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Duality

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Duality

From Wikipedia, Duality may refer to:In philosophy, logic, and psychology: Dualism, a twofold division in several spiritual, religious, and

philosophical doctrines Dualism (philosophy of mind), where the body and mind are considered

to be irreducibly distinct De Morgan's Laws, specifically the ability to generate the dual of any

logical expression.

In mathematics: Duality (mathematics), a mathematical concept

Dual (category theory), a formalization of mathematical duality Duality (projective geometry), general principle of projective geometry S-duality (homotopy theory)

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DualityIn science:

Wave-particle duality, a concept in quantum mechanics Duality (electrical circuits), regarding isomorphism of electrical

circuits Duality (mechanical engineering), regarding isomorphism of some

mechanical laws Duality (electricity and magnetism), regarding isomorphism of

physical laws

Physical dualities; S-duality T-duality U-duality

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Duality

Light Darkness Male Female Voltage CurrentYang (양) Yin (음) Husband Wife Controllability Observability

Angel Devil King Queen Potential Energy Kinetic EnergyHeaven Hell Father Mother Electricity Magnetism

Soul Body Summer Winter Wave ParticleGood Evil Hot Cold Day NightGod Satan Up Down Software HardwareSky Earth Positive Negative Capitalism Communism

Paradise World Friend Enemy Right Wing Left WingTruth Grace Teaching Enlightenment Right Brain Left BrainPeace War Law Love Faith ActionRight Left Logos Pathos Reason Emotion

We encounter many things appear in pairs which compete, conflict, contradict, coordinate, or cooperate each other. For example,

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Duality: Electrical Circuits

Duality (electrical circuits)In electrical engineering, electrical terms are associated into pairs called duals. A dual of a relationship is formed by interchanging voltage and current in an expression. The dual expression thus produced is of the same form, and the reason that the dual is always a valid statement can be traced to the duality of electricity and magnetism.

Here is a partial list of electrical dualities:voltage — current, parallel — serial (circuits) resistance — conductance, impedance — admittancecapacitance — inductance, reactance — susceptanceshort circuit — open circuittwo resistances in series — two conductances in parallel; Kirchhoff's current law — Kirchhoff's voltage law. Thévenin's theorem — Norton's theorem

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Duality in Electric Circuits

Duality in Electric CircuitsWe shall define duality in terms of the circuit equations. Two circuits are duals if the mesh equations that characterize one of them have the same mathematical form as the nodal equations that characterize the other. They are said to be exact duals if each mesh equation of the one circuit is numerically identical with the corresponding nodal equation of the other; the current and voltage variables themselves cannot be identical, of course. Duality itself merely refers to any of the properties exhibited by dual circuits.

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Duality TheoryDual Relationship of Two RLC Circuits

Single Node-pair: Use KCL

Ri Li Ci

0 CLRS iiii

)();()(1;)(0

0

tdtdvCitidxxv

Li

Rtvi CL

t

tLR

SL

t

tit

dtdvCtidxxv

LRv

)()()(10

0

Differentiatingdtdi

Lv

dtdv

RdtvdC S

12

2

Single Loop: Use KVL

Rv

Cv

0 LCRS vvvv

)();()(1; 00

tdtdiLvtvdxxi

CvRiv LC

t

tCR

dtdv

Ci

dtdiR

dtidL S2

2

SC

t

tvt

dtdiLtvdxxi

CRi )()()(1

00

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Duality TheoryRecall Diet Problem

0.3 0.1 0.450.2 0.8 1.200.1 0.1 0.3 2 4

Grain Fishmeal

Carbohydrates

Protein

Fat

Price

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Duality Theory

1 22 4z x x

1 20.3 0.1 0.45x x

1 20.2 0.8 1.20x x

1 20.1 0.1 0.3x x

1 2, 0x x

minimize

subject to

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Duality Theory

Pharmaceutical company producing nutrient pillscompetitive with real food.

What prices should the company charge for one unit of carbohydrates, protein, fat?

