Matematika | Diszkrét Matematika » Vektorterek, lineáris egyenletrendszerek és a szimplex módszer

Alapadatok

Év, oldalszám:2003, 17 oldal

Nyelv:magyar

Letöltések száma:828

Feltöltve:2005. április 11.

Méret:101 KB

Intézmény:
-

Megjegyzés:

Csatolmány:-

Letöltés PDF-ben:Kérlek jelentkezz be!



Értékelések

Nincs még értékelés. Legyél Te az első!


Tartalmi kivonat

    "! $#&%"!()*+$(,-"! .0/1235427698:<;>="?A@B:<;C&D*2AE6>FHGI2J;CHK 86ML"GNCO:<;QPG>R>8HGNCHSTRM=F ;U@9FHG78HVT@9CHECHEVWC8F XRM= [Y Z]H^` Haab`HcaHd5[^JbJdeM HfNbJg*H^JhWe"af> HdHcji"g>HckHZ"lWccH^HZ]H^`HmonlWhH^JpTHbrqHH^nlWhs ^`H^JcHlWZ"tdc uHpve"^rwWe"c&gNZ"bJpvm^JeIx&pvyzgNZ"e"f{^Je"hWe"hHwgNZ"e"fN|aaTqH H^r}>lWZM}< HdHc5jpve"hW~"fs }>~"gN~"€e"ZgNZ"cgN~"hWe"gNe"c ‚ Y 1CHKJƒ;j1H6<4@BRGI@B:„G nlWhH^JpTo‚ e"hHwWe"d e"hHwO€H^JpTHZ&†ˆ‡IqWe"c}>lWfNlWc‡U€H^JpTHZM‰IuHpve"^rwae"dŠ~"f‹}>e"^Jpve"Z„qWeUqHHd e"hHw ⊕sH^ŒNe"^JVW^r}]pk|oqWe"^Je„}†N‡„WgNgNZ"eMHz

Hg‡M‰0HZH^` Haab}>t^`ŒNzlWdg> HhWlWccH^7Ž †bA‰pvbJdze"d x, y ∈ V sBfNeHŽ x ⊕ y = y ⊕ x †cHlWpvpkto}<}>brqbr}< Hg<‰Iu †bJbA‰TpvbJdze"d x, y, z ∈ V sBfNeHŽ x ⊕ (y ⊕ z) = (x ⊕ y) ⊕ z †HgNgNZ"li"b`}>brqbr}< Hg<‰Iu †bJbJbA‰&qHHdlW^rwHd θ ∈ V uHpve"^rwfNe x ⊕ θ = x pvbJdze"d x ∈ V sBfNe†HZWgNgNZ"eMHz HgNf> d~"Z„qWejqHHd V sBae"dˆde"to}>f> H^JbJgQe"^Je"p‰Iu †brq‰OpvbJdze"d x ∈ V sB€e"ZqHHd‘lW^rwHd −x ∈ V u’€lWhHw x ⊕ (−x) = θ †pvbJdze"d sBae"^Jb“e"^Je"pvde"cvqHHdHzzbr}>”rqTbJdqWe"fNZ"e‰I‚ V • f‹}>e"^Jpve"Z„qWe–qHHd—pv~"h‘e"hHw˜‡„pk|oqWe"^Je„} ‡u[Hpve"^rwWe„}ŠgNcH^` HfNf>H^0qHH^Jy‘gNZ"lWfNZM HgNdHc €”rqtdc™~"gšHpve"^rw™pvbJdze"d (α,

x) m HfN€lWZHu&H€lW^ α ∈ R e"hHw›qHH^JyWgœgNZM Hp~"g u{e"hHw α ⊗ xsBe"^ŒNe"^JW^r} V sBae"^Jb{qWe"c}>lWf‹}TfNe"dze"^†HZ x qWe"c}>lWfvgNcH^` HfNgNZ"lWfNls x ∈ V g> }I‰ž~"g[Hpve"^rwde"cHZH^` Haab}>t^`ŒNzlWdg> HhHbqHHddHc Ž Ÿ bJdze"d α, β ∈ R ~"g x, y ∈ V sBfNeHu †‰ α ⊗ (x ⊕ y) = (α ⊗ x) ⊕ (α ⊗ y) u †a ‰ (α + β) ⊗ x = (α ⊗ x) ⊕ (β ⊗ x) u †i‰ (αβ) ⊗ x = α ⊗ (β ⊗ x) u †z ‰ 1 ⊗ x = x ‚ ~"^Jz HcTqHH^JyWgžqWe"c}>lWf‹}>e"fNe"cfNeH‚ ¡ ‚0Y¢qHH^JyWg gNZM Hp nsBe"gNe"ck}>e"fNe† nsBzbJpve"dZ"bJyWg*cHllWfNzbJd }<}>~"f<‰Iu€jHZ]WgNgNZ"eMHz Hg‹} ~"gˆ‘gNcH^` HfNf>H^[qHH^Jy‘gNZ"lWfNZM Hg‹}ˆcHlWpvmlWde"dgNe"dcH~"d}œze„£db` H^

ŒNtc¤~"gˆ$i"gNtm sBayW^“ H^J^JykgNZM Hp nsBe"g¦HZUWgNgNZ"eMHz Hg[de"to}>f> H^JbJgQe"^Je"pveHu ¡ ‚ "e hHwWe"d F HZ [a, b] bJd}>e"f‹qHH^J^JtpvlWdk~"f‹}>e"^Jpve"Z"e„}N}’qHH^JyWg*nhWhHqW~"dwWe"c5€H^JpTHZMo‚ Y[Z[WgNgNZ"eMHz Hg‹}]~"gQgNcH^` HfNf>H^qHH^JygNZ"lWfNZM Hg‹}Q~"f‹}>e"^Jpve"Z"Z"cvcHqWe„}>cHe"ZvcH~"ms me"d“Ž pvbJdze"d f, g ∈ F u α ∈ R ~"g x ∈ [a, b]sBfNej^Je"hHwWe"d (f ⊕ g)(x) = f (x) + g(x), (α ⊗ f )(x) = αf (x). Y[ZŠWgNgNZ"eMHz HgTde"to}>f> H^JbJg5e"^JpveOHZˆHZ"lWdlWg>Hd¤dt^J^`nhWhHqW~"dwW‚¦Wddw|¤ae"^` }>db7u €lWhHw F HZU”JhHwze„£db` H^r}]pk|oqWe"^Je„}>e"ccHe"^qHH^JyWgQqWe"c}>lWf‹}>e"fNe„} H^JcHlH}M‚ Y }>lqH

Haab`HcaHd i"g>Hcde"p pvbJdzbJhqHH^JyWgTgNZM Hp nsBe"gNe"c¤}>e"fN~„qWe"^[nlWhW^`H^JcHls Z"tdc~"g5de"p lWcHlWZOpTŒNz n~"^JfNe"~"f‹}>~"g‹}Mu €–HZOe"hHwgNZ"e"fN|gN~"hšcHe"zoqW~"~"f‹}qWe"c}>lWfNlWc WgNgNZ"eMHz Hg> Hf>5~"gQgNcH^` HfNf>H^ qHH^JykgNZ"lWfNZM Hg> Hf>5k€HhHwWlWpT HdwWlWg"uqHH^JyWgžgNZM HpvlWcvcHs fN~"ae"d pve"hWgNZ"lWcHlH}N }  gNZ"bJpkayW^JtpvlH}jbJ^J^Je„}NqWeTHZTe"hHwpT Hgpve"^J^J~&”Jf> Hg‹}€HgNZ"d H^ ŒNtc ‚ Y qWe"c}>lWfNlWc}UqHHg‹}<HhOae„}>|oqWe"^7u-de"to}>f> H^JbJg e"^Je"pve„}Mu“dt^J^ s9qWe"c}>lWf‹} os9qHH^nlWhMŒNtc ŒNe"^JW^Jdb7‚ e"hHwWe"d V e"hHw–qWe"c}>lWf‹}>~"f"u v , . , v ∈ V ~"g α , ,

α ∈ R ‚TY[Z α v + WgNgNZ"e"hWe„}Q G = {v , 1. , v }n qWe"c}>lWfNfNe"dz1gNZ"e"f α ,n , α gNcH^` HfN1lWc1cH^ .cH~"m+ α n vn Z"e„}N} K 2AE6>FHGI2J;Q4RSvL"2AEF „27ƒ "FH1EC 4 €”rqMŒNntc ‚ ¦e„qWe"Z"Z"cv G qW1e"c}>lWfNfNe"ndzgNZ"e"f K 2AE6>F  GI2J;L G>4WFHEC 4 ~"gŒNe"^JW^ ŒNc5^JbJd (G)s9qWe"^“ G qWe"c}>lWf>HbJdHcvWgNgNZ"e"g0^Je"€e„}>gN~"hWe"g]^JbJdeM HfNbJg cHlWpkabJd Hi"bJy Œ> }M ‚ ¦WddwWe"d¤^` }>€}>yu’€lWhHw‘^JbJd pTHhœbJgk^JbJdeM HfNbJg}>~"f†H^r}>~"f<‰ V sBae"d$ze„£db` H^r}5pk|oqWe"^Je„}>e"ccHe"^¦~"g&de"to}>f> H^JbJgk(G) e"^Je"pvpve"^7‚šY G = {v , . , v } 1 n qWe"c}>lWfNfNe"dzgNZ"e"f‹} K

