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    Lecture 3 The Dual Problem

    ua s an pro em t at s er ve mat emat ca yfrom a given primal model

    .

    eL

    P

    the primal is a problem the dual will be a

    problem and visa v

    max min

    er

    ,If LP

    sa.this lecture we introduce the of the dual problem

    based on the

    definition

    standard form

    ,

    .

    In

    such by defining the dual problem f standardrom thethe results will be

    formconsistentwith the info

    , ,Asrmation contained in

    the simplex tableau.

    automatically accounts for all the forms given in the othertreatments

    .

    LP

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    o f a p r im als tan da rd formT h e L P

    1 1 2 2 n n z c x c x c x

    1 1 2 2 n n1 1 1 1

    a x a x a x

    b

    1 1 2 22 n n2 2 2

    ., ,,

    1 1 2 2 n

    j

    m m m mn

    x 0 j = 1 n

    T

    0,A x = b x

    S u bject to

    ( ., ,

    1 2 n=

    x x x x

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    that the variables include the and , , , .Note jn x artifslacks icials

    t e purpose o construct ng t e ua , we arrange t ecoefficients of the primal schematically as shown in

    .or

    Table 1

    diagrThe am shows that the dual is obtained symmetrically

    from the primal according to the following rules:a dual variable

    a dual

    every primal c there is

    there

    onstraint

    every primal varia isble

    .

    For

    For

    1.

    2. constraint.

    constraint coefficients of a primal variable form the left -side coefficients of the corres ondin dual constraint;

    The3.

    and the objective coefficient of the same variable becomes

    t

    he right side of the dual constraint.( ) , e.g., the tinted column under .jee xS

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    Table 1

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    Table 2

    dual problemrules indicate that the willThesehave variables ( ) and constraints , , ... ,

    1 2 mnm y y y

    , , ... , .1 2 n

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    xample 1- con.

    Standard Primal

    1 2 3 4

    Subject to

    1 2 3 4

    x 2x x x 10

    .

    , , ,1 2 3 4

    1 2 3 4 x 0 x 0 x

    0 x 0

    N slackthat is a in the first constraint;otice 4x

    function and the second constrain .t

    ,

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    .Example 1- con

    Dual

    1 2810 Min y y w

    : 5

    Subject to

    x 2

    :2 1 2 12y x 2 y

    ::

    1 23

    4

    xx

    y y ( )implies that1 2 1 0 0y y y0

    , unrestricted1 2 y y

    t at " s om nate y

    dual constraint associated with .

    " unrestricted .

    bserve

    The1 1

    4

    y

    x

    y

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    , the should read asdual problemThus

    1 210 8Min w y y

    1 2 5

    2y y

    1 2

    1 2 4

    y 3y

    , unrestricted1 20y y

    the cha ng es in the du a l show n if its p r im alInd ica te

    E x e rc ise 1

    [ :

    m in im iz a tio n m a x im iz a tis ins tea d o f io n

    . C ha ng es are , fir s t th ree co ns tra in t s are

    .

    M a x im izA wen s

    ]o f th e typ e , a n d .1 0y

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    Example 2

    Primal Form

    1 2

    1.5

    nSubject to

    x x

    x x

    1 2 5 2x 3x

    x x 0

    Standard F rmo

    1 2

    S

    x x x xubj

    1ect

    .50to

    1 2 3 4 2x 3x x x

    x x x

    5

    x 0

    0

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    Example 3

    Primal 1 2

    Sub ect to

    1 2x 2x 5

    -1 2

    1 2 4x 7x 8

    unrestricted1 x

    2 x

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    .xample 3 - con

    wstandard rim

    erethe becomesal

    , , .etThen

    1 1 1 1 1x x x x x

    Standard primal

    1 1 2z 5x 5x 6xMax

    u ect to

    x x 2x x x00 5

    - 1 1 2 3 4x x 5x x x0 3

    1 1 2 3 4 4x 4x 7x x x 80

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    .Example 3 - con

    ua 1 2 35y 3y 8y wMin

    ( )implies that

    1 2 31 2 3- -

    y y yy y

    4 54 54 5

    y

    1 2 3 2 5 7 6 y y y

    2 2

    3 y 0

    that the first and second constraints can be

    replaced by the equation

    dual

    .

    Observe

    1 2 3y y y4

    5

    will always be the case when the primal variable is

    unrestricted, meaning that an unrestricted

    This

    primal variable

    uaw a ways ea to a equat on rather inequalihan tyt .

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    Exercise 2

    the changes in the dual just shownIndicate

    if the objective is minimization and the first

    " " .

    the type" ", and y ], .2 3y0 0

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    3. The dual problem in matrix form

    now provide a general matrix definitionWe

    standard

    primal model:LP

    T TPrimal

    I III II

    Su

    ect

    o

    I II

    Ax Ix b

    ,I IIx x

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    ( ) be the vector.. dual.., , ,Let 1 2 mY y y y

    ru es n y e e uao ow ng :T

    e a eMin w Y b

    T

    Subject to

    I

    I

    I Y C

    Y unrestricted vector

    the primal problem is changed to the dualminimization

    .