Carbohydrates Protein Fat

1y 2y 3y

Unit

cents/oz

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Duality Theory

1 2 30.45 1.20 0.30y y y

1 2 3

1 2 3

0.3 0.2 0.1 20.1 0.8 0.1 4

y y yy y y

0.45 1.20 0.30 y

0.3 0.2 0.1 2

0.1 0.8 0.1 4

y

maximze

subject to

max

profit, price

grainfishmeal

s. t.

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Duality Theory

Original Problem

2 4 x

0.3 0.1 0.450.2 0.8 1.200.1 0.1 0.30

x

min

s. t.

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Duality TheoryTc x

Ax bx 0

Tb yT A y c

y 0

max

min

s. t.

s. t.

Primal Problem

Dual Problem

y : dual variable; m – dimensional vectorcorresponding to constraint inequalities

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Linear program in standard formTc x

Ax b

x 0

minimize

s. t.

Equivalent formTc x

Ax b

x 0

minimize

s. t.

Ax b

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Linear program in standard form

Dual problem

minimize Tc x

A b uxA b v

subject to

maximize

subject to( )T b u v( )T A u v c

u 0v 0

Let : free variable y u v

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Linear program in standard form

minimize Tc x

Ax bx 0

subject to

Primal (P) Dual (D)

maximize Tb y

Ay cy

subject to

: free

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Duality Theorem

since andx 0

Lemma (Weak Duality Lemma): If and are feasible for (P) and (D) respectively, then

x yT Tc x b y

Proof. We have from (P) and (D)T T T T b y y b y Ax c x

.T Ty A c

Corollary. If and are feasible for (P) and (D), respectively, andif , then and are optimal for their respective problems.

0 0T Tc x b y

0y0x 0y

0x

Duality gap

Tc x

Tb y0

Tc x0

Tb yNo gap optimal

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Duality Theorem of Linear ProgrammingTheorem (Duality Theorem): If either of the problems (P) or (D) has a finite optimal solution, so does the other, and the corresponding values of the objective functions are equal. If either problem has an unbounded objective, the other problem has no feasible solution.

Proposition: Every T matrix looks like

1 T

πT

0 D

Proof Never pivot on the objective function

Induction

Proof. see text.

0

1

T0T

0 I

z

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Duality Theorem of Linear Programming

1

1

T

r

π

T0 D

Suppose

1r r r T P T

1

1 1

khk

hk

mkhk

c Ta

a

aa

π

0 0 D

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Duality Theorem of Linear Programming

1 1

T Tπ

0 β 0 D

'

'

11

TT T ππ D

0 βD 0 D

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Duality Theorem of Linear Programming

ˆ1 1 0

ˆ

T T Tπ c 0

0 A I b0 D

Let

ˆ ˆ ˆ1

ˆ ˆ ˆ

T T T Tc π A π π b

0 DA D Db

* Note: Primal problem is a max problem for correct sign1

m

1 1mn

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Duality Theorem of Linear Programming

πDual m variables

is a candidate for the solution to dual

ˆ 0Tπ

ˆ 0 T Tc π A

ˆ TA π c

Optimality condition

satisfies constraints and restrictions for dual, therefore is a feasible solution to the dual problem. It is also the optimal solution

π π

Restriction for dual variables

Constraints for the dual problem

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Duality Theorem of Linear Programming

ˆ Tπ bObjective function for the optimal solution is

In other words, if is the optimal solution to (P),x

ˆ ˆT Tc x π b

The value of the objective function for the dual is ˆTb π

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Duality Theorem of Linear Programming

ex) max

infeasible

10 1

0

yyy

10 1

0

xxx

min

infeasible

※ Both Primal and Dual may be infeasible.

