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Bartolomeo Stellato — Fall 2020 ORF522 – Linear and Nonlinear Optimization 5. The simplex method

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  • Bartolomeo Stellato — Fall 2020

    ORF522 – Linear and Nonlinear Optimization5. The simplex method

  • Ed forum• What is the geometric picture of the standard form? Given a standard form P, can we

    always convert it back to the version defined by halfspaces? (next slides)

    • Extent to which these methods are generalizable to infinite-dimensional restrictions e.g. linear difference equations when t goes to infinity.

    • Efficient way to deal with inverses? (next lecture)

    • How to pick index entering the basis? (this lecture)

    • Adjacent solutions why defined that way? (Same active constraints except 1)

    • Feasibility LP condition in no strong arbitrage from Arrow-Debreu theory (That’s

    correct! We will discuss feasibility in duality lectures)

    2

  • Recap

  • Standard form polyhedraDefinition

    4

    cTx

    Ax = b

    x ≥ 0

    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

    Standard form LP P = {x | Ax = b, x ≥ 0}

    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

    Standard form polyhedron

    x1

    AAACy3icbVE9bxNBEF0fX8F8JVBQQLEiskSDdWchkXSR3JjCKAicRMpZ1t56bK+8H6edPWOzuKSkhf9Bw5+gpOEn8C/YdSKhc5hqNG/ezHszRSkFujT93UiuXb9x89bO7eadu/fuP9jde3iCprIcBtxIY88KhiCFhoETTsJZaYGpQsJpMe9G/HQBFoXR792qhKFiUy0mgjMXSu+Wo2y0u5+2003Qq0l2mewfPf3549fj7p/j0V7jez42vFKgHZcM8TxLSzf0zDrBJaybeYVQMj5nUzgfL0SJminAoV9u5NZwzxTiShVr2lLMzXAbK3C1prRVGDmO+DYca/+lxoozRmJdTeUmB0MvdFk50LyupDBm7lgRGM0Wfa15uCIC7bMlXQB3xlKQEC1jgHMEx02lHVgfOvrMWbHsxnW+k9YHMGvZimLYAs3cgoYP3CjF9NjnGwidDbNma5+1O4EZqIPA6oFcQDgno1JMZ45OjHY1J9q8iB6HPiJYQjQTRCkmdKz4fwPeQBV+Er6cbf/0anLSaWcv24dvw7t75CJ2yBPyjDwnGXlFjkiPHJMB4WRKvpCv5FvSTzD5mHy6aE0al5xHpBbJ57/UV+T0

    x2

    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

    x3

    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

    A 2 Rm⇥n m n

    AAADA3icbVE9bxNBEF0fX8F8OVDSrLAt0WDdWUiQLiiNKYICwkmk3GHtrefslffjtLvn2Fpdye/gB9BFtJS0UPNv2DWW0DlMNZo3b+a9mbzkzNg4/t2Kbty8dfvO3t32vfsPHj7q7D8+NarSFMZUcaXPc2KAMwljyyyH81IDETmHs3xxFPCzJWjDlPxo1yVkgswkKxgl1pcmnYPem5RJnApi53nhPtSfnEgtE2CwrHt4TgwuKs6xVpdYE7nAPYFTDlj2Jp1uPIg3ga8nyTbpom2cTPZbV+lU0UqAtJQTYy6SuLSZI9oyyqFup5WBktAFmcHFdMlKI4nXkbnVxmYDd0QYsxZ5jftBudnFcrOuMe7nik8DvguH2n+poWKV4qapprLF68wxWVYWJG0qyZVaWJJ7RruP30rqr28AH5MVXgK1SmPgECwbD6cGLFWVtKCd7zgmVrPVUVjnhnFzANGarLHxW6CdapBwSZUQRE5duoGM1X7WvHbJYOiZnjr2rBHwJfhzEszZbG5xoaRtOJHqRfCYuYCYEoIZL0oQJkPF/RvwDir/E//lZPen15PT4SB5OTh4P+wejrb/3kNP0TP0HCXoFTpEI3SCxoiiL+gH+ol+RZ+jr9FV9O1va9Tacp6gRkTf/wARcfYr

    Assumption

    InterpretationP (n�m)