    ,If

    II

    problem sens maxime of optimization is changed izatto

    and t

    n

    h

    io

    e first two sets of constraints are changed to withY

    remaining unrestricted.

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    Duality - Conclusion

    DualPrimal

    I

    T T

    I II IIMinMax

    Sub ect to

    w z C X C X Y b

    I II

    AX IX b IA CY

    ,I II

    unrestricted vectorY

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    the ofDual nonstaa ndardFinding LP

    c x c x c x

    Max problem

    It s dual

    Subject to

    Min problem

    w b b b

    2 2 2 2

    n n

    1 1 2 2 n n

    a x a x a x b

    Subject to

    m m1 1 2 2a x a x

    1 1 2 2 m m

    m

    1 1 1 1

    n mn

    a x a

    =

    b

    1 1 2 2 m m2 2 2 2a a cy y y

    .

    It s dual

    1 1 2 2n n n nm m a ay y y

    =

    a c

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    Example 4

    Primal Dual

    1 2 1 z 5x 2x wMax Min 12y 9

    Subjec

    t to

    2 3y 20y

    Subject to

    1 2

    1 2 2

    13

    4

    x x

    x x

    12 y

    9 y

    1

    3

    1 2 3y y y 5

    2

    1

    3 4 8

    3 7

    1 2 38x x

    x 0

    y27 0

    , ,1 2 3y 0 y y

    Original constraints Dual constraints

    1 2

    1 2

    x x

    x x4 3

    1 2 3y y y 5

    91 3 7

    1 2x x 1 2 3

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    Examp le 5

    t e o ow ng pa r o pr ma an ua pro emsPrimal Prim

    onsal

    er

    1 2z 0.4x 0.5x z 0Min Min

    1 2.4x 0.5x

    1 20.3 0.1x x 2.7

    1 2 1y2.70.3 0.1x x- -

    1 2

    1 2

    0.50.6

    x xx x

    0.50.4

    66

    -

    - -

    1 2

    1 2

    2

    2

    y6y6

    0.50.5

    0.50.5

    x xx x

    ,1 2

    x x 0 1 320.0.6x x y

    x

    4 6

    0

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    .Dual

    Example 5 - con

    -( )1 2 1 2 2 3

    2.7y 6 y y 6ywz 0.4x 0.5x -MaxMi n

    1 2 10.1 y2.7x x0.3

    --

    u ec o

    -

    -

    ( )1 2 2 3

    y y y y 0.40.3 0- .5 0.6

    - - -

    1 2

    1 2

    2

    2

    0.5

    0.

    y60.5

    0.5

    x x

    x x5 y6

    -

    -

    , , ,

    1 2 2 3

    1 2 2 3

    y .

    0

    y y y. . .

    y y y y

    ,

    1 2

    2

    3

    1

    0.6x x

    x x 0

    0

    y6.4

    Dual

    1 2 32.7y 6yw - 6yMax

    1 20.3 0.5 0- y y

    -

    3

    0.1 0.5 0.4

    y. 0.46

    .50

    ,1 3 0yy

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    Primal DualRelationship Between And Objective Values

    satisfy the following relationship:

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    carefully that these say nothing about whichtwo resu tslObserve

    is the sense of optimization ( )

    .

    maximization and minimizationIt

    t at matters n t s case

    prove

    .

    To the validity of these results let ( ) and be the, ,I IIX YXfeasible primal and dual solutions corresponding to the primal

    -dual definitions given in matrix form.

    premultiplying the primal constra,Then ints by we get,Y

    I IIX X w( Y b Y

    we get

    ,T T T T T I I II II

    IIX A C X X XY Y C

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    that ; hence the direction of the inequality

    remains unchan ed

    ,

    .

    Observe I IIX 0 X 0

    adding the two constraints yields

    T T T T T

    ,Then

    the left - hand sides of th

    I II I II

    Since

    I II z

    e and identities above are equal,w zwe conc u e t at

    z w

    to show that at the optimum solutions observe

    .

    ,Now z w that isz

    means that seeks the highest value among all feasible ( )

    and seeks the lowest value amon all feas lib e

    ,

    .

    This IIz X

    w Y

    X

    ( ) for all feasible solutions including the optima , the two

    roblems will reach o timalit onl when .m

    z w

    wax

    Si ce

    min

    n

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    Example 6

    e o ow ng pa r o pr ma an pro emua sons er

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    feasible solutions given above are determined by inspectingThe

    objective value in the maximization problem ( ) is less

    .

    The dual

    result means that

    .

    This

    the range ( ) is relatively narrow we can actually

    to ,Since

    14 15

    think of the two feasible solutions above as being near optimalessence the given inequality can be used to test the goodness

    . , " "In

    two limit

    of the feasible solutions

    the ha en to be e ual the corres ondin solutionss

    .

    Ifare optimal.

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    the of the followin robldu emal.Write3

    1 2z x x 5 2 Max

    Subject to

    1 2

    1 2x x 93 4

    1 2x x 2087

    ,1 2


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