2AE6>FHGI2J;MCH E W886„@KJ6„EE6<4 de„qWe"Z"Z"c u€ α1 v1 + . + αn vn = 0 =⇒ α1 = = αn = 0 ¦WddwWe"dae"^` }>€}>yu €lWhHwU^JbJdeM HfNbJgnhWhWe„}>^Je"dgN~"h0ne"d}>bze„£d”Ji"bJy Œ> MqHH^We"cqbrqHH^Je"dg HZjH^` Haab“cH~„}[ze„£d”Ji"bJyŽ ¡ ‚ Ÿ bJdze"d v ∈ ^JbJd (G) e"hHwW~"f‹}>e"^Jpk|e"d H^J^J”r}>€}>yve"^  G qWe"c}>lWf>HbJdHcœ^JbJdeM HfNbJg cHlWpkabJd Hi"bJy Œ>HcH~"d}M‚ ‚ G e"hHwWe„}>^Je"djqWe"c}>lWf>gNe"p neBŒNe"Z"€e„} 0cbž}>WaabJe"c[^JbJdeM HfNbJgcHlWpkabJd Hi"bJy Œ>HcH~"d}M‚  Y›ze„£d”Ji"bJy&d~"€ HdwŠe"hHwgNZ"e"fN|cHqWe„}>cHe"Z"pv~"dwWeHŽ ¡ ‚ bJdeM HfNbJg>Hd–nhWhWe„}>^Je"d

fNe"dzgNZ"e"fde"p }<Hf‹}<H^JpTHZ"€}jHZ"lWdlWgjqWe"c}>lWfNlWc}~"g 5dt^J^qWe"c}>lWf‹}[gNe"pˆ‚ ‚0Y[Z e"hHwWe„}>^Je"dŠde"pvZ"~"fNy5qWe"c}>lWf‹}]}<Hf‹}<H^JpTHZ"yqWe"c}>lWfNfNe"dzgNZ"e"f¦^JbJdeM HfNbJg>Hdvnhs hWe„}>^Je"d“‚ ‚ Jb deM HfNbJg>HdOnhWhWe„}>^Je"dOqWe"c}>lWfNfNe"dzgNZ"e"f pvbJdze"dˆfN~"gNZ"fNe"dzgNZ"e"fNekbJg0^JbJdeM HfNbJg>Hd nhWhWe„}>^Je"d“‚ hHw G qWe"c}>lWfNfNe"dzgNZ"e"f‹}O V qWe"c}>lWf‹}>~"f 86„E6„GNFH@BRGIG>6„E3 ;>= 6„G>:„E6<4 €”rqMŒNtc už€ ^JbJd ‚  hHw qWe"c}>lWf‹}>e"fNe„} ‹3H2AST6„E="27ƒ;„EC 4 de„qWe"Z"dc u{€ŠqHHd qWe"c}>lWf‹} V }<Hf‹=}<H^JpTHZ"(G)

yžhWe"de"f> }>lWfNfNe"dzgNZ"e"fNeHuHzne a HfNpve"^rw nsBd~"^WcHe„qWe"gNe"aaqWe"c}>lWf‹}}<Hnf‹}<H^JpTHZ"y qWe"c}>lWfNfNe"dzgNZ"e"fNeUpT Hf de"p hWe"de"f> }>lWfNfNe"dzgNZ"e"f"‚  hHw V qWe"c}>lWf‹}>~"f G qWe"c}>lWfNfNe"dzs gNZ"e"fN~„} L<F="2J;„EC 4 †cHllWfNzbJd }<HfNe"dzgNZ"e"fNde"co‰]de„qWe"Z"Z"c u€ G ^JbJdeM HfNbJg>HdTnhWhWe„}>^Je"d hWe"de"f> }>lWfNfNe"dzgNZ"e"f"‚ ¦qWe„}>cHe"Z"pv~"dwWe"c Ž ¡ ‚ Ÿ bJdze"d qW~"hWe"gOzbJpve"dZ"bJyWg5qWe"c}>lWf‹}>~"fa HfNpve"^rw qWe"c}>lWf>e"hHwW~"f‹}>e"^Jpk|e"d cb s neBŒNe"Z"€e„} $a

HfNpve"^rw¤a HZ"bJg> HdHc¤^JbJdeM HfNbJgvcHlWpkabJd Hi"bJy Œ>HcH~"d}M‚ Y pve"hHne"^Je"^  gNZ"lWfNZ"yWc} C=kC3R@@QL<F="2J;„GNCv1HRECH@A4RM= ƒk4RRGN3H2AEFH@9F 4EC 4 de„qWe"Z"Z"c ‚ ‚0Y™qHH^JyWg]gNZM Hp nsBe"gNe"c}>e"fNe n zbJpve"dZ"bJyWg"‚ ‚  hHw nsBzbJpve"dZ"bJyWgŠqWe"c}>lWf‹}>~"f–a HfNpve"^rw—^JbJdeM HfNbJg>Hd—nhWhWe„}>^Je"d qWe"c}>lWfNfNe"dzs gNZ"e"fNecbJe"hW~"gNZ"”r}>€e„} &a HZ"bJgNg> o‚ qWe"cWs  ZM HpvlWgjmfNlWa^J~"pTpve"hWlW^Jz Hg>gNlWf> HdšgNZ"cgN~"hWdc qHHdHfNf>ou€lWhHw V }>lWf‹}>~"f e"hHwkHzlH}N}’a HZ"bJg> HfNyW^o }N}>~"fAŒNdce"hHwkpT HgNbJcja HZ"bJgNf>o‚

e"fNpv~"gNZ"e„}>e"gNe"dve"ZM H^r}<H^ a HfNpve"^rwqWe"c}>lWf e"gNe„}>~"ae"d cHllWfNzbJd }< HcbJg’pve"h nlWhWdHcqH H^r}>lWZ"db7‚ BhWe"dve"hHwgNZ"e"fN| HZ  ŒcHllWfNzbJd }< HcdHcj¦cbJgNZM Hpv”r}< Hg>ou €[HZ  Œ*a HZ"bJgfN~"hWbr} W^i"g>Hce"hHwWe„}>^Je"dqWe"cWs }>lWfNaHdœc^JWdaWZ"bJc ‚ Y a HZ"bJg[pve"hHqH H^r}>lWZ„}<}< Hg> }[e"ccHlWf 6„KJ6„S52QL<F="2J;„@GNCHEo;>= MRGISvF  „27ƒEC 4 de„qWe"Z"Z"c ‚ e"hHwWe"de"c B = {b , . , b } V nsBzbJpve"dZ"bJyWg’qWe"c}>lWf‹}>~"f]e"hHw a HZ"bJg>O~"g c 6= 0 e"hHwœ}>e„}>gNZ W^Je"hWe"gjqWe"c1}>lWf"u*Hpvne"^rwde"c– B

sBfNe5qWlWd}>cHlWZ"ycHllWfNzb s d }< Hb ~"gQ}>e"hHwcvne"^7uo€lWhHw ‚ e"hHwWe"dO}>lqH Haa e"hHwT}>e„}>gNZ W^Je"hWe"g qWe"c}>lWf"cu1 ,H.pv e",^rwcdn e"c B sBfNejqWlWd}>cHlWZ"yTcHclklW6=fNzbJ0d }< Hb x , , x ‚  xccHlWf b qWe"c}>lWf cbJi"gNe"fN~"^J€e„}  cs9qWe"^~"g{HZ  Œ’a HZ"bJg B 0 = {b , . , b 1 , c, b n , , b } ‚ kHe"^JW^ ŒNc s9qHH^-HZ x € HdwHzlWg‹}M‚  ccHlWf[HZ x qWe"c}>1lWf B 0 a Hk−1 Z"bJgNf>kqWlWk+1 d}>cHlWZ"yTcHnllWfNzbJd }< Hb7Ž δ  k ck x1 − δc1 , . , xk−1 − δck−1 , δ , xk+1 − δck+1 , , xn − δcn Y gNZM HpvlH} 86„E6„GNFHKJƒ6„KJ6„S5E6<4 de„qWe"Z"Z"c ‚&Y[Z5e"^ WaabJe"cH^`HmŒ> Hdœpve"h H^J^` s m”r}>c€k}>6=yu0€lWhHwŠa

HZ"bJgNaˆae"^J~"m ˆqWe"c}>lWfNdHcšˆa HZ"bJg‹}e"^J€HhHwWyˆqWe"c}>lWfNf>ˆqWlWd}‹s cHlWZ"ycHllWfNzbJd }< Œ>v†jhWe"de"f> H^Jyje"^Je"p‰ dt^J^` }>yW^c^JWdaWZcHe"^J^^Je"hHwWe"d“‚ ¦WddwWe"d ^` }>€}>yu{€lWhHw$e"ZgNZ"cgN~"hWe"gO~"gTe"^J~"hWgN~"hWe"gvne"^r}>~„}>e"^JeˆHddHc u]€lWhHw HZˆe"^Je"pvb]a s Z"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJykqW~"hWfNe"€Œ‹}>€}>yv^Je"hHwWe"d“‚  lWcgNZ"lWfnlWd}>lWg}>tzdb†m^7‚pT }>fNb xolWc bJdqWe"f‹}< H^` Hg> Hd H^7uH¦gNZ"bJpvm^JeIxjpvyzgNZ"e"fNd~"^A‰ ja HZ"bJgNayW^~"mme"dTcbJcHe"fN^r} b qWe"c}>lWfNdHcvHZ  Œ a

HZ"bJgNf>UqWlWd}>cHlWZ"ykcHllWfNzbJd }< Hbr}M‚ k u x = 1, x = 0, . , x = 0 g ”JhHw b sBdHcHZ  ccHlWf x = 0, . . . , x = 0 1 k−1 k+1 n k  Œ]a HZ"bJgNf>kqWlWd}>cHlWZ"yTcHllWfNzbJd }<k Hb7Ž H€lW^ ‚ −δc1 , . , −δck−1 , δ, −δck , , −δcn , a Hfs pve"^rwbJcpT HgNbJca HZ"bJgNao‚  Z„} &}>e"i<€dbJc }[^Je"€e„}[€HgNZ"d H^JdbHZtd“‚ 4RS[PCH@2L"2AK 2A@9F ; 12r=I;>8FHKrCH@GNC ‚ e"hHwcne"^7u“€lWhHw–HzlH}N}Ue"hHw G = {a , . , a } qWe"c}>lWfNfNe"dzgNZ"e"fkHZ sBzbJpve"dZ"bJyWg0qWe"c}>lWf‹}>~"fNae"d“u“}>lqH Haa OHZ nsBzbJpve"dZ"1bJyWg0qWe"c}>klWf‹}>~"fUe"hHw b qWe"c}>lWf>o‚ n Y[Z„}jgNZ"e"fNe„}>d~"dc e"^JzWd}>e"db7u-€lWhHwœne"dd H^J^ sBek

b ∈ ^JbJd G }<Hf‹}<H^JpTHZM Hg"uqHHhHwbJg b cbrneBŒNe"Z"€e„}sBe qWe"c}>lWf>HbJdHcTqHH^`HpvbJ^rwWe"dO^JbJdeM HfNbJgQcHlWpkabJd Hi"bJy Œ>HcH~"d}M‚ e"hHwc ne"^7u€lWhHw b cHllWfNGzbJd }< Hb B = {e , . , e } e"hHwgN~"hHqWe"c}>lWfNlWckH^JcHlH}N}<¦a HZ"bJgNf>5†HZ td“‚ @GI2A12FHK 2J;kL<F="2J;„GNC ‰ qWlWd}>cHlWZ"dH1c u }>e"€ } n b = b e + . + b e ‚ Y[^`Hc”r}>gNtc } pvlWg‹}QnlWcHlWZM}>lWg>Hd“upvbJdzbJhke"hHwqWe"c}>lWf‹}¦cbJi"gNe"fN~"^rqW1eT1†e"^Je"pvb“a HZ"nbJg‹}>nf>HdgNZ„nlWfNpT Hi"bJyH} €Œ‹}NqHQqW~"hWfNe‰- B a HZ"bJg‹} hHwWu€lWhHwU G qWe"c}>lWfNfNe"dzgNZ"e"fqWe"c}>lWf>HbcHWZ"^^Je"€e„} W^Je"h pvbJd~"^“}>Waae„}¦qWlWd

