/9 F G 780 Y 45 2 3 BC A / 1 0 ?@ / > . /0 - ; j | z{ gh y f M Basics of Linear and Discrete Optimization: Examples – p.2/30 ¢¡ w w c a bc r s d NO sx uwv t e PQ ¸± £ ´ ÏÓ ¯ ¶ · ¶¼ À± µ¶ ³ ´ ² ¶ °± ¾ ¶¿ ½ ® ¯ ± Ë ÌÍ ÉÊ Ù é â ý ü ö ø Þwá Û Þ Ú í êwï î ü ùwþ ý Øà ì í û ü Û ß ç ë ö ú Ò ÌÑ ÏwÐ Î õ ø î í ç â Ï ° ¹ Ú Í Ì ÏwÔ Ò ¯ ¸ Ú ÕÖ ° ¹ ¼ öÿ æ é ÎÖ Á ± ² ¶¼ º» bi RS ¤¥ ñ çð ¸Â ¸¯ | ² ³ u} UV T ;= < s| kl § ¦ } ~ m W n c j S RX D d W @ ?E 0 [ o D p Z ; / 46 gq \ µ± ¹» Ù è Þ å ê Ý ÷ ô ù é ø Basics of Linear and Discrete Optimization: Examples – p.2/30 ç è æ ö ÷ õ × Ø ã ä ò ó å ô Û Ü Ú Ù MIP Oswald/Reinelt ^ ¸ ] H ≥ 0 J µ x1 , x2 I : _ à ¶ ¨ ≥ 3 bi K 2x1 + x2 ` ! j L  ³ w kl "# ° ¹ © ≥ 5 $ m % &' d n c j x2 À± ¾ ª () ² x1 , x2 ≥ 0 * ± µ¶ x1 +, w MIP Oswald/Reinelt ¶ ³ ´ 2x1 + 2x2 ± Ä ≥ 4 Å x1 + 4x2 ¶¼ « min 3x1 + 5x2 ¬ 2x1 + x2 ≥ 3 ÀÇÆ ¬ 3x1 + 5x2 È 2x1 + 2x2 ≥ 5 Å x1 + 4x2 ≥ 4 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.1/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.1/30 A $% # R3S > ?@ PQ O M N _` ] ^ n3o l m Z [\ j k {}| yz w x ¶° ¡ ª Àà© µ° ° Ë Basics of Linear and Discrete Optimization: Examples – p.4/30 ÓÕÔ ÉË ÀÅÆ ¿ Ä «± Ö Á Ç ± × ÅÆ í Ì ÄË Basics of Linear and Discrete Optimization: Examples – p.4/30 ≥ 0 ≥ 5000 ≥ 10 ≥ 200 ≥ 90 ≥ 65 æ ã3æ ì ë ÉÍ ¢£ 3 î ÄËÆ © ¡ 3 É Ø ~ p áï É x1 , x2 , x3 í ê áé ö ö ç è Ê À3Á ¿ É ½¾ È ¾ Æ äå æç ã á à áâ MIP Oswald/Reinelt BDC q \ +7300x2 &' TDU 80x1 '( [ a EF ¢ +0.6x3 @ +2x2 A p 1.4x1 r bc +7x3 Ù Ê « ¡ ª ðñé É ¢ dDe +83x2 ¿Å Ê 8x1 ¤ ¥¦ í Ë ª® ¨ ´ §¨ © ) * ð òó ð +18x3 «3¬ ª íá Ú 3x2 s fg í Ä Î Í À Ë ¿ Ä ° æç +2x3 +, V GH Ì è ð Û Ì ° ¡ + *- Ü ¿ ·¹¸ ° ó ç +3x2 3® ¢ ¤ ª é 20x1 áô ã Ý +10x3 º¹¸ . tu Ï ÉÐ áé î Þß +15x2 h /0 ª ê ó 40x1 W IJ 132 Ð ÉÑÒ ðñ min À â å x3 «» 45 ÉÁ ¤ «± ª ¯° © ¡ 6 Æ É õ í x1 x2 v XY 7 89 ² i KL ¿Ë ¼ ª MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.3/30 MIP Oswald/Reinelt :; < ¡ ³ = ! Basics of Linear and Discrete Optimization: Examples – p.3/30 ´ MIP Oswald/Reinelt ¤ " ¢ model Marketplace uses "mmxprs" declarations NProd = 3 NIncred = 5 IP = 1..NProd II = 1..NIncred TAB: array(II,IP) REQ: array(II) of PRICE: array(IP) of x: array(IP) of end-declarations of real real real mpvar MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.5/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.6/30 Basics of Linear and Discrete Optimization: Examples – p.5/30 