j
(
x
)+
g
jT
(
z
−
x
)
>
0
8-21
Inequalityconstrainedproblem
C
isthesetofoptimalpointsofconvexproblem
minimize
f
0
(
x
)
subjectto
f
i
(
x
)
≤
0
,i
=1
,...,m
cutting-planeoracle
•
•
iiffxxiissnfoetasfiebale
fyas,elbisj
(x)>0,ewhave(deep)feasibili
f
j
(
x
)+
g
jT
(
z
−
x
)
≤
0
where
g
j
∈
∂f
j
(
x
)
,ewhave(neutral)objectivecut
g
0
T
(
z
−
x
)
≤
0
where
g
0
∈
∂f
0
(
x
)
ytcut
(or,deepcut
g
0
T
(
z
−
x
)+
f
0
(
x
)
−
f
¯
≤
0
if
f
¯
∈
[
p
⋆
,f
0
(
x
))
isknown)
Cutting-planemethods
19-2
(Conceptual)cutting-planealgorithm
given
initialpolyhedron
P
0
=
{
z
|
Az
b
}
knowntocontain
C
repeat
for
k
=1
,
2
,...
:
chooseapoint
x
(
k
)
in
P
k
−
1
andquerythecutting-planeoracleat
x
(
k
)
•
if
x
(
k
)
∈
C
,return
x
(
k
)
•
else,addnewcutting-plane
a
kT
z
≤
b
k
:
if
P
k
=
∅
,
Cutting-planemethods
quit
P
k
:=
P
k
−
1
∩{
z
|
a
kT
z
≤
b
k
}
01-21
geometry
P
k
giv
P−k1
esunce
Cutting-planemethods
rt
ak
)k(x
ainytofCafteriterationk
Pk
ak
)k(x
11-21