    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

  • Standard form polyhedraVisualization

    5

    x1

    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

    x2

    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

    x3

    AAACy3icbVE9bxNBEF0fX8F8JIGCAooVkSUarDuDBHSR3JjCKAicRMpZ1t56bK+8H6edPWOzuKSkhf9Bw5+gpOEn8C/YdSKhc5hqNG/ezHszRSkFujT93UiuXL12/cbOzeat23fu7u7t3ztGU1kOA26ksacFQ5BCw8AJJ+G0tMBUIeGkmHcjfrIAi8Lo925VwlCxqRYTwZkLpXfL0bPR3kHaTjdBLyfZRXJw+Ojnj18Pun+ORvuN7/nY8EqBdlwyxLMsLd3QM+sEl7Bu5hVCyficTeFsvBAlaqYAh365kVvDPVOIK1WsaUsxN8NtrMDVmtJWYeQ44ttwrP2XGivOGIl1NZWbvBx6ocvKgeZ1JYUxc8eKwGi26GvNwxURaJ8t6QK4M5aChGgZA5wjOG4q7cD60NFnzoplN67znbQ+gFnLVhTDFmjmFjR84EYppsc+30DobJg1W/us3QnMQB0EVg/kAsI5GZViOnN0YrSrOdHmafQ49BHBEqKZIEoxoWPF/xvwBqrwk/DlbPunl5PjTjt73n71Nry7R85jhzwkj8kTkpEX5JD0yBEZEE6m5Av5Sr4l/QSTj8mn89akccG5T2qRfP4L2SHk9g==

    x1=0

    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

    x2 = 0

    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

    x 3=0

    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

    x4 = 0

    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

    x5=0

    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

    Three dimensions Higher dimensions

    P = {x | Ax = b, x ≥ 0}, n−m = 2

    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

  • Constructing basic solution

    6

    m A AB(1), . . . , AB(m)xi = 0 i 6= B(1), . . . , B(m)

    Ax = b xB(1), . . . , xB(m)

    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

    xB = B−1b

    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

    B =

    AB(1) AB(2) . . . AB(m)

    , xB =

    xB(1)

    xB(m)

    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

    Basis matrix

    Basis columns Basic variables

    xB � 0 x

    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

  • Feasible directions

    7

    P = {x | Ax = b, x ≥ 0}

    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

    xB = B�1bxi = 0, 8i 6= B(1), . . . , B(m)

    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

    B =hAB(1) . . . AB(m)

    i

    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

    x

    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

    Conditions

    d

    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

    A(x+ ✓d) = b =) Ad = 0x+ ✓d � 0

    AAADv3icbVJbaxNBFJ5kvdS02lQffRlsAxUx7AZBfRAa+hLBSgXTFrJLmJ092QyZyzIzmyYu+0MFf4wzMV429Twdvu98554WnBkbht9b7eDe/QcP9x519g8ePznsHj29MqrUFMZUcaVvUmKAMwljyyyHm0IDESmH63Rx7vnrJWjDlPxq1wUkguSSzRgl1kHT7jJOIWeyYhYE+wb1RCrvGiiSTuw9fDI8XeFXOLZzsARnL/EHnOL4k5K5ZvncEq3VLR5mDg5Pfkv+FeA4hw0FMvtTZto9DvvhxvBdJ9o6x2hrl9OjdhRnipYCpKWcGDOJwsImFdGWUQ51Jy5dz4QuSA7VrOS8cE4DnWRLVhhJBJikWm0211QRYcxapDXuCWLnZpdLzbrGuJcqnnl+l/bYf6UesUpx0+ymtLN3ScVkUVqQtNlJqtTCktQpOj38UVJ3UAP4gqzwEqhVGgMHvwjj6NiApaqUFnTlIi6I1Wx17stVg7CZwJ2KrLFxVaATa5BwS5UQxJ0l3lDGapdrXldRf+CUTjp2qhHwJbglE8z9vfFMSduYRKrXfsak8owpwA/jmhKESY9UfxN8htLdpDEqyFL4p3CaZj+5YLKuYp9ZFVWsBd5iO5GU6KwZt0F8/+7Jot2XuutcDfrRm/77L4Pjs9H23fbQc/QCnaIIvUVnaIQu0RhR9KPVbu23DoJhkAcyKH6FtltbzTPUsGD9Ez6qOLM=

  • Feasible directions

    8

    Nonbasic indices

    Basic indices

    Ad = 0 =n∑

    i=1

    Aidi = BdB +Aj = 0 =⇒ dB = −B−1Aj

    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

    Computation

    Basic directiondj = 1dk = 0, 8k /2 {j, B(1), . . . , B(m)}

    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

    Non-negativity (non-degenerate assumption)xi = 0 di � 0

    xB > 0 9✓ > 0 xB + ✓dB � 0

    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

  • Stepsize

    9

    c̄j

    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

    xj

    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

    Unboundedd � 0 ✓? = 1

    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

    How far can we go?θ! = max{θ | θ ≥ 0 x+ θd ≥ 0}

    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

    d j

    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

    Boundeddi < 0 i

    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

    θ! = min{i|di

  • Moving to a new basis

    10

    B(`) 2 {B(1), . . . , B(m)} ✓? = �xB(`)dB(`)

    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

    xB(!) + θ"dB(!) = 0

    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

    Next feasible solutionx+ θ!d

    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

    New solutionxB(`) 0xj ✓?