MIP Oswald/Reinelt TAB := [20, 3, 2, 0, 3, 18, 8, 83, 7, 1.4, 2, 0.6, 80, 7300, 0] REQ := [65, 90, 200, 10, 5000] PRICE:= [40, 15, 10] MinPrice := sum(p in IP) PRICE(p) * x(p) forall(i in II) sum(p in IP) TAB(i,p) * x(p) >= REQ(i) minimize(MinPrice) MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.6/30 writeln("Objective: ", getobjval) forall(p in IP) write("Product",p,":",getsol(x(p))," writeln ") end-model Basics of Linear and Discrete Optimization: Examples – p.7/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.8/30 MIP Oswald/Reinelt 34$ (2 / 1, ( & '# , 0/' .) , +.) ( 0/'$ , ) ! ** () U M HF Q M O STR L 7 @ 86 D ! OP LE G HI X YZ ]\[ ^_ ` Va b TdV c EF y2 BC=@ A %'& %$ ?> NM #$ =<;: 7 89 56 y1 L<KJ "!! ( $ + ** , Basics of Linear and Discrete Optimization: Examples – p.7/30 MIP Oswald/Reinelt g hi ]kj lm pCno m \ VW y3 ¦ § ¡C¦ ¡ ¦ ¥ ¦ £ ¢ ¦ ¡ ¢ ¤T£ £ ¤ ¢ ¡ £ ±« ¶ ®³ ·° T¦ © ² ´C± ³ « µ° §¨£ £ ¤ ¢ ³ ¡ ± ¯° ¯ ¬® «ª MIP Oswald/Reinelt s T ¢¥ ¤ C qr t ~ y5 ~ s tu xyw v x z C}{ | g ef y4 Basics of Linear and Discrete Optimization: Examples – p.8/30 +200y3 +10y4 +5000y5 +8y3 +1.4y4 +80y5 ≤ 40 +7300y5 ≤ 15 max 65y1 +90y2 20y1 3y1 +3y2 +83y3 +2y4 2y1 +18y2 +7y3 +0.6y4 ≤ 10 y1 , y2 , y3 , y4 , y5 Basics of Linear and Discrete Optimization: Examples – p.9/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.9/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Examples – p.10/30 % "# ! $ MIP Oswald/Reinelt ≥ 0 &4 2( 01 . /. '& () +,)* .( 3 min {cT x | Ax ≥ b, x ≥ 0} ; B @A bT y0 ≤ cT x0 ?> : y0 ∈ {y | AT y ≤ c, y ≥ 0} C 9 < MIP Oswald/Reinelt 98 x0 ∈ {x | Ax ≥ b, x ≥ 0} 6= 65 7 max {bT y | AT y ≤ c, y ≥ 0} Basics of Linear and Discrete Optimization: Duality – p.10/30 " # ! > 8 C O P O R W [ U W MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Duality – p.11/30 Basics of Linear and Discrete Optimization: Duality – p.12/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Duality – p.12/30 d Basics of Linear and Discrete Optimization: Duality – p.11/30 m r q s | z y y {u j y x, y xw m ig ig j mo i m kl j g f ig uv i v t g f g p n e gh f b c MIP Oswald/Reinelt a` ^ U _ R Z \ Z^ ] J N M F ? 7 < M L F = −→ 0 ' /( + . 