ŒNtdcŠaekva HZ"bJgNao‚ e"fNpv~"gNZ"e„}>e"gNe"d–mlWd}>lWg>Hd–Hddwbr}¦}>tztdc ae„qWlWddb7uHpve"ddwb O^JbJd (G) H^r}>~"f zbJpve"dZ"bJy Œ>ou“Hpvbr} G GNCHE8 "FHEC 4 bJgUde„qWe"Z"€eIs }>dc ‚  G f>HdhMŒ> r ~"gž}>Wf‹}>~"de„}>e"gNe"d a , . , a ^JbJdeM HfNbJg>Hd&nhWhWe„}>^Je"de"cœ†^JbJd G e"hHwŠa HZ"bJg> }[Hz Œ> Hco‰Iu Hpve"^rwWe"cHe„} HZ e , . 1 , e qWe"rc}>lWfNlWccH^i"gNe"fN~"^r}>dcne"^7u HccHlWf HZœe"^Je"pvb¦a HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyWc‘gNlWfNlW1 ZM}<e"rhHw H = {a , . , a , e , , e } a HZ"bJg‹}e"fNe"zpv~"dwWe"Z"e„}N}M‚ Y b qWe"c}>lWfNdHc™HZe"fNfNe ¤a H1Z"bJgNf>$qWlWrd}>r+1 cHlWZ"y

cHllWnfs zbJd }< Hb0^Je"hHwWe"de"c d0, . , d0, d0 , , d0 ‚  d0 = = d0 = 0 u0HccHlWf 1 uqHHhHwrbJg r+1 ^JbJd n u€ r+1 qHH^`Hpve"^rw n 0 0 0 b = d a + . . . + d a b ∈ (G) d = 6 0 j 1 r e"gNe„}>~"d“1u]HccHlWf b ∈/r ^JbJd (G)  ‚  e"hHwc‘~"gNZ"fNeHu]€lWj hHw$ cHlWpvm}>bJabJ^J=br}< Hrg+q1,bJZ"gN.h H ^`,n} }>t^`ŒNzlWdcH~"mme"dHddHcO5pve"h H^J^`Hm”r}< Hg>ou€lWhHwŠHZ  1 δ^Je"= pvbHcak HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyWc0gNlWfNlWZM}< MqHH^Wa HfNpve"^rw[a HZ"bJgNayW^W }N}>~"fN€e„}>dc x 1 a1 + . + x k ak = b ^JbJdeM HfNbJg e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"cTqHHdsBe0pve"hWlW^Jz Hg>o‚  U qHHd“uoHccHlWf{e"hHw o}N}<H^ pve"hbJg €}< HfNlWZ"tdce"hHwWe„}MŽ x0 = (d01 , . , d0r , 0, , 0)

Y[Ze"hHwWe"d^Je„}>fNe"dzgNZ"e"fUe"hHwˆbJ^rwWe"d 0 pve"hWlW^Jz Hg> }MuHpve"^rwae"dœHZ 0 de"p5sBdt^J^` cHlWpvmlWde"dgNe"bJ€e"Zˆ}<Hf‹}>lWZ"yšqWe"c}>lWf>Hb x G qWe"c}>lWfNfNe"dzgNZ"e"fNde"c—š^JbJd x(G) a HZ"bJg> } H^JcHlH}Œ> Hc u L<F="2J;„ST698RKr3F ;„EC 4 de„qWe"Z"Z"c ‚  G pTHh–a HZ"bJg"u]HccHlWfv^JbJd (G) HZ e"hW~"gNZ nsBzbJpve"dZ"bJyWgUqWe"c}>lWf‹}>~"f~"gkˆ^JbJdeM HfNbJgUe"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"c i"g>Hc e"hHwWe„}>^Je"d  pve"hWlW^Jz Hg>TqHHd“ŽjHZ&e"fNfNevŠa HZ"bJgNf>qWlWd}>cHlWZ"ycHllWfNzbJd }< Hc ‚5Y cHlWpvm}>bJabJ^Jbr}< Hg q bJZ"gNh H^`}kpT

HgNbJcš€HgNZ"dlWg&pve"^J^J~"c}>e"fNpv~"cHeHu’€lWhHw e"hHw o}N}<H^Qpve"hW€}< HfNlWZ"Z"tc G qWe"c}>lWfNfNe"dzgNZ"e"f0f>HdhMŒ> }QbJg"‚¦~"Z"Z"dcpvlWg‹}Qe"hHwTm~"^Jz }]cHlWpvm}>bJabJ^Jbr}< Hg qbJZ"gNh s ^`}>f>o‚ e"hHwWe"d  a1  a2 1 1    G=  2   1  1  ,  −1 3 0  a3   a4  a5 1 1 1   −1   1   1  ,         1 ,  0 ,  1  0 1 2  b 9  4     1  5  Y gNZM Hpv”r}< HgNlWc}Ui"~"^JgNZ"e"fN| }< Ha^` HZM}>lWg nlWfNpT HaHd–e"^rqW~"hWe"Z"db7‚5Y }< Ha^` HZM}Ue"^Jg  lWgNZ"^JlWm HaHd5HZ0Hc}>t H^JbJg*a HZ"bJg‹qWe"c}>lWfNlWc} }>d}>e„}ŒNcne"^7uWHZ]e"^Jg

UgNlWfNaHd& a HZ"bJgNlWd c”rq^JbqWe"c}>lWfNlWc}Mu}< Ha^` HZM}Uae"^JgNeBŒN~"ae"dšme"zbJhŠOa HZ"bJgNlWd c”rq^Jb*qWe"c}>lWfNlWcdHc HZvHzlH}N}ja HZ"bJgNf>OqWlWd}>cHlWZ"ycHllWfNzbJd }< Hbr}M‚kY[Z  ŒjcHllWfNzbJd }< Hc}jHZ&e"^Je"pvb a s Z"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJy5gNZMHa H^rw MqHH^“€}< HfNlWZ"Z"tcOpve"h‚ Bdzt^Jy5}< Ha^` HZM}MŽ ¦e"pka HZ"bJgQqWe"c}>lWfNlWc ¡ ¡ a1 H Z"bJg qWe"c}>lWfNlWc  e1 e2 e3 e4 ¡ ¡ a4 a5 ¡ ¡ ¡ a2 a3 ¡ −1 −1 ↓ ¡ ¡ ¡ b ¡ ¡   ¦llWfNzbJd }< Hc Y[Z a , . , a qWe"c}>lWfNlWc‘cHWZ"^¦a HfNpve"^rwbJcšae"^J~"m€e„} a HZ"bJgNa † b qWe"c}>lWf‹} a HZ"bJgN1lWdšc”rq5^ }<Hf‹}ŒNtc

‰Iu*HfNf>ŠcHe"^J^ i"g>Hc hHwWe"^Jddc u €lWhHw hWe"de"f> H^Jyˆe"^Je"p †HZT cHllWfNzbJd }<ouHpve"^rw&a HZ"bJgNaae"^J~"m ~"gQa HZ"bJg‹}{e"^J€HhHwWyqWe"c}>lWfU‡„pve„}>gNZ"~"gNmlWd}‹s Œ> HaHd‡qHHd ‰ s9}>yW^*c^JWdaWZ"Z"~"c ‚ U H^`HgNgNZ"tcœm^7‚kOae"cHe"fNe„}>e"Z"e„}N} ¡ sBe"g‹}jhWe"de"f> H^Jy e"^Je"pvde"c ‚  ZOHZ„}¦ŒNe"^Je"d}>b7u €lWhHw HZ e qWe"c}>lWf5e"^J€HhHwMŒ>œa HZ"bJg‹}5~"g a cHe"fN^] 1 €e"^rwW~"fNeH‚’Y™gNZM Hpv”r}< HgNlWc}ži"~"^JgNZ"e"fN| hHwT e"^rqW~"hWe"Z"db7uo€lWhHwve"^ WgNZ"WfQhWe"de"f>1 H^Jyke"^Je"p gNlWf> }

qW~"hWbJhWlWgNZ„}ŒNtc–hWe"de"f> H^JyOe"^Je"pvpve"^7‚  ccHlWfjpvbJdze"d“u“pv~"hŠOa HZ"bJgNlWd–c”rq^ pTHf>Hzo}UqWe"c}>lWfNf>ˆpve"hWcHmŒNtc ‡ δ ‡5€ HdwHzlWg‹}M‚  Z"Z"e"^{HZ„}< Hd ŠhWe"de"f> H^Jyˆe"^Je"p   lWgNZ"^JlWm }kpve"hWgNZ"lWfNlWZ"Z"tc~"g&HZbJ^J^Je„}ŠqWe"c}>lWf&lWgNZ"^JlWm HayW^žHZ”JhHwšcHmlH}N}kqWe"c}>lWf‹} cbrqWlWd ŒNtc ‚ BhHwŠ5cHqWe„}>cHe"Z }< Ha^` HZM}>lWcvgNlWfNlWZM}< }0cHmŒNtc Ž a1 e2 e3 e4 a1 a2 e3 e4 a1 a2 a4 e4 a2 ¡ a3 a4 a5 s¡ ¡ −3 −2 −3 −3 −1 −1 −2 −1 −14 −8 −1 −22 a3 −2 a5 ¡ 1 ¡ ¡ a3 −2 −3 a5  a4   ¡ ¡ b b −5 14 H H b −5 −6 H −1 Y›gNZM Hpv”r}< HgNlWc}]de"p }>tz ŒNtcT}>lqH

HaaŠnlW^rw}<}>db7u€bJgNZ"e"d e sBe„}0de"p ^Je"€e„}¦gNe"p sH^7ugNe"p a sBe"^cbJi"gNe"fN~"^Jdb7‚ Ÿ brqWe"^ bsBde"cvHZ e sBfNe¦qWlWd}>cHlWZ"y54 cHllWfNzbJd }< Œ> u b a 4 ne"3^J”JfN€}>y5HZ5H^` HaabpvyzlWd“Ž b = −5a1 − 6a2 + 20a4 . dde"c–€e"^rwWe"gNgN~"hW~"f W^ pve"hŠbJgUhHw WZz€e„}>dc ‚vY G qWe"c}>lWfNfNe"dzgNZ"e"ff>HdhMŒ> ~"g ^JbJd (G) ‚ b∈  bsBde"cHZ e4 sBfNe qWlWd}>cHlWZ"y cHllWfNzbJd }< Œ>¦dt^J^` }>yW^Wc^JWdaWZ ^Je"ddeHuWHccHlWf ^JbJd ^Je"ddeH‚ b∈ / Z"Z"e"(G) ^jgNZM Hpv”r}< HgNg>H^e"hHw o}N}<H^ pve"hW€}< HfNlWZ„}>tc5jcHqWe„}>cHe"Z5e"hHwWe"d^Je„}>fNe"dzgNZ"e"f  e"hHwš†a