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Duality Theorem of Linear Programming

1 225 5z x x

Ex. For the primal, find the dual problem.

max

1 2 1

1 2 2

1 2 3

1 2

2 103

3 2 14

0, 0

x x yx x yx x y

x x

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Duality Theorem of Linear Programming

1 2 3

1 2 3

1 2 3

1 2 3

10 3 143 25

2 2 50, 0, 0

w y y yy y yy y y

y y y

1 2

1 2 3

( , ) (3, 2)85

( , , ) (4,5,6)139

x xz

y y yw

Dual

min

Always

Weak Duality Lemma: Always the value of the objective function of the minimization problem is bigger than that of the maximization problem

Any feasible solution

z w

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Duality Theorem of Linear Programming

1 25 5 0 0 0 00 1 2 1 0 0 100 1 1 0 1 0 30 3 2 0 0 1 14

1 0 0 0 7 6 1050 0 0 1 4 / 5 3 / 5 40 1 0 0 2 / 5 1/ 5 40 0 1 0 3 / 5 1/ 5 1

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Duality Theorem of Linear Programming

1 0 7 60 1 4 / 5 3 / 50 0 2 / 5 1/ 50 0 3 / 5 1/ 5

T

0 1 4 / 5 3 / 57 , 0 2 / 5 1/ 56 0 3 / 5 1/ 5

y D

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Duality Theorem of Linear Programming

1 2

1 2 3

4, 1, 1050, 7, 6, 105

x x zy y y w

1 1

2 2

1 1

2 2

3 3

4 01 04 00 70 6

x tx ts ys ys y

Claim: For optimal solutions for the primal and dual problems,always one of the pair is zero

Eureka!

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Tc xAx bx 0

T

T

b yA y cy 0

min

max

(D)

(P)

Duality Theorem of Linear ProgrammingClaim: For optimal solutions for the primal and dual problems, always one of the pair is zero

Complementary Slackness Condition (C. S. C.)

Note:

Primal problem is a max problem for correct sign

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Duality Theorem of Linear Programming

T T T T Tc x y Ax x A y b y

ˆ ˆ ˆ ˆ ˆ ˆ ˆ T T T T T Tc x y Ax x A y b y c x

whenever and are feasible in P and D.

Suppose and are optimal for P and D,then, from WDL we have

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( ) 0 ( ) 0

T T T T T

T T T T T

T T T

c x y Ax x A y b yx c x A y y Ax y bx A y c y b Ax

becomes

Proof.

x y

x y

Optimality

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Two complementary slackness conditions

ˆ ˆ( ) 0 T Tx A y cˆˆ TA y t c ˆ t 0

ˆˆT A y c t

ˆˆ 0Tx t

1

ˆˆ 0

n

i ii

x t ˆˆ , 0i ix t

Surplus variable

ˆˆ 0, i ix t 1, 2, ...,for i n

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Two complementary slackness conditions

ˆ ˆ 0 Ty b Ax

ˆˆ Ax s b

ˆˆ 0Ty s

1

ˆ ˆ 0,

m

i ii

y s

ˆ ˆ 0 i iy s

From

ˆ ˆ, 0i iy s

1, 2, ...,for i m

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Complementary slackness theorem

Theorem Complementary slackness theorem

Proof Omitted.

A pair of feasible solution and to P and D,respectively are optimal, if and only if thecomplementary slackness conditions hold.

x y

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Dual transform from general primal

max Tc xAx bx 0

min Tb yT A y c

y 0

1 1 2 2 3 3min c x c x c x

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

a x a x a x ba x a x a x ba x a x a x b

1 2 30, 0, is freex x x

Example

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Dual transform from general primal

1 1 2 2 3 3 3 3m in c x c x c x c x

11 1 12 2 13 3 13 3 1

21 1 22 2 23 3 23 3 2

31 1 32 2 33 3 33 3 3

a x a x a x a x ba x a x a x a x ba x a x a x a x b

1 2 3 30, 0, 0, 0 x x x x

31 1 32 2 33 3 33 3 3 a x a x a x a x b

a b a ba b

21 1 22 2 23 3 23 3 2

21 1 22 2 23 3 23 3 2

a x a x a x a x ba x a x a x a x b

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Dual transform from general primal

