Affirmative Action Programs

Description

• fiche de synthèse - matière potentielle : job announcements for distribution

WSDOT Equal Employment Opportunity and Affirmative Action Program 2007 – 2011 Fiscal Years Office of Equal Opportunity 310 Maple Park Avenue SE P.O. Box 47314 Olympia, WA 98504-7314

• 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

Sujets

• Petersen
• Classe

Informations

• Publié par: phewyag
• Nombre de lectures: 23
• Langue: English

Extrait

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methods

12.Cutting-plane

•

•

•

•

EE236C(Spring2010-11)

1-21

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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 ∈

•

returnsasepraa C ti , ng ro hyperplanebewtee

a T z ≤ b for z ∈ C,

oracleprovidesblack-boxdescriptionof C

Cutting-planemethods

nxandC:a

T b≥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 j T ( z − x ) ≤ 0

proof: f j ( x )+ g j T ( z − x ) > 0 implies z 6∈ C because

Cutting-planemethods

f j ( z ) ≥ f j ( x )+ g j T ( z − x ) > 0

8-21

Inequalityconstrainedproblem

C isthesetofoptimalpointsofconvexproblem

minimize f 0 ( x ) subjectto f i ( x ) ≤ 0 ,i =1 ,...,m

cutting-planeoracle

•

•

i i f f x x i i s s n f o e t as f i e b a le

fyas,elbis j

(x)>0,ewhave(deep) feasibili

f j ( x )+ g j T ( 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 k T z ≤ b k :

if P k = ∅

,

Cutting-planemethods

quit

P k := P k − 1 ∩{ z | a k T z ≤ b k }

01-21

geometry

P

k

giv

P −k1

esunce

Cutting-planemethods

rt

a k

)k( x

ainytofCafteriterationk

P k

a k

)k( x

11-21