HZ"bJg<‰žpve"hWlW^Jz Hg> }MŽ  x1 + x 2 + x 3 + x 4 2x1 + x2 − x3 + x4 x1 − x 2 + x 3 3x1 + x4  + x5 + x5 + x5 + 2x5 = = = = 9 4 1 5 Y[Z u u u u i"g>HcthHwHd e"hHwœa HZ"bJgNpveIs hWlW^Jzx H1g"‚*=Y[−5 fNfNyW^7ux€2lWhH=w&−6 pve"hWlW^Jxz3 H=g’e"0hHwgNZ"xe"4fN|O=a20 e"€e"^rwWxe„5}N}>=e"gN0”r}>~"gNgNe"^“hHw WZz€e„}>dc&pve"h‚ Y[Z„} me"zbJhu €lWhHwˆa HZ"bJgNpve"hWlW^Jz Hg¦HaayW^^` }>€}ŒNtc u €lWhHwHZ x , x , x de"pvdt^J^` qH H^r}>lWZ"yWc€lWZ }<Hf‹}>lWZ"y a , a , a qWe"c}>lWfNlWcO^JbJd (G) a HZ"bJg> }¦H^JcHlH1}Œ> Hc 2 ‚ 4  5ne"d}>be"hHwWe"d^Je„}>1fNe"d2zgNZ"4e"fNde"cŠde"pvi"g>Hcve"hHwpve"hWlW^Jz Hg> }žgNZ"e"fNe„}>d~"dcOpve"hs

€}< HfNlWZ"db7u€Hde"p qHH^`Hpve"ddwbr}MuoHccHlWf]bJgže"^Je"hWe"dz vne"d}>bcHlWpvm}>bJabJ^Jbr}< Hg’qbJZ"gs h H^`}>lH}[e"^rqW~"hWe"Z"db7‚Y cHlWpvm}>bJabJ^Jbr}< Hg]qbJZ"gNh H^`}Ue"fNe"zpv~"dwWe"cH~"d}thHwHdbJg GsBde"c ša HZ"bJgNašaede"p cHe"fN^r}TqWe"c}>lWf>HbJdHc H a HZ"bJgNf> qWlWd}>cHlWZ"y‘cHllWfNzbJd }< Hbr}&bJg pve"hW€}< HfNlWZ„}>tc ‚ e"hHwWe"de"cŠe"Z"e"c dj (i = 1, . , r; j = r + 1, , k) qHHhHwbJg i  aj = dj1 a1 + . + djr ar ccHlWf[HZUe"hHwWe"d^Je„}>fNe"dzgNZ"e"f[”JhHwbJg]”JfN€}>yŽ Ÿ brqWe"^ j = r + 1, . , n  r+1 a1 x1 + . + ar xr + dr+1 1 a1 + . + dr ar xr+1 +  + dk1 a1 + . + dkr ar xk = d01 a1 + + d0r ar H a

HZ"bJgNf>kqWlWd€}>yvcHllWfNzbJd }< HcTe"hHwW~"f‹}>e"^Jpk|e"c u e"Z"~"f‹} xi + dr+1 xr+1 + . + dki xk = d0i i (i = 1, . , r) xi = d0i − dr+1 xr+1 − . − dki xk i (i = 1, . , r) qHHhHwO }>fNe"dze"Z„qWe Y[Z x , . , x qH H^r}>lWZ"yWc} 4/@B/@@L<F="2J;U1FHK @BRM= ƒ4 dHc uHZ x , , x qH H^r}>lWZ"yWc} ^r}>lWZ"yWcdHcœk }>e„}>gNZW^Je"hWe"g ;>=CL<C13 † E6„SvrL<F="2J; ‰ 1FHK @BRM= ƒ4EC 4 de„qWe"Z"Z"c ‚–Y gNZMHaHz qH Hr+1 ~"f‹}>~"cHe„}HzoqH–~"gvHZ„}< Hd –cHH}>H}N}&qH H^r}>lWZ"yWc}5œne"d}>bQnlWfNpkt^` MqHH^žcbJgNZM Hpv”r}NqH–HZ e"hHwWe"d^Je„}>fNe"dzgNZ"e"f0e"hHwTpve"hWlW^Jz Hg> }U† PCHGI@2J4 KrFHGI2J;UST698RKr3F ; ‰’cHmŒNtcv~"g0HZ

e"hHwWe"ds ^Je„}>fNe"dzgNZ"e"fQpvbJdze"dTpve"hWlW^Jz Hg>je"^  H^J^J”r}>€}>y bJ^rw5pvyzlWd“‚  gNZMHaHzš†de"pka HZ"bJg<‰ qH H^r}>lWZ"yWcdHcdt^J^`U~"f‹}>~"cHe„}QHztdc uoHccHlWf{HZ0”JhHwkcHmlH}N}[†mHf‹}>bJct^` HfNbJg<‰pve"hWlW^Jz Hg e"hHwOa HZ"bJgNpve"hWlW^Jz Hg"‚ Y gNZM Hpv”r}< HgNb*e"^ Œ> Hf> HgUthHwHdHZHupvbJd}UOcHlWpvm}>bJabJ^Jbr}< Hg]qbJZ"gNh H^`}>d H^7‚  Z"lWc Hg HZ"lWdaHd bJ^rwWe"dcHlWfkŠqWe"c}>lWfNlWc}&ˆ€lWZ"ZM }<Hf‹}>lWZ"yqH H^r}>lWZ"yWccH^ fNe"mfNe"Z"e"d}< H^Jdb7u’HZ e"hHwgN~"hHqWe"c}>lWfNlWc} me"zbJh&de"p ”JfAŒNtccb7‚ Y gNZM Hpv”r}< HgNlWc}]”JhHwO&cHqWe„}>cHe"Z }>WpvWf nlWfNpT

HaHdqW~"hWe"Z"€e„}ŒNcŠe"^7Ž x1 x1 x2 x4  2 1 3 x2 x3 x4 1 1 1 1 −1 1 −1 1 0 0 0 1 x5 1 1 1 2 b 9 4 1 5 x2 1  −2 −3 x3 x4 x5 b 1 1 1 9 −3 −1 −1 −14 0 −1 0 −8 −3 −2 −1 −22  x3 −2 3 6 6 x4 0 1  1 x5 0 1 2 2 b −5 14 20 20 x 3 x5 −2 0 −3 −1 6 2 0 0 b −5 −6 20 0 Y[Zje"hHwWe"d^Je„}>fNe"dzgNZ"e"f H^r}<H^` HdlWgžpve"hWlW^Jz Hg>oŽ  m^7‚ˆHZ x1 = −5 + 2x3 x2 = −6 + 3x3 + x5 x4 = 20 − 6x3 − 2x5 u ~"f‹}>~"cHe"cHe„}vHz ŒNtcšgNZMHaHz qH H^r}>lWZ"yWcdHc u HccHlWf5HZ mHf‹}>bJct^` HfNbJg¦pve"hWlW^Jz Hg‹} cHmŒNtc u x3 = 1 x 5 = 1 u u u u x 1 = −3 x2 = −2 x3 = 1 x4 = 12 x5 = 1 Hpv b}>e"fNpv~"gNZ"e„}>e"gNe"dœde"p a HZ"bJgNpve"hWlW^Jz Hg"‚ Y cHlWpvm}>bJabJ^Jbr}< Hg qbJZ"gNh H^`}>lH}U†~"g]HZ

e"^Je"pvb a HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyWg{}>e"i<€dbJc }I‰ ™ pT }>fNb xOf>HdhMŒ> HdHcOpve"hW€}< HfNlWZM Hg> H€lWZj~"g[pT }>fNb xŠbJdqWe"fNZ"~"de"cˆcbJgNZM Hpv”r}< Hg> H€lWZjbJg ^Je"€e„}[€HgNZ"d H^Jdb7‚]Y[Ze"^ WaabHZ"~"f‹} ^Je"€e„}>gN~"hWe"g"upve"f‹}[vpT }>fNb xŠf>HdhMŒ>vHZlWgNZ"^JlWms qWe"c}>lWfNfNe"dzgNZ"e"fN~"de"cˆf>HdhMŒ>qHHhHwbJgQHZ[lWgNZ"^JlWmqWe"c}>lWfNlWcTcHWZ"^“a HfNpve"^rwva HZ"bJgNa ae„qWlWd€}>yŠqWe"c}>lWfNlWc–pT xobJpT H^JbJg¦gNZM HpTo‚5Y ne"d}>b’e"hHwWe"d^Je„}>fNe"dzgNZ"e"fe"hHwo}N}>€}>y pT }>fNb x HdHc¦f>HdhMŒ>

u€bJgNZ"e"dmlWd}>lWg>Hde"ddwbr}}>tzo}>tdcjHZ lWgNZ"^JlWmqWe"c}>lWfNlWcUcHWZ"^ 5a HZ"bJgNa5ae„qbJddb7‚ nsBfNe"dz| A pT }>fNb xbJdqWe"fNZ"eme"zbJhvHZ Ax1 = e1 . Axn = en zHf>Hae"hHwWe"d^Je„}>fNe"dzgNZ"e"f e"hHwW~"f‹}>e"^Jpk|kpve"hWlW^Jz Hg> HayW^†€[HZžbJdqWe"fNZQ^J~„}>e"Z"bJco‰-cHms n €}>yŽ A−1 = [x1 , . , xn ] Y[Z   1 1 1 A= 0 1 1  1 0 1 pT }>fNb x bJdqWe"fNZ"~„}’HZ{H^` HaabWgNZM Hpv”r}< HgNlWccH^€}< HfNlWZ"€}ŒNtc pve"h0e"^Je"pvba HZ"bJg‹}>f>HdgNZIs nlWfNpT Hi"bJyWc&gNlWfNlWZM}< MqHH^7Ž a1 a1 a2 a3  0 1 a2 1 1 0 a3 1 1 1 e1 1 0 0 e2 0 1 0 e3 0 0 1 a2 1 a3 1 1 −1 0  e1 e2 1 0 0 1 −1 0 e3 0 0 1 a3 0 1  e1 e2 1 −1 0 1 −1 1 e3 0 0 1 e1 1 1 1

e2 −1 0 1 ↓ A−1 e3 0 −1 1 [Y Z AA−1 = I e"hHwWe"d^WgN~"hše"^J^Je"d WfNZ"~"gN~„qWe"^]hHw W Zz€e„}>dc‘pve"h HfNfNyW^7u’€lWhHw qHH^JyWaHdŠHZ A bJdqWe"fNZ"~„}[gNZM Hpv”r}>lH}N}>tcvcb7‚ "!()*+ (,-"! † F="KrCH@BR;2J;„ST6„GI@B6„@B:<; ‰ e"cbJd}>gNdcTe"hHw K 2AE6>FHGI2J;{PG>R>8HGNCHSTRM=F ;„2 M 6„KrC3CH@BR@ HZ td“‚ 4WCHERE2J4  ;jCHKrC 4WL<CHE Ž x≥0 †¡‰ Ax = b c0 x − max H€lW^’ne"^r}>e"gNgNZ"c u’€lWhHwHZ A m × nsBe"gpT }>fNb xšgNlWf>Hb ^JbJdeM HfNbJg>Hd nhWhWe„}>^Je"de"c ~"g u0qHH^`HpvbJd}T€lWhHw¢HZ Ax = b e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"c qHHd ^Je"hH^` Haa¤e"hHw b ≥ 0 ^Je"€e„}>gN~"hWe"gTa HZ"bJgNpve"hWlW^Jz Hg>o‚ Y[Z