    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

    New basisB̄ =

    [AB(1) . . . AB(!−1) Aj AB(!+1) . . . AB(m)

    ]

    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

  • An iteration of the simplex method

    11

    x B =hAB(1) . . . , AB(m)

    i

    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

    c̄j = cj � cTBB�1Aj j 2 N

    c̄j � 0 x

    j c̄j < 0

    dB = �B�1Aj

    dB � 0 �1

    ✓? = min{i2B|di

  • An iteration of the simplex method

    12

    c̄j = cj � cTBB�1Aj j 2 N

    c̄j � 0 x

    j c̄j < 0

    dB = �B�1Aj

    dB � 0 �1

    ✓? = min{i2B|di

  • Today’s agenda

    • Find initial feasible solution

    • Degeneracy

    • Complexity

    13

  • Find an initial point in simplex method

  • Initial basic feasible solution

    15

    cTx

    Ax = b

    x ≥ 0

    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

    x B

    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

  • Finding an initial basic feasible solution

    16

    cTx

    Ax = b

    x ≥ 0

    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

    b � 0 �1

    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

    x = 0, y = b

    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

    Minimize 
violations

    Possible outcomesy? = 0 x?

    y? > 0

    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

    Auxiliary problem

    1T y

    Ax+ y = b

    x ≥ 0, y ≥ 0

    AAADO3icbVFNbxMxEHUWaEv4SuHIxaJKhQREuxEScEAq6iUcigpq2krdUNnOJDHxx7L2hizWih/Cv+CO+APcEWduiCtHJOw0EtqUudiaN29m3huaCW5sHH9vRBcuXlpb37jcvHL12vUbrc2bh0YXOYM+00Lnx5QYEFxB33Ir4DjLgUgq4IhOdwN+NIPccK0ObJnBQJKx4iPOiPWp09bblMKYK0fynJSVE6JqppLquZNcccnfQ4W3cSqJndCRS6rXB2WaLitMQd8As9jqUPNsju/h8in18Daep2PA8X1cLt7AADVczjhtbcWdeBH4/CdZfrZ2Xn2svvQ+fN0/3Wx8SoeaFRKUZYIYc5LEmR34dpYzAX7hwkBG2JSM4WQ445lRRIIZuPnCnhruiDSmlLTC7aDJrGLUlBXGbarFMOCrcMj9lxoyVmth6tsUdvR44LjKCguK1TehWk8toZ7RbOPnivmrGcB7ZI5n3lWdYxAQJBsPpwYs04WykDtfsUdszue7YZzrxvUGC5Ox8VOgmeag4B3TUhJvf7qAjM19r0nlkk7XMz2171k9EDPwdhIs+Hhi8UgrW1Oi9IOgceACYjIIYvxSknAVMu5fgxdQ+Jv4KyerNz3/Oex2koedJy/9uXvoLDbQbXQH3UUJeoR2UA/toz5i6Bv601hrrEefox/Rz+jXWWnUWHJuoVpEv/8C2TIOyA==

