4 duality theory

8/11/2019 4 Duality Theory

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Duality Theory


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One of the most important discoveries in the

early development of linear programming wasthe concept of duality.

Every linear programming problem is associated

with another linear programming problem calledthe dual.

The relationships between the dual problem and

the original problem (called the primal) prove

to be extremely useful in a variety of ways.


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The dual problem uses exactly the same parameters

as the primal problem, but in different location.

Primal and Dual Problems

Primal Problem Dual Problem

Max

s.t.

Min

s.t.

n

j

jjxcZ1

,

m

i

iiybW1

,

n

j

ijij bxa1

, m

i

jiij cya1

,

for for.,,2,1 mi .,,2,1 nj

for .,,2,1 mi for .,,2,1 nj ,0jx ,0iy


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In matrix notation

Primal Problem Dual Problem

Maximize

subject to

.0x .0y

Minimize

subject to

bAx cyA

,cxZ ,ybW

Where and are row

vectors but and are column vectors.

c myyyy ,,, 21

b x


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Example

Max

s.t.

Min

s.t.

Primal Problemin Algebraic Form Dual Problemin Algebraic Form

,53 21 xxZ

,18124 321 yyyW

1823 21 xx

122 2x41x

0x,0x 21 522 32

yy

33 3 y1y

0y,0y,0y 321


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Max

s.t.

Primal Problemin Matrix Form

Dual Problemin Matrix Form

Min

s.t.

,5,32

1

x

xZ

18

12

4

,

2

2

0

3

0

1

2

1

x

x

.0

0

2

1

x

x .0,0,0,, 321 yyy

5,3

2

2

0

3

0

1

,, 321

yyy

18

124

,, 321 yyyW


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Primal-dual table for linear programming

Primal Problem

Coefficient of:RightSide

Righ

t

SideD

ualProblem

Coefficient

of

:

my

y

y

2

1

21

11

a

a

22

12

a

a

n

n

a

a

2

1

1x 2x nx

1c 2c ncVI VI VI

Coefficients forObjective Function

(Maximize)

1b

mna2ma1ma

2b

mb

Coefficients

for

Obj

ectiveFunction

(Minimize)


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One Problem Other ProblemConstraint Variable

Objective function Right sides

i i

Relationships between Primal and Dual Problems

Minimization Maximization

Variables

Variables

Constraints

Constraints

0

0

0

0

Unrestricted

Unrestricted


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The feasible solutions for a dual problem are

those that satisfy the condition of optimality for

its primal problem.

A maximum value of Z in a primal problem

equals the minimum value of W in the dualproblem.


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Rationale: Primal to Dual Reformulation

Max cx

s.t. Ax b

x 0L(X,Y) = cx - y(Ax - b)

=yb + (c - yA) x

Min yb

s.t. yA c

y 0

Lagrangian Function )],([ YXL

X

YXL

)],([=c-yA


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The following relation is always maintained

yAx yb (from Primal: Ax b)

yAx cx (from Dual : yA c)

From (1) and (2), we have (Weak Duality)cx yAx yb

At optimality

cx* = y*Ax* = y*b

is always maintained (Strong Duality).

(1)

(2)

(3)

(4)


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Complementary slackness Conditions are

obtained from (4)

( c - y*A ) x* = 0

y*( b - Ax*) = 0

xj* > 0 y*aj= cj , y*aj> cj xj* = 0

yi* > 0 aix* = bi , ai x*


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Any pair of primal and dual problems can be

converted to each other.

The dual of a dual problem always is the primal

problem.


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Min W = yb,s.t. yA c

y 0.

Dual Problem

Max (-W) = -yb,s.t. -yA -c

y 0.

Converted toStandard Form

Min (-Z) = -cx,s.t. -Ax -b

x 0.

Its Dual Problem

Max Z = cx,s.t. Ax b

x 0.

Converted toStandard Form


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Min

s.t.

64.06.065.05.0

7.21.03.0

21

21

21

xxxx

xx

0,0 21 xx

21 5.04.0 xx

Min

s.t.

][y64.06.0

][y65.05.0

][y65.05.0

][y7.21.03.0

321

-

221

221

121

xx

xx

xx

xx

0,0 21 xx

21 5.04.0 xx


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Max

s.t.

.0,0,0,0

5.04.0)(5.01.04.06.0)(5.03.0

6)(67.2

3221

3221

3221

3221

yyyy

yyyyyyyy

yyyy

Max

s.t.

.0,URS:,0

5.04.05.01.04.06.05.03.0

667.2

321

321

321

321

yyy

yyyyyy

yyy


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Application of

Complementary Slackness Conditions.

Example: Solving a problem with 2 functional

constraints by graphical method.

0,,

104

3043..

372max

321

321

321

321

xxx

xxx

xxxts

xxxZ Optimal solution

x1=10

x2=0

x3=0

4 duality theory

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