H^r}<H^` HdlWgNg> Hh–cHlWfN^` }>lWZM Hg> d~"^Jc^]ne"^r}>e"€e„}ŒNc u €lWhHwœHZ x0 = [x0 , x0 ] de"pvde"h}>”rqa HZ"bJgNpve"hWlW^Jz Hg hHwmHf‹}>”Ji"bJlWd H^J€}>yu€lWhHw }<Hf‹}<H^JpTHZ"ZMvBva HNZ"bJg‹qH H^r}>lWZ"yWc}k†cHH}>H}N}[bJgNpve"fNe„}>^Je"de"co‰Iu x me"zbJhvde"pka s xB Z"bJg‹qH H^r}>lWZ"yWc}†gNZMHaHzbJgNpve"fNe„}>^Je"de"co‰I‚  dde"cŠpve"hHne"^Je"^ e"dŠmHNf‹}>”Ji"bJlWd H^ ŒNtcTHZ A pT }>fNb xolH}MuqHH^`HpvbJd}Q c qWe"c}>lWf‹}[bJg"Ž Y B pT }>fNb xlWgNZ"^JlWmHb}>e"€ }[HZ e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNa W ^7uHZ A = [B, N] , c0 = [c0B , c0N ] . A lWgNZ"^JlWmqWe"c}>lWf‹}>e"fN~"de"cŠe"hHwOa

HZ"bJg> }0Hz Œ> Hcg]”JhHw BxB + NxN = b xB + B−1 NxN = B−1 b e"cqbrqHH^Je"dg’e"hHwWe"d^Je„}>fNe"dzgNZ"e"f‹}žcHmŒNtc uHpve"^rwa W ^HZ dt^J^` HdHc&qH H^`HgNZ„}NqH&HZ xB = B−1 b xN = 0 xN † ‰ gNZMHaHz&qH H^r}>lWZ"yWc~"f‹}>~"cH~„} a HZ"bJgNpve"hWlW^Jz HgN€lWZ¦ŒNto}>tdc  ‚ e"^r}>~„}>e"^Je"Z"~"gNdc–gNZ"e"fNbJd} −1 ug”JhHw–e"€€e"Z a HZ"bJgNpve"hWlW^Jz HgN€lWZ}<Hf‹}>lWZ"yOi"~"^rnhWhHqW~"dwW~"f‹}>~"c Ž c0 B−1Bb ‚ b ≥B0−1b > 0 uHccHlWf gNZ"yWaHd nlWfNhWyOa HZ"bJgNpve"hWlW^Jz Hg‹} E6„Sv36986„E6„GNFHK @EBC 4 ue"^J^Je"dcHe"ZŠe"gNe„}>ae"d 36986„E6  GNFHK @EC 4 de„qWe"Z"Z"c ‚žY † ‰ e"hHwWe"d^

WgN~"hWe„}Qne"^J€HgNZ"d H^rqHHZ −1 −1 qWe"c}>lWf‹}[HZT† ¡ ‰ži"~"^rnhWhHqW~"dwW~"ae€e"^rwWe„}N}>e"gN”r}NqWeHZH^` Haabxe"Bcq=brqHHB^Je"dg0b^J−bJdeMB HfNbJgžNx mfNlNs hWf>HpvlWZM HgNbne"^`Hz}>lH}0cHmŒNtc Ž xB ≥ 0, xN ≥ 0 xB + B−1 NxN = B−1 b † ‰ c0B B−1 b + (c0N − c0B B−1 N)xN − max Ÿ bJd}>€lWhHw[ne"^r}>e„}N}>c u €lWhHw B−1b ≥ 0 u ~"gpvbrqWe"^ x ≥ 0 u c0 −c0 B−1N ≤ 0 e"hHwWe"d^H}>^Je"dgN~"hne"dd H^J^` Hg>Še"gNe„}>~"d$HZ x = B−1b u Nx = 0 aN HZ"bJgNpvB e"hWlW^Jz Hg>ˆ † ‰kne"^`Hz}>dHc uQg”JhHw šqWe"^Jee"cqbrqHH^Je"dBg–† ¡ ‰9sBde"c$bJgvNlWmo}>bJpT H^JbJg5pve"hWlW^Jz Hg>o‚—Y ne"^`Hz}[}>e"€ }HZHu-€lWhHwœcHe"fNe"gNgNdc HZ Ax = b

e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"c e"hHwœlW^rwHd a HZ"bJgNpve"hWlW^Jz Hg> }MužHpve"^rwfNeŠ −1 ~"gT 0 e"hHwWe"d^ H}‹s 0 −1 ^Je"dgN~"hWe"c †lWmo}>bJpTH^Jbr}< HgNbcfNbr}>~"fNbJBtpvlWbco‰¦≥e"hHw0gNZ"e"fNfNeTc}>Ne"^ ŒN−e"gNc^JBdBe"c ‚TNY[Zv≤H^` H0aab’g‹}>f>}>~Is hWb` }QcHqWe„}ŒNc Ž UbJbJdzt^Jtdcve"hHwlW^rwHdŠa HZ"bJgNpve"hWlW^Jz HgNayW^7uHpve"^rwfNej B−1b ≥ 0 }>e"^ ŒNe"gN^]~"gT€ c0 − c0 B−1N ≤ 0 de"p H^J^žne"dd“u]HccHlWfT }N}>~"fNdc‘e"hHw$lW^rwHd a HZ"bJgNpve"hWlW^Jz HgNf>oNuHpve"B^rwŠi"g>Hcˆe"hHwa HZ"bJg‹qWe"c}>lWfNaHd–c^JWdaWZ"bJcˆHZe"^ WZ H} W^7u}>ls qH

Haaf>Jgde"pvde"h}>”rq$ze €lWZ"ZM }<Hf‹}>lWZ"y‘i"~"^rnhWhHqW~"dwW~"f‹}>~"c¢dqWe"cgNZ"bJc †AqHHhHw ^Je"hH^` HaabJg{de"p i"gNWccHe"d ‰I‚  Z„}¦5a HZ"bJgNayW^cbJ^J~"m 5~"g¦ka HZ"bJgNa5ae"^J~"m kqWe"c}>lWfNlWc hWlWdzlWgpve"hHqH H^`HgNZ„}< Hg> MqHH^H~"fAŒNcUe"^7‚ e"hHwc ne"^7u €lWhHwj j sBe"zbJc[de"pka HZ"bJg‹qWe"c}>lWfNf> ‚  ccHlWf0cH~„}[e"gNe„}[^Je"€e„}>gN~"hWe"g"Ž (c0N − c0B B−1 N)ej > 0 ¡ ‚ B−1Ne ≤ 0 ‚ e"cbJd}>gNcšpvlWg‹}HZ x = λe u x = B−1b − λB−1Ne ^Je"€e„}>gN~"hWje"gžpve"hWlW^Jz Hg‹}MuH€lW^ λ > 0 ‚  N€€e"ZU jλ(c0 B− c0 B−1N)e i"~"^rnhWhjs

qW~"dwW~"f‹}>~"cŠ}<Hf‹}>lWZ"bJc uHpve"^rwTpvbJdze"dŠ€}< HfNlWdŠ} ^-Nd u€B λ − ∞j ‚  ZHZ„} ŒNe"^Je"d}>b7uo€lWhHwŠˆ† ‰žbJ^J^7‚ HZT† ¡ ‰{ne"^`Hz}>dHcTdbJdi"g]lWmo}>bJpT H^JbJgžpve"hWlW^Jz Hg>o‚ ‚ e0 B−1Ne > 0 qHH^`Hpve"^rw ksBf>o‚ k e"hHwWe"d Kj = {i | e0 B−1Ne > 0} ‚ 0wbJ^rqH Hd K 6= ∅ ‚ e"hHwWe"d i ∈ K e"hHw j lW^rwHdŠbJdzeIxu€lWhHw i e0i B−1 b e0k B−1 b = . min k∈K e0k B−1 Nej e0i B−1 Nej ¡ † ‰  Z"to}< Hd›qW~"hWfNe"€Œ‹}>tdc™e"hHw™a HZ"bJgNi"gNe"fN~„}¤†e"^Je"pvba HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyH}I‰IŽ HZ Bis bJc a HZ"bJg‹qH H^r}>lWZ"y–e"^J€HhHwMŒ>š a HZ"bJg‹}v~"gT€e"^rwW~"fNe j sBe"zbJc de"pka HZ"bJg‹qH

H^r}>lWZ"y cHeIs fN^7‚Ue"de"f> H^Jy e"^Je"p HZ e0 B−1Ne > 0 e"^Je"p ^Je"gNZH‚ Y †  ‰qH H^`HgNZ„}< HgNb]gNZMHa H^rw †N‡„gNZ"|cˆcHe"fNe"gNZ„}>pve„}>gNZ"e„} ‡kgNZ"ie"fNbJd}>bqH Hj ^`HgNZ„}< Hg<‰0abJZ„}>lWgN”r}Œ>ou €lWhHwHZ  Œ[a HZ"bJgNpve"hWlW^ s z Hg bJg’^Je"€e„}>gN~"hWe"g{^Je"gNZHu (c0 − c0 B−1N)e > 0 e"hHwWe"d^ H}>^Je"dgN~"hme"zbJhHZ„}Mu€lWhHw HZ  Œa HZ"bJgN€lWZ5}<Hf‹}>lWZ"yˆi"~"^rNnhWhHqWB~"dwW~"f‹}>~"cšjde"p ^Je"gNZvfNlWgNgNZMHaa c0 B−1bsBd~"^7ug H} €}< HfNlWZ"lH}N}<HdkŒNlWaa^Je"gNZHu€ e0 B−1b > 0 ‚  Zto}>yWaabe"gNe„} pvbJdzBbJh5ne"dd H^J^7u€T a HZ"bJgN€lWZU}<Hf‹}>lWZ"y&a

