PRIMAL & DUAL PROBLEMS,THEIR INTER RELATIONSHIP

PREPARED BY: 130010119042 –KHAMBHAYATA MAYUR130010119045 – KHANT VIJAY 130010119047 – LAD YASH A.

SUBMITTED TO:PROF. S.R.PANDYA

CONTENT: Duality Theory Examples Standard form of the Dual Problem Definition Primal-Dual relationship Duality in LP General Rules for Constructing Dual Strong Dual Weak Dual

Duality Theory The theory of duality is a very elegant and important

concept within the field of operations research. This theory was first developed in relation to linear programming, but it has many applications, and perhaps even a more natural and intuitive interpretation, in several related areas such as nonlinear programming, networks and game theory.

The notion of duality within linear programming asserts that every linear program has associated with it a related linear program called its dual. The original problem in relation to its dual is termed the primal.

it is the relationship between the primal and its dual, both on a mathematical and economic level, that is truly the essence of duality theory.

Duality Theory (continue…)

Examples There is a small company in Melbourne which has recently

become engaged in the production of office furniture. The company manufactures tables, desks and chairs. The production of a table requires 8 kgs of wood and 5 kgs of metal and is sold for $80; a desk uses 6 kgs of wood and 4 kgs of metal and is sold for $60; and a chair requires 4 kgs of both metal and wood and is sold for $50. We would like to determine the revenue maximizing strategy for this company, given that their resources are limited to 100 kgs of wood and 60 kgs of metal.

Standard form of the Primal Problem

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

a x a x a x bx x x

n n

n n

m m mn n m

n

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

1 2 0

. . .. ..

. .. .. . . . . .. .

. .. .. . . . . .. .. ..

, , .. .,

maxx j j

j

nZ c x

1

Standard form of the Dual Problem

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

a y a y a y cy y y

m m

m m

n n mn m n

m

11 1 21 2 1 1

12 1 22 2 2 2

1 1 2 2

1 2 0

.. .. . .

. .. .. . . . . .. .

. .. .. . . . . .. .. ..

, , .. . ,

miny

ii

miw b y

1

Definition

z Z cx

s tAx bx

x*: max

. .

0

w* :minxw yb

s.t.yAcy 0

Primal Problem Dual Problem

b is not assumed to be non-negative

Primal-Dual relationship

x1 0 x2 0 xn 0 w=y1 0 a11 a12 a n1 b1

Dual y2 0 a21 a22 a n2 b2(min w) ... ... ... ... ...

ym 0 am1 am2 amn bn Z= c1 c2 cn

Example

5 18 5 158 12 8 8

12 4 8 102 5 5

0

1 2 3

1 2 3

1 2 3

1 3

1 2 3

x x xx x xx x xx x

x x x

, ,

maxxZ x x x 4 10 91 2 3

x1 0 x2 0 x3 0 w=y1 0 5 - 18 5 15

Dual y2 0 8 12 0 8(min w) y3 0 12 - 4 8 10

y4 0 2 0 - 5 5 Z= 4 10 - 9

Dual

5 8 12 2 418 12 4 105 8 5 9

0

1 2 3 4

1 2 3

1 3 4

1 2 3 4

y y y yy y yy y yy y y y

, , ,

minyw y y y y 15 8 10 51 2 3 4

d x eii

ki

1

d x e

d x e

ii

ki

ii

ki

1

1

d x e

d x e

ii

k

i

ii

k

i

1

1

Standard form!

Primal-Dual relationship

Primal Problem

opt=max

Constraint i : <= form = form

Variable j: xj >= 0 xj urs

opt=min

Dual Problem

Variable i : yi >= 0 yi urs

Constraint j: >= form = form

Duality in LP

In LP models, scarce resources are allocated, so they should be, valued

Whenever we solve an LP problem, we solve two problems: the primal resource allocation problem,

and the dual resource valuation problem

If the primal problem has n variables and m constraints, the dual problem will have m variables and n

constraints

Primal and Dual Algebra

Primal

j jj

ij j ij

j

c X

. . a X b 1,...,

X 0 1,...,

Max

s t i m

j n

Dual

ii

ij ji

i

Min b

s.t. a c 1,...,

Y 0 1,...,

i

i

Y

Y j n

i m

'

. .0

Max C Xs t X b

X

'

'. .0

Min bY

s t AY CY

Example

1 2

1 2

1 2

1 2

40 30 ( ). . 120 ( )

4 2 320 ( ), 0

Max x x profitss t x x land

x x laborx x

1 2

1 2 1

1 2 2

1 2

( ) ( )120 320

. . 4 40 ( )2 30 ( )

, 0

land laborMin y ys t y y x

y y xy y

Primal

Dual

General Rules for Constructing Dual1. The number of variables in the dual problem is equal to the number of

constraints in the original (primal) problem. The number of constraints in the dual problem is equal to the number of variables in the original problem.

2. Coefficient of the objective function in the dual problem come from the right-hand side of the original problem.

3. If the original problem is a max model, the dual is a min model; if the original problem is a min model, the dual problem is the max problem.

4. The coefficient of the first constraint function for the dual problem are the coefficients of the first variable in the constraints for the original problem, and the similarly for other constraints.

5. The right-hand sides of the dual constraints come from the objective function coefficients in the original problem.

Relations between Primal and Dual

1. The dual of the dual problem is again the primal problem.

2. Either of the two problems has an optimal solution if and only if the other does; if one problem is feasible but unbounded, then the other is infeasible; if one is infeasible, then the other is either infeasible or feasible/unbounded.

3. Weak Duality Theorem: The objective function value of the primal (dual) to be maximized evaluated at any primal (dual) feasible solution cannot exceed the dual (primal) objective function value evaluated at a dual (primal) feasible solution.

cTx >= bTy (in the standard equality form)

Relations between Primal and Dual (continued)

4. Strong Duality Theorem: When there is an optimal solution, the optimal objective value of the primal is the same as the optimal objective value of the dual.

cTx* = bTy*

Weak Duality

• DLP provides upper bound (in the case of maximization) to the solution of the PLP.

• Ex) maximum flow vs. minimum cut• Weak duality : any feasible solution to the primal linear

program has a value no greater than that of any feasible solution to the dual linear program.

• Lemma : Let x and y be any feasible solution to the PLP and DLP respectively. Then cTx ≤ yTb.

Strong duality : if PLP is feasible and has a finite optimum then DLP is feasible and has a finite

optimum. Furthermore, if x* and y* are optimal solutions for

PLP and DLP then cT x* = y*Tb

Strong Duality

Four Possible Primal Dual Problems

Dual Primal Finite optimum Unbounded Infeasible

Finite optimum 1 x x

Unbounded x x 2

Infeasible x 3 4