  • Two-phase simplex method

    17

    Phase I

    Phase II y

    x B

    <latexit sha1

    _base64="XBC7kbiKACIk8VG

    E7yCa+h/eIXc=">AAAEZHicb

    VPbbtNAEHXaACUUaKl4QkIrm

    koFqZYTUQFvFX0pD0XlkrZSH

    IX1epKsuhdrdx0SrfzC5/EH/

    ADfwWwaqXXKPo3nzJy5HWeF4

    NYlyZ/G2nrz3v0HGw9bjzYfP

    3m6tf3s3OrSMOgxLbS5zKgFw

    RX0HHcCLgsDVGYCLrKr44BfT

    MFYrtV3Ny9gIOlY8RFn1KFru

    PU7zWDMlQdVSjDUQdVXmjuQF

    opBKw0W+QpMIwfRhmMoFaQwG

    vkl2c+NLsiUGk7x25L2vE2oy

    okB67SBmwSmrXu9ZPumxRTuc

    llHjeNqTEZGS+ImQKwWZWiSt

    GfXtNxZgrNyLPSxHbdSUPmtv

    lvDrd0kThaP3DU6S2M3Wr6z4

    fbarzTXDAmUY4Ja2+8khRv40

    AgTyLhH0hL3QNkVHYMflUIUa

    FStW95+PuWFVVSCHfjZ4h413

    FNp7VxmFdmT1E3sKpbZeUXIX

    qZFHvBVOPj+mxo8Tmth692Ub

    vR+4LkqSgeK1TvJtL5yNMMMH

    OyTYigTC+SUzsgUGN6LAN4BV

    2HD3BYc06VyYDxGnFJn+Ow4l

    PPdpE5AjaFzYrEKtFIDCn4yL

    SXey6cLyDqDXJPKd+IuZmJqD

    7NOAFWAa6ZE8PHEkZFWrjaJ0

    gdhxoEPiC0gDINNScpV8Pgbg

    s9Q4k0WqAA1dhOfFihJleMsl

    U+YDOCtPQTJBCUiYb3ZseSq8

    mkoqwufGkmWvpVIRk1ej1t4a

    lFvfIq/jMuowUhTCugfdGA28

    El8WGBX3fgQZlVYB6q2s6rRu

    8Z5N+68jT986e4enSz1uxG9i

    F5F+1EnehcdRSfRWdSLWCNpn

    DeGjR/rf5ubzZ3m8+vQtcYyZ

    yeqvebLfxTieYY=</latexit

    >

    b � 0(x, y) = (0, b)

    0

    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

  • Big-M method

    18

    cTx+M1T y

    Ax+ y = b

    x ≥ 0, y ≥ 0

    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

    Very large
constant

    Incorporate penalty in the costy = b � 0

    y

    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

    M

    AAAEQXicbVNbaxNBFN40Xmq0ttVHXwbbgggNu9HiBYVCXypYiWDaQjaU2dmTZMhclpnZmDjsi3/GH+Kv8Cf4Jr764pkYtJs6D8vhfN937psVglsXx98aa81r12/cXL/Vun1n4+7m1va9U6tLw6DHtNDmPKMWBFfQc9wJOC8MUJkJOMsmRwE/m4KxXKsPbl7AQNKR4kPOqEPXxdaXNIMRV547kPwTVH2lg2mhGLTSYBGfZkPSNfplRRRATpwmVospEK3EHD9A3nYvU4+0Qip3hFuitCNA7TyICs4mZPdkl1CVB1jSCVjixkAKo7FcSTKaY0TLqMA0rRRU/res1sXWTtyOF49cNZKlsRMtX/die+1zmmtWSlCOCWptP4kLN/DUOM4ERtwjaYltUjahI/DDUogCjap1ydvPp7ywikqwAz9bDLuGeyqtncusInuSurFdxTI7rwjZy7TIA74KB99/pcHjtBa2Xk3phs8HnquidKBYvZJM64mjGSqwsTeK4Q1YICd0RqbAnDYEcMI4Chv6tuCYLpUD45FxQp3hs6OQznfiegBqDMWVYBZopQYUfGRaStygTxeQdQZjjSuftDuoRGkPVceA94FjpkTw0diRoVau1onS+6HHgQ+ILSA0g0VJylXw+H8B3kGJO1mgAtTIjX1aUMNVjr1UPmYygJfmAKqU4WgwYL3YkeSq8mlIqwufGkmWvhUmoyav8xaeGuuxT/Gfchk1yDSlgP5+ArOBj9sHBVbVaR/ArArjwKtNVm/0qnHaaSdP2y/ed3YOj5f3ux49iB5Gj6IkehYdRsdRN+pFrLHReNJ41Xjd/Nr83vzR/PmHutZYau5Htdf89RufZ222

    Remarks

  • Degeneracy

  • Degenerate basic feasible solutions

    20

    x̄ |I(x̄)| > n

    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

    P

    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

    x

    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

    y

    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

  • Degenerate basic feasible solutions

    21

    xB = B�1bxi = 0, 8i 6= B(1), . . . , B(m)

    AAADunicbVJbaxNBFJ4k2tZ4aaqPvgymgRbSsBsEFRVK+hIfKhVMW+jGMDs5SYbMZTszGxOH/Z3ijxGcicGyqefp8H3nO/c048zYKPpVqdYePNzZ3XtUf/zk6bP9xsHzS6NyTWFAFVf6OiUGOJMwsMxyuM40EJFyuErnZ4G/WoA2TMmvdpXBUJCpZBNGifXQqHGbpDBl0jELgv2A4kaq4BrIhvUkePhwOXK9An/EvW/uJC7SwzuceTRqJ++TidKEc8xwIuEW947i43YyVta0e0fi2AtAjv9VGDWaUSdaG77vxBuniTZ2MTqoxj4ZzQVISzkx5iaOMjt0RFtGORT1JPftEjonU3CTnPPMOyX0ZrxgmZFEgBm65XppZRURxqxEWuCWIHZmtrnUrAqMW6ni48Bv0wH7rzQgViluyt3kdvJ26JjMcguSljtJlZpbknpFvYU/SepvaQCfkyVeALVKY+AQFmE8nRiwVOXSgnY+4pxYzZZnoZzrRuUERGuywsZXgXqiQcJ3qoQg/izJmjJW+1yzwsWdrld66cCr+sAX4JdMMGfTmcUTJW1pEqlOwoxDFxiTQRjGNyUIkwFxdwk+Q+5vUhoVZC7CU3hNuZ+pYLJwScisMpdogTfYViQlelyOWyOhf/9k8fZL3Xcuu534defdl27ztL95tz30Er1CRyhGb9Ap6qMLNEAU/US/KzuV3dqHWlpjtfnf0Gplo3mBSlazfwC9HTgL