HZ"bJgNpve"hWilW^Jz HgQde"pvze"hWe"de"f> H^r}M‚ B Y ne"d}>bj^J~"m~"gNe"cHe„}–e"Z"to}< Hd pve"hWbJgNpv~„}>e"^ ŒNc HZ  ŒŠa HZ"bJgNpve"hWlW^Jz HgŠe"gNe„}>~"ae"d“Ž e"^J^Je"d WfNbJZ"Z"c$HZlWmo}>bJpTH^Jbr}< Hg‹}5bJ^J^7‚ ne"^`Hz}Tpve"hWlW^Jz€}>yWg> Hh }MužpTŒNz¤e"hHw a s Z"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyH}qW~"hWZ"dce"^Qne"d}>bžgNZMHa H^rwWlWcšgNZ"e"fNbJd}M‚ Ÿ brqWe"^Q^Je"€e„}>gN~"hWe"g a HZ"bJgNlWcš†~"g[”JhHwŠva HZ"bJgNpve"hWlW^Jz HgNlWco‰žgNZM HpT&qW~"hWe"g[~"g[ze"hWe"de"f> Hi"bJyTpve"d}>e"g[e"gNe„}‹s ae"d–vi"~"^rnhWhHqW~"dw~"f‹}>~"cHekd

uze"hWe"de"f> Hi"bJyTpve"d}>e"g e"gNe„}>ae"dœqW~"hWe"g gNZM Hp e"^Je"pvb a HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJy—†gNZ"bJpvm^JeIx¢br}>e"f> Hi"bJy‰&e"^rqW~"hWZ"~"gNešto}< Hd—e"^ cHe"^J^ ŒNto}>dtdc¢e"hHw lWmo}>bJpT H^JbJga HZ"bJgNpve"hWlW^Jz HgN€lWZQqHHhHw&HddHckpve"h H^J^`Hm”r}< Hg> H€lWZHuH€lWhHw& ne"^`Hz}>dHc dbJdi"g0lWmo}>bJpT H^JbJg{pve"hWlW^Jz Hg>o‚ Ÿ e"hMŒNe"hHwWe"Z"Z"c u{€lWhHw œa HZ"bJgNaae"^J~"m œbJ^J^7‚ˆcbJ^J~"m ˆqWe"c}>lWfNlWc pve"hHqH H^`HgNZ„}< s g> HdHc£dlWpv”r}< Hg> MqHH^U†pve"hHne"^Je"^ œgNZMHa H^rwWlWcne"^` H^J^J”r}< Hg> MqHH^{HfNf> HZŠe"gNe„}>fNeHuž€š qH H^`HgNZ„}<

Hg{de"p e"hHwW~"f‹}>e"^Jpk| ‰{jqW~"hWe"gNgN~"h&HZ[ H^r}<H^` HdlWg’e"gNe„}>ae"d“ujde"pvze"hWe"de"f> H^r}‹s g> Hh5ne"^r}>~„}>e"^Je"Z"~"gNed~"^Jc^“bJg]abJZ"lWdw”r}>€}>y‚  zzbJhvi"g>HcˆHZ„}[HZe"gNe„}>e„} d~"Z„}>c uHpvbJcHlWf]bJgNpve"f‹}>dce"hHwŠbJdzt^Jy&^Je"€e„}>gN~"hWe"g a HZ"bJgNpve"hWlW^Jz Hg‹}M‚ [HhHwWlWdcHWddw|–e"hHwˆbJdzt^Jyvpve"hWlW^Jz Hg‹}¦Hzdb7u €5ne"^`Hz} HZ H^` Haab“H^`Hc “Ž x ≥ 0, u ≥ 0 Ax + u = b † ‰ c0 x − max ~"g b ≥ 0 ‚  Z„}&Šne"^`Hz}>lH} ERGISvFHK M6„KrC3CH@EC 4 €”rqMŒNtc u*~"gkHZ x = 0 u u = b a HZ"bJgNpve"hWlW^Jz Hgj†HZ u &a HZ"bJg‹qH H^r}>lWZ"yWc uHZ x &de"pka

HZ"bJg]qH H^r}>lWZ"yWcvqWe"c}>lWf>‰]e"hHw H^JcH^JpTHg{bJdzt^Jy5pve"hWlW^Jz Hg"‚  vkne"^`Hz}Q}>Wf‹}>~"de„}>e"gNe"dde"p bJ^rwWe"dˆH^`Hc “uHccHlWf qbJgNgNZMMqWe"Z"e„}>€e„}ŒNcŠHZ5†  ‰’nlWfNpT Hf>m^7‚’HZ td“‚¦‡„cH~„}Nn HZ"bJg OgNZ"bJpvm^JeIxvpvyzgNZ"e"fNfNe"^B‡‚ e"hHwcne"^7u€lWhHwŠ5pve"hWlW^JzHdzykne"^`Hz}05cHqWe„}>cHe"Z Ž x ≥ 0, u1 ≥ 0 A1 x + u 1 = b 1 A2 x = b 2 ¡W¡ † ‰  c0 x − max Y[Z H^r}<H^` HdlWgNg> Hhjpve"hWgN~"f‹}>~"gNe[d~"^Jc^ ne"^r}>e"€e„}ŒNc uo€lWhHw ~"g ‚¦† Ÿ bJds ze"d^JbJdeM HfNbJg mfNlWhWf>HpvlWZM HgNb ne"^`Hz}jbJ^rwWe"dH^`Hcf>O€lWZ"€b}>1y≥‚TY[0 Z axb2≤≥ b0u b ≥ 0 H^`Hc 

ne"^r}>~„}>e"^Je"c HZ cbJe"hW~"gNZ"”r}ˆqH H^r}>lWZ"yˆgNe"hW”r}>gN~"hW~„qWe"^ + u = b H^`Hcf> €lWZ"€}>yWc~"g ”JhHwkHZ]e"u^Jg U≥ne"0^r}>~„}>e"^i"gNlWmlWf‹}>aHd5qHHddHc ‚ Y[Z ax ax e"hHwWe"d^ WgN~"hWe"c& = b pT HgNlzbJcaHd“upvbrqWe"^br}N}0 b ≥ 0 gNZ"cgN~"hve"gNe„}>~"d −1sBhHwWe"^“qHH^Jykae"gNZ"lWfNZM HgNg>H^-pvbJds zbJhabJZ„}>lWgN”r}>€}>y‚jY[Z ≥ b u b ≥ 0 ne"^r}>~„}>e"^Je"cˆme"zbJhŠ v ≥ 0 ne"^Je"gN^Je"hqH H^r}>lWZ"y gNe"hW”r}>gN~"hW~„qWe"^ ax − v =ax H^`Hcf>€lWZ"€}>yWcv~"gQ”JhHwpT HgNlzbJc5ne"^r}>~„}>e"^i"gNlWmlWf‹}>aHd b qHHddHc ‚ ‰ Y † ‰]ne"^`Hz}>lH} STƒM3R;„?A@BR@@QERGISvFHK

M6„KrC3CH@EC 4 €”rqMŒNtc ‚ 0e"dze"^ ŒNcˆ€lWZ"ZM  † ‰{ne"^`H z}>€lWZj5cHqWe„}>cHe"Z  ‡„gNe"hW~"zone"^`Hz}>lH} ‡uHpve"^rwe"hHwOdlWfNpT H^ne"^`Hz}MŽ  x ≥ 0, u1 ≥ 0, u2 ≥ 0 A1 x + u 1 = b1 A2 x + u2 = b2 10 A2 x − max † ‰ Y †  ‰Qne"^`Hz}¦i"~"^rnhWhHqW~"dwWekcHlWfN^` }>lWg"u €bJgNZ"e"d 10b vi"~"^rnhWhHqW~"dwˆe"hHwŠne"^Jg TcHlWfs `^ }Œ>o‚BhHwš†  ‰9sBde"cvqHHdŠlWmo}>bJpT H^JbJg{pve"hWlW^Jz Hg>o‚ e"2hHwWe"d † x0, u0 , u0 ‰Qˆ†  ‰{ne"^`Hz} e"hHwOlWmo}>bJpT H^JbJg{a HZ"bJgNpve"hWlW^Jz Hg>Š†bJ^rwWe"dŠpvbJdzbJhk^J~„}>e"Z"bJc ‰I‚  ccH1lWf¦cH2~„}¦e"gNe„}¦qHHd“Ž ¡ ‚ 10A x0 = 10b uHpvbJa W^ A x0 = b u u0 = 0 cHqWe„}>cHe"Z"bJc u g]”JhHwœ†

x0, u0 ‰žHZ 1 e"fNe"z2e„}>b{† ‰{ne"^`2Hz}0e"hHwO^Je"€2e„}>gN~"hWe"g¦2a HZ"2bJgNpve"hWlW^Jz Hg>o‚  ‚ 10A x0 < 10 b u[†AqHHhHw Hpvb{thHwHdHZHu u0 sBdHc qHHd‘^Je"hH^` Haae"hHw mlWZ"br}>”rq cHlWpv2mlWde"dgNeHu{H2pvb{gNZ"cgN~"hWcH~"mme"d¢a HZ"bJg‹qH2 H^r}>lWZ"y‰I‚  aae"d¤HZe"gNe„}>ae"d¤HZ e"fNe"ze„}>b{† ‰{ne"^`Hz}>dHcTdbJdi"g0^Je"€e„}>gN~"hWe"g¦pve"hWlW^Jz Hg>o‚   ~"^JgNZ"e"fN|š–†  ‰]ne"^`Hz}>lH}[bJgUTgNZ"bJpvm^JeIxpvyzgNZ"e"fNfNe"^*pve"hWlW^JzHdb7‚  Z„}UgNZ"lWc Hg ‡„e"^Jg On HZ"bJgNdHc‡5de„qWe"Z"dbž~"ge"ZvcbJZM HfNyW^`HhŠ† ‰[ne"^`Hz}je"hHw ^Je"€e„}>gN~"hWeva HZ"bJgNpveIs hWlW^Jz Hg>

HdHc‘pve"hW€}< HfNlWZM Hg> Hf>šgNZ"lW^Jh H^7‚—Y[Zœ e"^Jg n HZ"bJgNaHd¤de"p bJgv}>Wf zdc i"~"^rnhWhHqW~"ddwWe"^7udlW€&e"dde"cŠqH H^r}>lWZM Hg> }0i"~"^JgNZ"e"fN|œdwWlWpvlWdˆcHqWe„}>db HZe"hHwWe"g cbr}>0e"xf> Hi"bJyWcgNlWf> Hd“‚  \HZˆe"^Jg  n HZ"bJgvHZ"ZMH^]qW~"hWZ zbJc u]€lWhHw$gNbJcHe"fN^r}&}<H^` H^JdbQe"hHw ^Je"€e„}>gN~"hWe"g¦a HZ"bJgNpve"hWlW^Jz Hg‹}† ¡ ‚’e"gNe„}I‰Iu HccHlWf0bJdde"dˆcHe"Z"zoqWekHZ d“‚[‡„pT HgNlzbJc&n s Z"bJgNaHd‡{e"aa W^ a HZ"bJgNpve"hWlW^Jz HgNayW^cbJbJdzt^rqHnlW^rw}<}ŒNtcHZQlWmo}>bJpTH^JbJZM H^` Hg‹}-pvlWg‹} pT Hf HZ 10 A x pT HgNlz^`HhWlWg

i"~"^rnhWhHqW~"dw}{£hHwWe"^Jpve"d&c”rq^€HhHwqHj~"g qbJgNgNZM}>~"f‹qWe 2 ¡ HZ0e"fNe"ze„}>b 0 i"~"^rnhWhHqW~"dw&pT xobJpTH^JbJZM H^` Hg> Hf>o‚ e"fNpv~"gNZ"e„}>e"gNe"dŠHfNf> qbJhHw HZ"dtdc cHe"^J^7u€lWhHwHcZ{xe"^J~"f‹}Q‡„^Je"€e„}>gN~"hWe"gNgN~"hWe„} ‡{qW~"hWbJh¦pve"h WfNbJZ"Z"c ‚  dde"cje"hHwgNZ"e"fN|5pvyz Œ> HZHu€HZ u qH H^r}>lWZ"yWc}[dt^J^`&~"f‹}>~"cHe"d fNWhWZ"”r}ŒNc u“HpvbTgNZ"bJpvm^JeIxˆpvyzgNZ"e"fNae"d }>e"i<€dbJcHbJ^`Hh 2 hHwOqHH^JyWgN”r}>€}>ykpve"hu€lWhHwŠHZ u qH H^r}>lWZ"yWbr}QqW~"hWbJhv&a HZ"bJgNlWdŠc”rq^ }<Hf‹}ŒNtc ‚O† Ÿ e"hMŒNe"^JW^ ŒNc