2 %1 % ;5 −→ K @ J J R Z X Y X W V T UT −→ [ ( E F 5 7 IH D E G F C 47 9: 8 + ,+ -. % −→ E ( *) ' 5 65 A CDB @ S R Q −→ & % $ 4 3 −→ Ax +By ≤ a Cx +Dy = b | y y { u z y u, v x w m ig g f ig i h v x ≥ 0 uT A +v T C ≥ 0 uT B +v T D = 0 u ≥ 0 T u a MIP Oswald/Reinelt T +v b < 0. # " ' Basics of Linear and Discrete Optimization: Duality – p.14/30 G QUT u op n lm j y f d u z ∗ = +∞ s s { ^i G B 6 FK DJ I 5 3 4 H FG DE C AB \g c bg | \_ s f \ _e } © ¤ ¥ ¡£ ¡ ¡¢ « ¡ ® ¬° ¯ « ¬ §© }~ t ¢ © ¥« ¤ y ]_ \^ ¥© ± u [ ]\ Z ¡¢ | c d `a ¨ ¦ ¤² ® ¦ ] \b ¨ ¡£ ¡ t u w u w { jk © ª ¤ ¥ | x h L B C y q Q P 12 ¥« ∀i. x | H MN x r s bj OO N ¡¦ ¤ ui > 0 ⇐⇒ Ai. x = bi /0 Basics of Linear and Discrete Optimization: Duality – p.14/30 u IGW ¥ ¤§¨ w ∀i. G O S t v t su x T (D) min{uT b | uT A = cT , u ≥ 0}. N Q w ¡ ui > 0 =⇒ Ai. x = bi < RS u∗ = −∞ % $ AE M ³´ rx s v | w y { x z vy ¨ © ª ¥© ¹º ·¸ ¶ MIP Oswald/Reinelt XY (P ) max{cT x | Ax ≤ b} MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Duality – p.13/30 MIP Oswald/Reinelt !( = 798 D=∅⇒P =∅ −∞ z > OV *+ ) P =∅ µ : P =∅⇒D=∅ u & ?@ ,- . # z +∞ ∗ @ ; u∗ = −∞ ⇒ P = ∅ P ∗ * + ∗ ,- . $ & u∗ D ! z ∗ = +∞ ⇒ D = ∅ ! MIP Oswald/Reinelt $ ⇐⇒ u∗ Basics of Linear and Discrete Optimization: Duality – p.13/30 −∞ < z ∗ = u∗ < +∞ ⇐⇒ z ∗ +dT y Ax +By ≥ a Cx +Dy = b x ≥ 0 ≤ b Ax max cT x cT x max max cT x = b x ≥ 0 Basics of Linear and Discrete Optimization: Problem Formulations – p.15/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Problem Formulations – p.16/30 MIP Oswald/Reinelt & ! ;17 3 56 4 9 /3 1 2: 84 - , + ( * "! #$ "% )( Ax ≤ b JH B Ax ≤ b Q P G C \Z y≥0 c b WY V Ax + Iy = b SW a` UT R TZ Ax ≥ b S @ T W_ X V WX YR −→ Ax ≤ b [R G G @ CI D EE ? − x+ i , xi ≥ 0 ON H A@ E −→ Ax = b =4 < > x+ i 7 −→ − x− i 17 3 56 FE CD CM I B A@ L@ K SR UT ^T ] MIP Oswald/Reinelt ( ! ! #$ 1< yi = 7 −→ ! # "! 4 /3 /0. 1 21 xi ≥ 0 ' Basics of Linear and Discrete Optimization: Problem Formulations – p.15/30 MIP Oswald/Reinelt Ax Basics of Linear and Discrete Optimization: Problem Formulations – p.16/30 ! " $# max {dT y | By = a, Dy ≤ b, y ≥ 0} max {cT x + dT y | Ax + By = a, Cx + Dy ≤ b, y ≥ 0, x ≥ 0, x 43 0 /2 01 ,+ / -. 0. &% ) '( *( } 65 9 78 :8 max {cT x | Ax = a, Cx ≤ b, x ≥ 0, x Basics of Linear and Discrete Optimization: Problem Formulations – p.17/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Problem Formulations – p.17/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Problem Formulations – p.18/30 > N @M LM i lCj k m j i on [_ hh fg d e] ]d }~ {| xyz bc [ YZ ]\ ^] [ wu ^_ a` O RCP Q TS VU X WP KJ D P = (A, b) = {x ∈ Rn | Ax = b, x ≥ 0} qp sCr ut wCq v GB J@ E H @I @? BCA ED FD = < ; MIP Oswald/Reinelt } £ ¢ ¡ ¡ C y, z ∈ P = (A, b) ¥ ¤ C x ∈ P = (A, b) x, y ∈ P = (A, b) =⇒ λx + (1 − λ)y ∈ P = (A, b) ∀ 0 ≤ λ ≤ 1 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Polyhedrons – p.18/30 ) + ( .