    B =hAB(1) . . . AB(m)

    i

    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

    x

    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

    xB = 0

    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

    x1=0

    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

    x2 = 0

    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

    x 3=0

    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

    x4 = 0

    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

    x5=0

    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

    x6=0

    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

    Definition

  • Degenerate basic feasible solutions

    22

    Example

    Degenerate solutions

    B = {1, 2}

    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

    x = (0, 1, 0)

    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

    x1 + x2 + x3 = 1

    �x1 + x2 � x3 = 1x1, x2, x3 � 0

    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

    B = {2, 3}

    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

    y = (0, 1, 0)

    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

  • Cycling

    23

    Stepsize

    θ! = min{i∈B|di

  • Pivoting rulesChoose the index entering the basis

    24

    c̄j = cj � cTBB�1Aj j 2 N

    c̄j � 0 x

    j c̄j < 0

    dB = �B�1Aj

    dB � 0 �1

    ✓? = min{i2B|di

  • c̄j = cj � cTBB�1Aj j 2 N

    c̄j � 0 x

    j c̄j < 0

    dB = �B�1Aj

    dB � 0 �1

    ✓? = min{i2B|di

  • Bland’s rule to avoid cycles

    26

    Theoremj

    i

    AAAEL3icbVPLbhMxFJ00PEp4tbBkY2groUJHMxEVsKvUTbsoKhJpK3WiyuPcZEz8GNmeNJE1G/6BD+FrEBvElm9gw3UaiU6KV1fn3Me5D+el4NYlyY/WSvvW7Tt3V+917j94+Ojx2vqTE6srw6DHtNDmLKcWBFfQc9wJOCsNUJkLOM3H+4E/nYCxXKtPblZCX9KR4kPOqEPoYu3r4ZBcAqksEFcA8Vk+JFZSIcA6wtUApsRUAmoy1IawQmvL1Yjk2hVz/83PmwSUAxPQAKAWbglVgyuabxIBdNJgXwdTXZVSmrAZE4G/5EIQzVhl6vhibSOJk/kjN410YWxEi3d8sb7yJRtoVkkUwwS19jxNStf31DjOUH5ni2TYY0nZmI7ADyshSjTqzjX0fDDhpVVUgu376Xy0Dd5Tae1M5jXZktQVdpnL7awmZCvXYhD4ZTpg/w0NiNNa2Kaayg3f9T1XZeVAsaaSXOuxozlGYGOHiuHGcYFHdEomwBxuCgSEUdjQtwXHdBWW5NHjiDrDp/uhnO8mzQTUGDojFqtAJzOg4JJpKXGZPptT1hnMVdQ+jbsYiaE9jDoAMQEcMyWCjwqHl6JcoxOld0KPfR8YW0JoBkVJylVA/L8EH6DCncxZAWrkCp+V1IQzVK72CZOBvDYHUJXkDhDOmmJHkqvaZ6GsLn1mJFlgS56MmkHTb440vLZ9hj/I5dSgZ/gM5zspTPs+iXdLVNWNd2Fah3Hg1abLN3rTOOnG6Zv4/cfuxt7B4n5Xo2fRi+hllEZvo73oIDqOehGL/rSet7Zbr9rf2t/bP9u/rlxXWouYp1HjtX//BbmPZzE=

    Proof idea (left as exercise)• Assume that Bland’s rule is applied and there exists a cycle with different

    bases.

    • Obtain same basis.