WcHe„}Mum~"^Jz Ht^*e"hH w MsBhH2^~"gj”JhHwpv~"hˆHccHlWfUgNe"p qbJgNgNZ"c WcHe„}5Ša HZ"bJgNaou*€ˆmlWZ"br}>”rq–i"~"^rnhWhHqW~"dwše"hHwo}N}>€}>y ŒNtc–qHHd“‚  ZHZvlWmo}>bJpTH^Jb s }< HgUcfNbr}>~"fNbJtpvlH}UbJgUpve"hHqH H^r}>lWZ„}<}Œ>oŽ[pvbJdze"d–de"p i"gNbJ^J^`HhWlWg[qH H^r}>lWZ"yW€lWZ}<Hf‹}>lWZ"y i"~"^rnhWhHqW~"dwŠe"hHwo}N}>€}>yWdHcde"p mlWZ"br}>”rqdHcTcHe"^J^“^Je"ddbJeH‚ ‰ Y gNZ"bJpvm^JeIxˆpvyzgNZ"e"f[pvbJdzcH~„}[n HZ"bJg> Hd H^TgNZM Hpv”r}< HgNlWcO^Je"alWdwWlW^J”r}< Hg> }¦pve"hs cHWddw”r}>b7u€e"hHwWe"d^Je„}QfNe"dzgNZ"e"fNe"cTpve"hWlW^Jz Hg> Hd H^€HgNZ"d H^r}ž}< Ha^` HZM}>lWg’nlWfNpT HaHd

fNe"dze"Z"Z"cve"^HZ¦e"hHwo}N}>€}>ykpT }>fNb xolH}Mu‡9ŒNlWaaTlW^JzH^r} ‡j† b ~"g b qWe"c}>lWfNlWc}I‰ ~"g Ti"~"^rnhWhHqW~"dwe"hHwo}N}>€}>yTqWe"c}>lWf>Hbr}5† c0 ~"gUHZ 10 A qWe"1c}>lWfNlWc2}M‚  e„qWe"Z"e„}‹s ŒNc5pv~"h z ~"g t qH H^r}>lWZ"yWc}MuHpve"^rwWe"c&HZ0e"^Jg z^Je"hWe"g ~"gžpT2 HgNlz^`HhWlWg i"~"^rnhWhHqW~"dw ~"f‹}>~"cH~„}UnlWhMŒ> HcŒNe"^JW^Jdb7‚ BhHwœ †  ‰]ne"^`Hz}¦ne"^r}>~„}>e"^Jbe"hHwWe"d^Je„}>e"bJ€e"Z5i"g>}>lW^ ŒNtcŠpv~"h ~"g¦ −z + c0 x = 0 −t + 10 Ax = 0 e"hHwWe"d^Je„}>e"cHe„}M‚ Y −z ~"g −t qH H^r}>lWZ"yWc} †bJ^J^Je„}NqWeœ €lWZ"ZM

ŒNtc¤}<Hf‹}>lWZ"y e ~"g lWgNZ"^JlWmqWe"c}>lWfNlWc}I‰ pvbJdzbJhjUa HZ"bJgNaHd5nlWhMŒNtc}<Hf‹}<Hdb~"g ”JhHwkcHWddw|Tm+1 ^` }>db7u em+2 €lWhHwm~"^Jz Ht^HZže"^Jg z^Je"hWe"gi"~"^rnhWhHqW~"dwgNlWf> HaHd5 0 0 B−1N qWe"c}>lWf ŒNe"^Je"dbJc pve"h‚ e"€ }*a HZ"bJgNaae"^J~"m ]qH H^r}>lWZ"yWdHc HZ e"^Jg 0n HZ"bJcgNaNH−c dHBZ“ŒNWdjgNZ"yWaouWHpve"^rw€e"Z mlWZ"br}>”rqœcHlWpvmlWde"dgU}<Hf‹}>lWZ"bJc– −t gNlWf> HaHd“‚vY pT HgNlzbJcŠn HZ"bJgNaHd¢†HpvbJcHlWf −t gNlWf> } pT Hf¦e"^bJg[^Je"€e„} €HhHwdbA‰0HZ5&qH H^r}>lWZ"yT^J~"m€e„} ae5Ta HZ"bJgNaouHpve"^rwbJcOde"p i"gNbJ^J^`HhWlWgž~"g −z gNlWf>

HaHdˆmlWZ"br}>”rqTcHllWfNzbJd }< Œ>qHHd“‚ Y˜}< Ha^` HZM}¦e"^Jg TlWgNZ"^JlWm HavTa HZ"bJg‹qH H^r}>lWZ"yWc}Mue"^Jg TgNlWf> HaTva HZ"bJgNlWdc”rq^Jb qH H^r}>lWZ"yWc}0”JfAŒNtc u5}< Ha^` HZM}¦e"^Je"pve"b-me"zbJhTHZHzlH}N}[a HZ"bJgNf>5qWlWd}>cHlWZ"ycHllWfNzb s d }< Hc}[pkto}<}Œ> Hc ‚]Y pvyzlWgN”r}>lH}N}0dlWfNpT H^“ne"^`Hz}[pve"hWlW^Jz Hg> }]}>e"€ }UHZkH^` Haab }< Ha^` HZM}>ayW^bJdz”r}ŒNtc Ž u1 ∗u2 −z −t x0 A1 b 1 A1 b 2 c0 0 0 1 A2 0 Y[Z u qH H^r}>lWZ"yWc}0‡„pve"hWi"gNbJ^J^`HhWlWZ„}>tc‡ue"Z"Z"e"^ ŒNe"^Je"Z„qWe0HZ„}Mu€lWhHw€ e"Z"e"cke"^J€HhHwMŒ> Hc a HZ"2bJg‹}MuHccHlWfžlzjpT Hf gNe"pvpvb

cHWfN^Jpv~"dwWe"c&cHWZ"H}N}Qde"p }>~"fN€e„}>de"c5qbJgNgNZMouoe"^ s ^Je"d}>~„}>ae"dv }>WaaboqH H^r}>lWZ"yqHH^7uWHpve"^rwWe"ckHc Hf*}>WaagNZ"Wf bJg pve"€e„}>de"ckae0Ua HZ"bJgNa ¡ ~"gž€HhHw€}Œ> Hc5bJgžHZ„}že"^7‚  HZ¦e"^Jg n HZ"bJg qW~"hW~"dš†jpT HgNlz^`HhWlWg i"~"^rnhWhHqW~"dwvlWms }>bJpktpT Hd H^A‰ qHH^`Hpve"ddwbi"gNbJ^J^`HhWlWgQqH H^r}>lWZ"yTe"^J€HhHw}<Tva HZ"bJg‹}MuHccHlWfUHZe"fNe"ze„}>b ne"^`Hz}¦e"hHwˆ^Je"€e„}>gN~"hWe"g a HZ"bJgNpve"hWlW^Jz Hg> Hd H^“qHHhHwtdc u“Hpve"^rwa W^-cbJbJdzt^rqH5pvlWg‹} pT

HfHZve"fNe"ze„}>bži"~"^rnhWhHqW~"dwšgNZ"e"fNbJd}nlW^rw}<}ŒNtc–HZTlWmo}>bJpTH^JbJZM H^` Hg‹}M‚  HZve"^Jg  n HZ"bJg0lWmo}>bJpT H^JbJgžpve"hWlW^Jz Hg> HaHdŠqHH^`Hpve"^rwTi"gNbJ^J^`HhWlWgQqH H^r}>lWZ"y&mlWZ"br}>”rqu HccHlWf[&ne"^ s Hz}>dHcTdbJdi"g0^Je"€e„}>gN~"hWe"g¦pve"hWlW^Jz Hg>o‚ B^J^JtgNZ„}>f> Hi"bJyWdHcvd~"Z"Z"cˆ5cHqWe„}>cHe"Z m~"^Jz }MŽ x1 , x 2 , x 3 , x 4 , x 5 ≥ 0 3x1 + x2 + x4 + x5 = 13 x1 + x3 − x4 + x5 ≤ 7 2x1 + x2 + x3 + x4 − x5 = 17 6x1 + 3x2 + 2x3 + x4 − 4x5 − max Y[ZjbJdzt^Jyk}< Ha^` HZM}05cHqWe„}>cHe"ZŽ  x1 x2 x3 x4 x5 ¡ 1 13 ∗u1 3 1 0 u2 1 0 1 −1 1 7 ∗u3 2 1 1 1 −1 17 −z 6 3 2 1 −4 0 −t 5 2 1 2 0 0 €€e"Zk}< Ha^` HZM}>€lWZjHZjH^` Haab“a

HZ"bJgNpve"hWlW^Jz Hgž}<Hf‹}>lWZ"bJc Ž x1 = x2 = x3 = x4 = x5 = 0 , u1 = 13, u2 = 7 , u3 = 17, z = 0, t = 0 U H^`HgNgNZ"tcOae"pve"d &qH H^r}>lWZ"yWdHcOHZ x Bs e„}M‚žY ‡„gNZ"|cŠcHe"fNe"gNZ„}>pve„}>gNZ"e„} ‡gNZM Hpv”r}< Hg> s €lWZ HZUH^` Haab € HdwHzlWgNlWc}]cH~"me"Z"Z"c Ž ~"g uHpve"^rwWe"cvcHWZ"^“ ¡ cbJgNe"aa“u }>e"€ } u ^Je"gNZvOcbJpve"d qH H^r}>lWZ"y‚Š† { H^Jy Œ> HaHd HZ (1, −1, 1, 1, 2) qWe"c}>lWfj^J~"m aeHu 4 ~"g¦HZ 13 1 1 (1, 0, 0, 0, 0) 17 1 ^J~"mŠcb7‚ ‰kY›hWe"de"f> H^Jy&e"^Je"pve„}0ae"cHe"fNe„}>e"Z„}>c ‚žY[Z  Œ]}< Ha^` HZM}MŽ x1 x2 x3 ∗u1 x5 x4 3 1 0 1 1 13 u2 4 1 1 1 2 20 ¡ −1 ∗u3 −1 0 −2 4 −z 3 2 2 −1 −5 −13 −t −1 0 1 −2 −2 −26