( $ 4 ( * < 8 9 A 7 @ G 6 L 7 5 6 H B 89 JK Y \ `a VZ Y V X S SUT R Y VW g n i Basics of Linear and Discrete Optimization: Polyhedrons – p.20/30 { w~ y h ^ \ Z w x ~{ x v w x w~ { | w û óô ' #& ò ÿ #% Ï ë é ö ý ú æ ù ú §¨ Ë ÊDÌ ÉÊ È à æ ã ÷ø É Þ è éê  À Á ¾ ¿ ½ È Æ ûüý å å ãä Ç õ ô óô ñò = 8< > óö Þâ Å Æ ! ôþ ëì ç à "# ü éíî Î Æ Í ¦ $# õ ö ÷ Ð © A−1 B b>0 x ê èï ð Basics of Linear and Discrete Optimization: Polyhedrons – p.20/30 xj ô õ ÉÑ Ä S _ X x xN = 0 A { | \ n V o X É Æ ª« ~ z{ \ \W [ ¡ () ü Ò ¬ ¡ , x z | pq `a l S [ z{ p f b ¢ Ó xj j ∈ B iW Sh X AB | b \] £ ø Ó Æ A−1 B b V iW [ AN y k ^_ ¥ ¤ ö õ ü ÔÕ § AN \ k ª ® z{ ý Ö× ¯ [ ó ØÙ ¯ ° y \W VW I r V | w { | w \ `a M 76 X l ú Ú w x z *+ ÖÛ ±¬ ./ ,- 8;: 98 78 - ,, 2 6 .5 ( (1 MIP Oswald/Reinelt | w { w~ } { s ²³ w z~ û ó ØÕ AB A−1 b ≥ 0 B $ 89 j w ~ ú Ö AB j∈N ) bdc t ´µ { w × Ü Ý ³¶ { z x t ·¸ * 23 01 ÷ xB = M : | w | w { Þß x = (xB , xN ) AB ; 6 7 NO x1 à ¹¶ AB . P = (A, b) 6= ∅ P = (A, b) <6 w x u ´ .2 (4 á » º T V Y ¼ A ∈ Rm×n b ∈ Rm B ⊂ {1, . . . , n} A |B| = m A.i i ∈ B A ! 7= P [ V MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Polyhedrons – p.19/30 MIP Oswald/Reinelt " ( % x ∈ P (A, b) >,? Sm X MIP Oswald/Reinelt # = %/ . 6@ x P (A, b) 6= ∅ P = (A, b) 6= ∅ P = (A, b) . ( %& $ " AB 7 + * '( A 6 % Q max{cT x | Ax = b, x ≥ 0} + <DC . ( ( $ 9 Ad = 0 A ) 10 \ e Vh ; 6 E $( l X S F 89 2 $ ( " * 0 λi = 1 d ≥ 0 & P $( Y λi ≥ 0 ,( + ' λi vi + d 3 # - @d ≥ 0 $ $ vi ∈V ) S h X x= V Ad = 0 # " P = (A, b) 6= ∅ = ^W x ∈ P = (A, b) Basics of Linear and Discrete Optimization: Polyhedrons – p.19/30 f ! 6A "# P = (A, b) .,/ ) , *" + <; 9 A8 6 9 6?@ *) %( ' %&$ "# Basics of Linear and Discrete Optimization: Simplex – p.22/30 Y 9C9 9 <>;= 9: 78 4 56 PQ @J AD B 5A _ Z` A = 99 A@ <B 5D 7D ? N 5 LM H F 5H B [h \h \ Yg Wf Z ] e\ X b Xac 5I 5A O K bl j Y Xj Zi R _^ ` ]X [\ Z W XY V \ k Xp nZ \\ X d ]Y _\ o ` X _` \ Za i «¬ ¹ µ³ v w~ wv rwv t u s qr ®¯ tuv l º » >y rz r wxs Xj h Zl ]n _m rv{ Xhc \ z w } r uv }r w~ } wz w| w } z  Á>À ¾Â ½¼ ¾¿ h X n l X X ¡ \i Xj £ ¡ ¢ Xo _b Xhc \ 7D P (A, b) *" x S ; = AD ¦¥ XYh i À ¾É ÇÈ Ã ¿ ¾ ÁÆ ¾ Ä ÃÅ ¤ ¢ kl W T § Basics of Linear and Discrete Optimization: Simplex – p.22/30 <B 9 78 ; + : p U 5E ` ZY n m 9 A8 rank(A) = m < n x \a ]X ]\ ³ª ¹ ¶µ ·¸ « ® ´ ³¬ °²± ®¯ «¬ ©ª MIP Oswald/Reinelt F 5= B B 5A ¨ B "' . )2 # .0 % " # * %+ "1 # 0+ x∈P ) MIP Oswald/Reinelt 8 C G ,3 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.21/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.21/30 B / 3 2 % " ! " 0/ !$ # ! " ! ( 1/ # 10 8 &6 : .+9 8 &. ,+ ( )* &' ( &7 4 1 3 16 1 . ; ! HK J C GI HB AB G >F E> CD AB w xy QRd u?t v rs Q N ? ` pq |}~ ^_ no L] m kl | Basics of Linear and Discrete Optimization: Simplex – p.24/30 º¿½¾ ÄÅ Ä» ¶ ¶µ ª ä N Ï ¹ ¿ÓÕ ÖÔ äè ÒÓ ä á éâ Ñ ä µ ½ É Ñ ¶È À º¼¹» ½À ¶¸µ · Ä ÐÎ ´ Ï » Å ´ ½Å Ç Í ÍÎ Ì ßéê äã à¼á ß ü çî í ç ø ù ö êâ ç Þ â éê ë ì ß çì ñ ô ö ö ÿõ õ÷ö ñ òô ö ó ö û øö õ û ÿ¼õ û ÿü ö ÿú þ õ ó øö õ ø ÿõ ó ò¼óñ þ õû è ö ó õû ú¼û å â ö ýþ â ç ÿ¼õ û ùö ÿó ã ù ÿ ö ç ä â åNæ ã ñ õ ù á å Î ú ö ó ñ éâ õ þ õö ÿ û ú ã ß Ç ¹ º¿½¾ àê Ø çð àï ü ý î ì å éë Î × Ê» ½À Ï ¹ ¶ ì æ Ó ØÙ Ë º¿¹Á Ó »Ä ½À »¹Â Ô ÚÛ ºÊ ¹Ã « ÄÆ ñþ é Basics of Linear and Discrete Optimization: Simplex – p.24/30 µ ¯ ¬ { z{ ° ® r ab ± k ¶µ ß äà ÿ¼ø ó ö π ¡¢ ~ Y Lc \ £¤ B A−1 b B i mt O QZ ? ½À åé à c X [?j iM Ohg TfMVe E = λ0 \d ¥ ¦¥ bi di P QR O LNM §¨ ²³ ø Þ Ï ¸Ü Ý ó MIP Oswald/Reinelt 4! ?T QUS Õ B N 65 U © MIP Oswald/Reinelt # 10 OM di > 0 =?>@ VW r (2 2$ #1 <0 1 !0 22 ! 40 10 M d := A−1 B A.s A b c max{cT x | Ax = b, x ≥ 0} V cs > 0 Z YT O X s cj := (cTN )j − π T AN ej π T := cTB A−1 B B A−1 b B j = 1, . . . , n − m \[ λ0 := min{ dbi | di > 0, i = 1, . . . , m} MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.23/30 MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.23/30 % ( & #'& % 789 !$ 46 3 UV KV O X M La PQ O ` # !" 45 3 T L 48 A J Y_ F I ACB GH O M X L T T R ] s o v e d cd T _ W s ~ syz w Basics of Linear and Discrete Optimization: Simplex – p.26/30 x q pSx q p s | v « « « ® ³ ¡ £ © §¨ ¦ Basics of Linear and Discrete Optimization: Simplex – p.26/30 Å Â ¨ ¨ Á ¿ ¿ÀÁ ´ § Ï Âà ª « ´ Ð ÁÄ ¨ «µ Ç Á ¸ µ « ÁÄ Í © ± Ê Ç ÅÆ ª ® ¥ ² ¯ ´ «¬ ª Ç Ñ ¨© ÀÁ ³ ¶ ®¯ ÇÆ ÄÈ ¹»º ¦ ¦ ¬ ¨ ÉÊ Á ¯ ª ¨ Ê ½ ¯ ± ¾ Ê Ç § ³ ÁÄ ¾ É ¶ ®° ¡ « £ fg Y U EF ?