  • Perturbation approach to avoid cycles

    27

    x1=0

    AAADnXicbVLdahNBFJ5m/anxr9ULLxRcLAERuuwGi3ohFIIQwZYKpilkQ5idPUmGzM8yMxs3DHvjC3irT+JLeOmNj+BbOJMW7KbO1eH7znfO+c6crGBUmzj+vdUKrl2/cXP7Vvv2nbv37u/sPjjVslQEBkQyqc4yrIFRAQNDDYOzQgHmGYNhtuh5frgEpakUn8yqgDHHM0GnlGDjoEE1Sd7Gk529OIrXL7waJBfB3uGTnz9+Per9OZnstr6kuSQlB2EIw1qPkrgwY4uVoYRB3e6EaamhwGSBZ2CnJWOFC+r2JXSUL2mhBeagx7ZaG2nwFnOtVzyrww7HZq43uUyv6jDsZJLlnt+kPfZfqUeMlEw3pynN9PXYUlGUBgRpTpJJuTA4cwpn7L0gbr8awiNchUsgRqoQGPhVaO9bgyGyFAaUdRlH2Cha9Xw7242bBbBSeBVq1wXaqQIBn4nkHIvcpmtKG+VqzWubRF2ndNKBU/WBLcGtGYeMzuYmnEphGk6E3Pcex9YzugBvxg3FMRUesf8KHEPp/mTNMhAzM7dpgRUVufNS25hwT17aA4iSUwMOTpvDzjgVtU19W1nYVPHwAtvIJFjlzbw10sh6YVN3rybDymWqksFoP4FqbOPooHBTdaMDqGq/Dne1yeaNXg1Ou1HyMnrz0Z1vH52/bfQYPUPPUYJeoUPURydogAii6Cv6hr4HT4N3wYfg+Dy1tXWheYgaLxj+BWKoNO8=

    x2 = 0

    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

    x 3=0

    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

    x4 = 0

    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

    x5=0

    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

    x6=0

    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

    x1=0

    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

    x2 = 0

    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

    x 3=0

    AAADnXicbVLdahNBFJ5m/anxp61eeKHgYgmI0GU3WtQLoRCECLZUME0hG8Ls5CQZMj/LzGzcMOyNL+CtPokv4aU3PoJv4UwasJs6V4fvO9855ztzspxRbeL491YjuHb9xs3tW83bd+7e29ndu3+mZaEI9IhkUp1nWAOjAnqGGgbnuQLMMwb9bN7xfH8BSlMpPpllDkOOp4JOKMHGQb1y9OJtPNrdj6N49cKrQbIO9o8e//zx62Hnz+lor/ElHUtScBCGMKz1IIlzM7RYGUoYVM1WmBYackzmeAp2UjCWu6BqXkIH4wXNtcAc9NCWKyM13mKu9ZJnVdji2Mz0JpfpZRWGrUyysec3aY/9V+oRIyXT9WkKM3k9tFTkhQFB6pNkUs4NzpzCGXsviNuvhvAYl+ECiJEqBAZ+Fdr71mCILIQBZV3GMTaKlh3fzrbjegGsFF6G2nWBZqpAwGciOcdibNMVpY1ytWaVTaK2Uzppz6m6wBbg1oxDRqczE06kMDUnQh54j0PrGZ2DN+OG4pgKj9h/BU6gcH+yYhmIqZnZNMeKirHzUtmYcE9e2gOIglMDDk7rw045FZVNfVuZ21TxcI1tZBKsxvW8FVLLem5Td68mw8plqoLB4CCBcmjj6DB3U7WjQygrvw53tcnmjV4NztpR8jJ689GdbxddvG30CD1Fz1CCXqEj1EWnqIcIougr+oa+B0+Cd8GH4OQitbG11jxAtRf0/wJpFjTx

    x4 = 0

    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

    x5=0

    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

    x6=0

    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

    Ax = b+ !

    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

  • Complexity

  • Complexity

    29

    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

    Estimate complexity of an algorithm

    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

  • Complexity

    30

    Notation

    Polynomial
Practical


    
Exponential
Impractical!

    0 20 40 60 80 100n

    0

    20

    40

    60

    80

    100

    f(n

    )

    2n

    n2

    n

    n log(n)p

    n

    log(n)

    g(x) ⇠ O(f(x)) c > 0 x0

    |g(x)| cf(x), 8x � x0

    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

  • 31

    P NP

    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

    P ⇢ NPNP

    AAAES3icbVNNb9NAEHWaFkr4aAsHDlxWtJUQUiM7ogIOiEpFUA6tgkTaSnFUrTeTZJX9sHbXacLKF/4RP4QfwO/ghjgwG6JSp+xp9N7MeN6bcZYLbl0c/6it1FfXbt1ev9O4e+/+g43NrYenVheGQYdpoc15Ri0IrqDjuBNwnhugMhNwlo0PA382AWO5Vp/dLIeepEPFB5xRh9DF5rc0gyFXnjuQ/AuUXaVDaCHvNdIQkTMgY6UviRtRR3ZSSd2IUeHbJUltkVlw5Ao7aZc7i6p3GizhjsAUNRBKci1mSktOBXFcAqFiqA13I0kG2lxrG1rsjajpk9xoFCHtW9JIQfWvJrzY3I6b8fyRm0GyCLajxWtfbK18TfuaFRKUY4Ja203i3PU8NY4zAWVjl6QFCqZsTIfgB4UQOQZl4xra7U94bhWVYHt+Ore9wnsqrZ3JrCS7QYld5jI7KwnZzbToB36ZDth/SwPitBa2Ok3hBq96nqu8cKBYdZJM67GjGVagsI+K4TVYIMd0SibAHJoN6CpaYYNuXB/ThXJgPGYcU2f49DB8zrfiagNqDJ0Ri1+BRmpAwSXTUlJcTDqnrDPYa1T6pNnCSiztYNURiAmgzZQIPhw5XLZyFSVK7wWNPR8Ym0MQg0NJylVA/L8GJ1DgTuasADV0I5/m1HDVRy2lj5kM5DUfQBUy3Aw2rA47lFyVfn5xOvepkWSBLWUyvMJq3hypZD33Kf5dLqMGM00hoLuXwLTn4+Z+jlO1mvswLYMdeLXJ8o3eDE5bzeRF8/Wn1vbB0eJ+16Mn0dPoWZREL6OD6ChqR52I1R7X3tTe1z7Uv9d/1n/Vf/9NXaktah5Flbe69gc5SHNQ