1 1 2 2 3 3 3 3max c x c x c x c x

11 1 12 2 13 3 13 3 1

21 1 22 2 23 3 23 3 2

21 1 22 2 23 3 23 3 2

31 1 32 2 33 3 33 3 3

a x a x a x a x ba x a x a x a x ba x a x a x a x ba x a x a x a x b

1 2 3 30, 0, 0, 0x x x x

1z

2z

3z

4z

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Dual transform from general primal

1 1 2 2 2 3 3 4b z b z b z b z

11 1 21 2 21 3 31 4 1

12 1 22 2 22 3 32 4 2

13 1 23 2 23 3 33 4 3

13 1 23 2 23 3 33 4 3

a z a z a z a z ca z a z a z a z ca z a z a z a z ca z a z a z a z c

1 2 3 40, 0, 0, 0z z z z

equality

2 3 2y z z

Dual problem

min

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Dual transform from general primal

min 1 1 2 3 4b z b y b z

11 1 21 31 4 1

12 1 22 32 4 2

13 1 23 33 4 3

a z a y a z ca z a y a z ca z a y a z c

1 40, : , 0freez y z

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Dual transform from general primal

Let 1 1 1, 0y z y

max 1 1 2 3 4b y b y b z

11 1 21 31 4 1

12 1 22 32 4 2

13 1 23 33 4 2

a y a y a z ca y a y a z ca y a y a z c

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Dual transform from general primal

max 1 1 2 2 3 3b y b y b y

11 1 21 2 31 3 1

12 1 22 2 32 3 2

13 1 23 2 33 3 2

a y a y a y ca y a y a y ca y a y a y c

Let 3 4 3 2 free, 0, :y z y y y

1 2 30, : , 0freey y y

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Dual transform from general primal

General Dual General Primal

constraint= constraint

constraint

variablefree variable

variable

variable variable

free variable

constraintconstraint

= constraint

0

00

0

max Tc xAx bx 0

min Tb yT A y c

y 0

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Dual transform from general primal

max

1 2 1

1 2 2

1 2 3

53 7

1

x x yx x yx x y

1 22x x

1 20, 0x x

Example

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Dual transform from general primal

min 1 2 35 7y y y

1 2 3

1 2 3

3 12

y y yy y y

1 2 30, 0, 0y y y

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Dual transform from general primal

1 1 2 0 0 0 00 1 1 1 0 0 50 3 1 0 1 0 70 1 1 0 0 1 1

Simplex tableau for the example (Primal problem is a max problem)

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Dual transform from general primal

1y 2y 3y

1 3,x 2 2x

1 0 0 3 2 0 1 2 70 0 0 2 1 1 40 0 1 1 2 0 1 2 20 1 0 1 2 0 1 2 3

Final tableau

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Dual transform from general primal

1 3 / 2 0 1/ 20 2 1 1

=0 1/ 2 0 1/ 20 1/ 2 0 1/ 2

T

Ty

1 2 33 / 2, 0, 1/ 2y y y

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Dual transform from general primal

1 2

1 2

1 2

1 2

1 2

25

3 71

0, 0

x xx xx xx x

x x

1 2 3

1 2 3

1 2 3

1 2 3

5 73 1

20, 0, 0

y y yy y yy y y

y y y

P : min

D : max

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Dual transform from general primal

min maxP D

max minP D

1

2

3

32012

y

y

y

Dual solution : negative of

Dual solution : correct sign

1)

2)

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Dual transform from general primal

1 2

1 2 1

1 2 2

1 2 3

1 2

25 52 10

33 2 14

0, 0

x xx x yx x yx x y

x x

1 2 3

1 2 3

1 2 3

1 2 3

10 3 143 25

2 2 50, 0, 0

y y yy y yy y y

y y y

min

max

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Dual transform from general primal

1 0 0 0 7 6 1050 0 0 1 4 / 5 3 / 5 40 1 0 0 2 / 5 1/ 5 40 0 1 0 3 / 5 1/ 5 1

1 1

2 1

1 1

2 2

3 3

4 01 04 00 70 6105 105

x tx ts ys ys yz w

ComplementarySlackness Condition