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Duality in Linear ProgrammingAbstract| Outline| Posted:June 23rd, 2012 | Last Modified:September 24th, 2013 | Prerequisites: Optimization

by Linear Programming| Tags:Linear Programming, Mathematic, Optimization| Views:331! |"o #omment $My artic%e on %inear programmingi a prere&uiite to thi artic%e'

(ua%ity i a concept )rom mathematica% programming' *n the cae o) %inear programming, dua%ity

yie%d many more amazing reu%t'

The dual linear program

+he dua%ity theory in %inear programming yie%d p%enty o) etraordinary reu%t, becaue o) thepeci)ic tructure o) %inear program' *n order to ep%ain dua%ity to you, *-%% ue the eamp%e o)

the mart robber * ued in the artic%e on %inear programming'.aica%%y, the mart robber /ant to

tea% a much go%d and do%%ar bi%% a he can' e i %imited by the o%ume o) hi bacpac and the

maima% /eight he can carry' "o/, %et- notice that /e can /rite the prob%em a )o%%o/'

+he prob%em /e hae /ritten here no matter /hich e&uia%ent )ormu%ation /e ued i /hat /e

ca%% the prima% %inear program' *t- no/ time )or you to %earn about the dua% %inear program5 +he

dua% program /i%% tota%%y change our undertanding o) the prob%em, and that- /hy it- o coo%' *hope you are a ecited a * am5

*n the prima% program, contraint had contant number on their right' +hee contant number

are our reource' +hey ay /hat /e are ab%e to do concerning each contraint' +he dua%

prob%em conit in ea%uating ho/ much our combined reource are /orth' *) the o))er meetthe demand, our reource /i%% be 6ut a much a their potentia%, /hich i the /orth o) the

robbery' S/eet, right7

8eource are a concept * hae come up /ith' *t- not a tandard concept'

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Let- go more into detai%' *n the dua% prob%em /e /i%% attribute a%ue to the reource a in

9ho/ much they-re /orth:' +hee a%ue are ca%%ed the dua% ariab%e' *n our cae, /e hae t/o

contraint, o /e /i%% hae 2 dua% ariab%e' +he )irt dua% ariab%e, %et- ca%% it

re)er to the a%ue o) one unit o) o%ume' ; you hae )igured it out, the econd dua% ariab%e

re)er to the a%ue o) one unit o) /eight' eem %ie a right name )or it, right7

"o/, * bet you can /rite the a%ue o) the robbery /ith thee t/o ne/ ariab%e< Let- ee i) /e

get the ame reu%t'

+hat- nice but ho/ are the a%ue per o%ume and per /eight determined7

*) * /anted to e%% my reource, potentia% buyer are going to minimize the a%ue o) my

reource' So their a%uation are the minimum o) the tota% a%ue' .ut a a e%%er, * /i%% argue

that each o) my reource i /orth a %ot, becaue it enab%e the robbery o) more go%d and morebi%%'

Obiou%y, the a%ue o) reource depend on the actua% a%ue o) go%d and bi%% per o%ume'

Let- hae a thought about the a%ue o) go%d and then you-%% be ab%e to app%y the ame reaoning

to bi%%' *) the contraint enab%ed u to tea% one more o%ume o) go%d, then incrementa% a%ue o)the robbery /ou%d be at %eat the a%ue o) thi one o%ume o) go%d, right7 *t cou%d be more, i) /e

ue the ne/ contraint to tea% omething e%e than go%d that- /orth more' =hat *-m aying ithat, i) the tota% o%ume enab%ed u to tea% one more unit o) o%ume o) go%d, and i) /e cou%dcarry one more unit o) /eight o) one o%ume o) go%d, then the a%ue o) thi incrementa% tea%

/ou%d be at %eat the a%ue o) one more o%ume o) go%d' Let- /rite it'

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*-%% %et you /rite the imi%ar contraint )or bi%%' *n )act, /e can do thi reaoning /ith any

ariab%e o) the prima% prob%em and a oure%e the &uetion> 9*) /e add one unit o) the

ariab%e, ho/ /i%% it a))ect the tota% a%uation7: ?rom that /e deduce a contraint on dua%ariab%e' ; you ee, any ariab%e in the prima% prob%em i aociated /ith a contraint in the

dua% prob%em and ice@era'

=e are a%mot done' Let- notice the )act that i) /e increae the tota% o%ume, then /e hae more

poibi%itie )or the prima% ariab%e, /hich are the o%ume o) to%en go%d and bi%%' +here)ore,the a%ue o) a unit o) o%ume cannot be negatie' +hi add t/o more contraint on the ign o)

dua% ariab%e' "o/, /e-re done and /e can /rite the dua% prob%em'

ey, that %oo %ie a %inear program