    P = NP

    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

    P

    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

    NP

    <latexit sha1_base64="DfhA1tNrpX7B8boSP7vixmkZL+w="

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    fjN+GWF4BaT5EdnpXvn7r3Vtfvxg4ePHj9Z33h6anVpGAyZFtqcZ9SC4AqGyFHAeWGAykzAWXZ1EPizGRjLtfqMVQEjSSeKjzmj6K

    HL9W9pBhOunOSKF3QCtev3drmqY5dmY3KgZSFgzrEiTFBra7KVSopTRoU7Pqm30jT+wGegCCWMqpznFIFYLcrQ/DXBKRggvt6i9ZJC

    i0ppyakgyCUQKibacJxKgpp4k3xcEY69OAWVXxu6XN9MeklzyO1Lf3HZjBbn5HJj5Wuaa1ZKUNi4vugnBY4cNciZgDreJmlpoaDsy

    vd341KI5qH4BnqRz3hhFZVgR27erLnFOyqtrWRWk+2wD7vMZbaqCdnOtMgDv0wH7L+lAUGthW27KXH8buS4KkoExdpOMq2vkGa+wg/

    2UTH/+xbIEZ37lTLUhoCAsAob5raATJcKwTivOKJo+PwgPOcGSbsBNYZWxPpXIE4NKPjCtJT+k13aUBaN7zUNeRn4Sl869FWHIGbg

    10yJ4JMpkrFW2JpE6Z0w48gFxhYQhvGmJOUqIO66wTGU/k8aVoCa4NSlBTVc5X6W2iVMBvLGHkCVkiN4OG2bnfgo1a7JrS5caiRZYE

    tKRk3e1jVIS/XKpT6omFHjlaYUcLHTh/nIJb3dwrsa9HZhXod1+NT2lzN6+3I66PXf9N5/GmzuHy7yuxY9j15EL6N+tBftR4fRSTS

    MWGe1s9N529nrfu/+7P7q/v4nXeksap5FrdP98xe9nW3O</latexit>

    NP

    NP

    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

    We don’t know any
polynomial time 
algorithm

  • Example of worst-case behavior

    32

    Complexity of the simplex method

    Innocent-looking problem−xn0 ≤ x ≤ 1

    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

    2n/2 1

    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

    2n/2 0

    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

    2n

    AAADrnicbVLLbhMxFHUbHiU8mgI7NhZpJITU0UxEBewqdRMWRUUiaaXMEHk8N4kVP0a2JyRYs+FL2MIf8TfYIRKdFK+uzjn3ca5vXnJmbBz/3ttv3bl77/7Bg/bDR4+fHHaOno6MqjSFIVVc6eucGOBMwtAyy+G61EBEzuEqX5wH/moJ2jAlP9t1CZkgM8mmjBLroUnn+XH/izzGLs2n2Osso2DqSacbR/Hm4dtBsg26aPsuJ0f739NC0UqAtJQTY8ZJXNrMkVCQQ93u4bQyUBK6IDNw04rz0gd1+wY6LpasNJIIMJlbbZw1eEeEMWuR17gniJ2bXS436xrjXq54EfhdOmD/TQ2IVYqb5jSVnb7LHJNlZUHS5iS5UgtLcp/hjX2Q1C/cAL4gK79CapXGwCGswgTfBixVlbSgnVdcEKvZ6jy0c/24WYBoTdbY+C7QTjVI+EqVEEQWLt1Qxmpfa167JOr7TJ869FkD4EvwayaYs9nc4qmStuFEqpPgMXOBMSUEM34oQZgMiPtX4CNU/k82LAc5s3OXlkQzWXgvtYupCOSNPYCsBLPg4bQ57EwwWbs0tFWlS7XAW2xHSYkumroN0lC9dmk4zJxor9QVh/FJAqvMxdFp6afqR6ewqsM6/NUmuzd6Oxj1o+RN9P5Tv3s22N7vAXqBXqJXKEFv0RkaoEs0RBR9Qz/QT/SrFbdGraw1+Svd39vmPEON15r/AWK4N1Q=