¡ Y›€lWZ"ZM 5}<Hf‹}>lWZ"y&a HZ"bJgNpve"hWlW^Jz Hg"Ž x1 = x2 = x3 = x5 = 0, x4 = 13, u1 = 0, u2 = 20, u3 = 4, z = 13, t = 26 lWg‹}{e"hHwW~"f‹}>e"^Jpk|“u€lWhHw&HZ x cHe"fN^ja HZ"bJgNaj~"g u e"^J€HhHwMŒ>jHZ„} †  ¡ € Hdw s 3 3 Ÿ  zlWgQcbJgNe"aa“upvbJd}] H ¡ ‰I‚  Z"Z"e"^-pvbJdzcH~„}]i"gNbJ^J^`HhWlWg qH H^r}>lWZ"ycbJcHe"fN^r}0ka HZ"bJgNayW^ ~"gQ”JhHwTcHqWe„}>cHe"Z &}< Ha^` HZM}QHZ e"fNe"ze„}>bne"^`Hz}že"hHwT^Je"€e„}>gN~"hWe"gQa HZ"bJgNpve"hWlW^Jz Hg> } Hz Œ>o‚’Y›pT HgNlz^`HhWlWgQi"~"^rnhWhHqW~"dwŠgNlWf> }0pvlWg‹}0pT HfQcbgNe"p ”JfAŒNtc ‚ x1 x2 ∗u3 ∗u1 x5 ¡ 0 1 1 13 x4 3 u2 5 1 −1 2 16 4 x3 −1 0 1 −1 −2 4 −z 5 2 −2 1 −1 −21 €€e"Zk}< Ha^`

HZM}>€lWZ }<Hf‹}>lWZ"y5a HZ"bJgNpve"hWlW^Jz Hg  x1 = x2 = x5 = 0, x3 = 4, x4 = 13, u1 = u3 = 0, u2 = 16 J^ e"€e„}>gN~"hWe"g"u zeOde"p lWmo}>bJpT H^JbJg"u€bJgNZ"e"d HZ x ~"g x sB€WZT}<Hf‹}>lWZ"y–to}>lW^JgNyœgNlWfNae"^Jb e"^Je"pve"c mlWZ"br}>”rqHHc u ”JhHw‘e"Z"e"c$a HfNpve"^rwbJcHeOae"1^J~"m€e„}2 a HZ"bJgNao‚›†  e"hHwc‘~"gNZ"fNeHu €lWhHw¢HZ u H^`}N}^J~„q ‘to}>lW^JgNy‘gNlWfNae"^Jb¦e"^Je"p bJgmlWZ"br}>”rqu]ze–e"dde"c¢e"^J^Je"d~"fNe u 1 de"p ^J~"m€e„}k aeˆa HZ"bJgNaou*pvbrqWe"^ i"gNbJ^J^`HhWlWg qH H^r}>lWZ"y‚ŠY[Z x s9}qH H^`HgNZ„}NqHŠae"^J~"m  1 qH H^r}>lWZ"yWdHcŠHZ x nlWhMŒ>TTa HZ"bJg[e"^J€HhHwdbQ† ¡  ¡  ¡

¡ ‰]~"2g TcHqWe„}>cHe"Z upvlWg‹}  4 pT HfQlWmo}>bJpT H^JbJg }< Ha^` }0cHmŒNtc Ž  x1 x4 ∗u3 ∗u1 x5 x2 3 1 0 1 1 13 u2 2 −1 −1 1 3 3 x3 −1 0 1 −1 −2 4 −z −1 −2 −2 −1 −3 −47 aa W ^k}< Ha^` HZM}>ayW^^Je }>tz ŒNtclW^rqHHgNdb-HZjlWmo}>bJpT H^JbJg{a HZ"bJgNpve"hWlW^Jz Hg‹}MŽ x1 = 0, x2 = 13, x3 = 4, x4 = 0, x5 = 0, u1 = 0, u2 = 3, u3 = 0, z = 47 . Y Ng Z"bJpvm^JeIxœpvyzgNZ"e"f&†e"hHwbJco‰¦e"^WdwWeHu-€lWhHw–HZkto}>lW^JgNyulWmo}>bJpT H^JbJg]}< Ha^` HZM}>ayW^* zt H^“ne"^`Hz}0e"hHwŠlWmo}>bJpT H^JbJgžpve"hWlW^Jz Hg> }]bJg0cb-}>tz ŒNtclW^rqHHgNdb7‚ Ÿ lWg‹} &a HZ"bJgNaHd ^J~„q  us9qH H^r}>lWZ"yWc€lWZ’}<Hf‹}>lWZ"y[zt H^qH H^r}>lWZ"yWcU~"f‹}>~"cHeQ^Je"gNZ u}>e"€ } u = 0 uW¦a HZ"bJgNlWd c”rq^Jb u qH

H^r}>lWZ"yWc[~"f‹}>~"cH~„} HZ’to}>lW^JgNy0gNlWfNaHdde"h}>”rq[e"^  ŒNe"^J^Je"^}<H^` H^ 2ŒNtc¦pve"h‚ e"€ } ‚ u1 =¦W1,dduwW3e"d› = e"2^J^Je"d WfNbJZ"€e„}ŒNc u[€lWhHw HZš”JhHw cHmlH}N}ŠmfNbJpT H^ sBzt H^[pve"hWlW^Jz HgNm Hf cbJe"^J~"hW”r}>b“5ztH^Jbr}< HgNb}>~„}>e"^Je"cHe„}M‚ ¡ VWC 4RGIKJƒ M6„KrC3CH@BR4 ¡ ‚  }< HfNlWZ"Z"tcœpve"hŠOcHqWe„}>cHe"Z–e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNe"cš H^r}<H^` HdlWg[pve"hWlW^Jz Hg> } e"^Je"pvba HZ"bJg‹}>f>HdgNZ„nlWfNpT Hi"bJyqHH^7Ž ‰ a‰ x−y+z−u = 1 x−y−z+u = 0 −2x + 2y + 4z − 4u = 1 x +z+u = 2 x+y + u = −3 y + z + u = −1 ‚   }< HfNlWZ"Z"tc pve"h¤HZ ¡ ‚

ne"^`Hz}>aHd—gNZ"e"fNe"m^¤e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNe"c qHH^` s pve"ddwb-a HZ"bJgNpve"hWlW^Jz Hg> }  ‚0Y[ZŠH^` Haabže"hHwWe"d^Je„}>fNe"dzgNZ"e"fTgNZMHaHzg> HhHnlWc–cHe„}N}‚  }< HfNlWZ"Z"tcpve"hšHZ e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNae"d gNZ"e"fNe"m^  ~"g mHf>Hpv~„}>e"fNe"cŠ~"f‹}>~"cH~„}MupTŒNz”JfAŒNtc ne"^-HZje"hHwWe"d^Je„}>fNe"dzgNZ"e"f H^r}<H^` Hda,lWgžb pve"hWclW^Jz Hg> }M‚ x1 + x2 + 2x1 + x2 −x1 + ax2 x3 + x4 x3 + 2x4 + 3x5 + 2x3 − 2x5 − 2x3 + x4 + bx5 = = = = 2 1 2 c  ‚  Jb Z"lWdw”r}>gNtcUaeHuH€lWhHwj€¦e"hHwj^JbJdeM HfNbJg-e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"cqHHdcH~„} c^JWds 

aWZmlWZ"br}>”rq&pve"hWlW^Jz Hg>ouHccHlWfžqHHdOHZ[e"hHwWe"d^Je„}>fNe"dzgNZ"e"fNde"cde"p mlWZ"br}>”rq pve"hWlW^Jz Hg>kbJg"‚ ‚  }< HfNlWZ"tcTpve"hv5cHqWe„}>cHe"Z OpT }>fNb xolWcvbJdqWe"fNZ"~„}[e"^Je"pvba HZ"bJg‹}>f>HdgNZ„nlWfNpT s i"bJyqHH^7Ž ‰   1 1 2  0 1 1  1 1 3 a‰   −2 5 1  3 −1 4  2 0 7 i‰   1 1 1 1  1 1 −1 −1     1 −1 1 −1  1 −1 −1 1 ‚ ^Jz ŒNtcpve"hT&gNZ"bJpvm^JeIxŠpvyzgNZ"e"fNfNe"^HZH^` Haab-^JbJdeM HfNbJgQmfNlWhWf>HpvlWZM HgNb“ne"^ s  Hz}>lWc}MŽ a) x 1 , x2 , x3 , x4 , x5 ≥ 0 x1 + 2x2 + 3x3 + x5 ≤ 100 x2 + x3 + x4 + x5 ≤ 80 x1 + x3 + x4 ≤ 50 2x1 + x2 + 3x3 + x4 + 2x5 − max ¡  b) x 1 , x2 , x3 , x4 ≥ 0 x1 + x 2 + x 3 ≤ 8 2x1 − x2 +

x4 = 12 7x1 + 3x2 + 4x3 + 6x4 − max c) x 1 , x2 , x3 ≥ 0 x1 + x2 + 2x3 ≤ 5 2x1 + x2 + x3 ≥ 2  ‚ BfAŒNtc‘ne"^¦ x1 + x2 + 3x3 − max ‚ ne"^`Hz}>aHd¤gNZ"e"fNe"m^  sBc$zt H^JbJg>Hbr}&~"gT€}< HfNlWZ"Z"tc$pve"h HZŠlWmo}>bJpT H^JbJggNZ"bJpvm^JeIx }< Ha^` HZM}>lWcayW^]–zt H^JbJgkne"^`Hz}>lWce"hHw‘lWmo}>bJpT H^JbJg a HZ"bJgNpve"hWlW^Jz Hg> }M‚ ‚  bJZ"lWdw”r}>gNtcaeHu{€lWhHw‘€–e"hHw‘^JbJdeM HfNbJgmfNlWhWf>HpvlWZM HgNb{ne"^`Hz}5^Je"€e„}>gN~"hWe"g }<Hf‹}>lWpT Hdwjde"pkfNe"gž~"g{cHlWfN^` }>lWg"uHccHlWfQjzt H^JbJg> HdHc5^Je"€e„}>gN~"hWe"g{}<Hf‹}>ls pT Hdw5de"p cHlWfN^` }>lWg"‚ ¡