    −xn! ≤ x1 ≤ 1!xi−1 ≤ xi ≤ 1− !xi−1, i = 2, . . . , n

    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

    Perturb unit cube

    x2

    AAACy3icbVE9bxNBEF0fX8F8JVBQQLEiskSDdWchkXSR3JjCKAicRMpZ1t56bK+8H6edPWOzuKSkhf9Bw5+gpOEn8C/YdSKhc5hqNG/ezHszRSkFujT93UiuXb9x89bO7eadu/fuP9jde3iCprIcBtxIY88KhiCFhoETTsJZaYGpQsJpMe9G/HQBFoXR792qhKFiUy0mgjMXSu+Wo85odz9tp5ugV5PsMtk/evrzx6/H3T/Ho73G93xseKVAOy4Z4nmWlm7omXWCS1g38wqhZHzOpnA+XogSNVOAQ7/cyK3hninElSrWtKWYm+E2VuBqTWmrMHIc8W041v5LjRVnjMS6mspNDoZe6LJyoHldSWHM3LEiMJot+lrzcEUE2mdLugDujKUgIVrGAOcIjptKO7A+dPSZs2LZjet8J60PYNayFcWwBZq5BQ0fuFGK6bHPNxA6G2bN1j5rdwIzUAeB1QO5gHBORqWYzhydGO1qTrR5ET0OfUSwhGgmiFJM6Fjx/wa8gSr8JHw52/7p1eSk085etg/fhnf3yEXskCfkGXlOMvKKHJEeOSYDwsmUfCFfybekn2DyMfl00Zo0LjmPSC2Sz38B1rzk9Q==

    x1

    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

    x3

    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

    x!

    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

  • Example of worst-case behavior

    33

    Complexity of the simplex method−xn! ≤ x1 ≤ 1!xi−1 ≤ xi ≤ 1− !xi−1, i = 2, . . . , n

    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

    x2

    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

    x1

    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

    x3

    AAACy3icbVE9bxNBEF0fX8F8JIGCAooVkSUarDuDBHSR3JjCKAicRMpZ1t56bK+8H6edPWOzuKSkhf9Bw5+gpOEn8C/YdSKhc5hqNG/ezHszRSkFujT93UiuXL12/cbOzeat23fu7u7t3ztGU1kOA26ksacFQ5BCw8AJJ+G0tMBUIeGkmHcjfrIAi8Lo925VwlCxqRYTwZkLpXfL0bPR3kHaTjdBLyfZRXJw+Ojnj18Pun+ORvuN7/nY8EqBdlwyxLMsLd3QM+sEl7Bu5hVCyficTeFsvBAlaqYAh365kVvDPVOIK1WsaUsxN8NtrMDVmtJWYeQ44ttwrP2XGivOGIl1NZWbvBx6ocvKgeZ1JYUxc8eKwGi26GvNwxURaJ8t6QK4M5aChGgZA5wjOG4q7cD60NFnzoplN67znbQ+gFnLVhTDFmjmFjR84EYppsc+30DobJg1W/us3QnMQB0EVg/kAsI5GZViOnN0YrSrOdHmafQ49BHBEqKZIEoxoWPF/xvwBqrwk/DlbPunl5PjTjt73n71Nry7R85jhzwkj8kTkpEX5JD0yBEZEE6m5Av5Sr4l/QSTj8mn89akccG5T2qRfP4L2SHk9g==

    Theorem

    2n � 1

    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

    Remark

    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

    x!

    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

  • Complexity of the simplex method

    34

    We do not know any polynomial version of the simplex method,
no matter which pivoting rule we pick.

    Still open research question!

    Worst-case

    n m O(2n)

    AAAEknictVPNTxNBFN9iVaxfoN68TKQkaEKzbSQqJwwXTEQxoUDCFjI7+9pOmK/MzJY2k70Yr/r/+Z949E1tIgtefckmb3+/9/3e5EZw59P0Z2PpVvP2nbvL91r3Hzx89Hhl9cmR06Vl0GdaaHuSUweCK+h77gWcGAtU5gKO84vdyB9PwDqu1aGfGRhIOlJ8yBn1CJ2v/DocgwVC8TNWo5ckXDlPFQNHLuecHwNxXBoBUyLBj3VBLrkQxJaKUEVClg8JTI1WoDyngqhS5mCJHhLuwc7zuAqjEvyTLuIxYsElq