@ b P M 46 4 5 Y^Z h ®° ± q { q p ® ³[· ² ¯ e'i T @ ; = u f R J ; D t jk ª ¨ Ð Ä Ë Æ^ ³ Ì ¯ ¯ y e d Y Í ³ ® ¨ n| eSl j « À Á¾ ¯ ± m O Q Z K « t OSR p s no § ³ £ n| s ²³ ª ¿ ¯ } y nq p TM OSR 8 _ LM = ~ ` N : r zn ¤ ÇÆ ¨ ¾ ¼½ Ê Á Æ ÂÎ MIP Oswald/Reinelt 4; n ¡ ÉÆ « | Ax + s = b, x, s ≥ 0} s = b, x = 0 s 6= 0 x ) V ¶ i si *+, st v ® ³ P % 3 + & ) ) * ;'< q £ ¢ min{ -. pi > ci 5 & ⇒ / 3 < < i Y n p L MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.25/30 MIP Oswald/Reinelt ) u XY[Z N W Basics of Linear and Discrete Optimization: Simplex – p.25/30 % v L V O T Y =8 0 s YN Z P > 12 K v MIP Oswald/Reinelt ⇒ \ !( '# %& U I GJ B EI H @< FGH >? DE <= b i k ld m L P K g kh ij ah y |~} xyz ¥¤ « Basics of Linear and Discrete Optimization: Simplex – p.28/30 » ¹º · ¶¸ ³ ´µ û Å· »´ ½» ¶ Ç · ¸¶ ·¶ Ƶ Ì ÍËÎ Ê Ã³ ÈÉ ×Õ áÙËâ ×Ì Ú Ë ÿ ç ý ñô ë íóñ ò ïÙîð íì éë é ç åè é åæ ê õþ ÿ û üúý û õö ) ! ( ' " & % "% $ # ÿ ý ù ÷ø Basics of Linear and Discrete Optimization: Simplex – p.28/30 = eTr A−1 B AN = Ar. ü ×Õ ÚÕ ÔË Í ÞÍ ÏÜ üÿ ß Ôà ÛÝ cs ws B AB ¿ ¼½ ¾ ¸ ÀÁ © ¡ª ¢ ÏÑÐÒ Â ·Á ã ÔÕÓ Ó Â £ ËÍÒ Ì ´ Á ¯ ä λ0 = ° ¤ « y{ ×Ì Í Ö þ s M à Áľ ½ Á¶ À À Õ c K : @AB > FP ML ¦ þ ü ! " ÷ õ MIP Oswald/Reinelt $& K ÍÎ Å ÷ d = A−1 B A.qs )* QQ E ¡ ¼½ ¾ br < 0 OPI N + § ± ² Ú ×ÙÌ Ø λ0 := min{ wjj | wj < 0, j = 1, . . . , n − m} LV § ¬ wN ≥ 0 M j XYW ¢ £ ÛÖ Å¸ Ä ¼ ÁÄ T wN ,- A−1 B b ./ AB ih QR G FI 0 ¥¤ «  r A [\ Z ] MS G ¤ a ¥ b = A−1 B b≥0 ll f } Z^ ¤ «® ¦ A b c max{cT x | Ax = b, x ≥ 0} cn § MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.27/30 MIP Oswald/Reinelt X_ K Basics of Linear and Discrete Optimization: Simplex – p.27/30 51 34 21 MIP Oswald/Reinelt 67 ] ` [ o ¤¦¨ AB 24 qr p ab T u C ≥0 c = cTN − cTB A−1 B AN ≤ 0 s cd ¥ P = {u | uT A ≥ cT } P u 89 : pt uT = cTB A−1 B uv ef xN = 0 ;< ! " x xB = A−1 B b $# ¢ s w q ! ( )* %$ AB +, @ ; Y[ V M Qb R[ U[ 4 ; Z YZ VZ 9 Q Y VXW U ^[_ TM 7 48 1 567 4 0213 J RQS O C OP LNM OY H E Y W jp Basics of Linear and Discrete Optimization: Simplex – p.30/30 m j e no o n kml upv rn l x z }~ { {| rs n d s ko feg a p j hi \ GK I kp i rs eq fp U np k rs t [ [ 5: 8 ! upv V [\ => < j r w je y Q \ O ? 3 ! "# U c ] s np k OU \ O 8 %&$ = ` ^[_ "# '( Y[ V CD )- E u lp ¯ Æ ¾Å ÀÄ Á Âà ¤® À¿ Á ¾ ¡ ¨ ¤ ¦ ½ ¥ ¬¤ ¼³ ¦ º» ¥ ¬« ·¸¹ ² ¶ ¤ª µ ¢© ²´ ¡¨ ²³ ¤§ ¬ ± ¦ ¥ ¤ ¢¤° ¢£ ¥ ¤ ¡ ¢ ¯¡ § ª Þ ÷ óí õmö ÙÌ ñ ó Ñ óô ð ÒÓ í î êìë &% % $ é ÉÊ àá ÏÐÑ æçè É ÌÎ à ãå Ì ã Í îï Ñ ÖÔ Õ ' Ø ×Ñ ðòï ( $ É ÛÝÜ ÉÐÏ ÉÚ ) Basics of Linear and Discrete Optimization: Simplex – p.30/30 )$/ . $ %$ \ Ç "# ! ùú ùú ü ÿ ä ú ýþ ËÌ âã ûü ÉÈ Ê È àß á ß ùø ú ø MIP Oswald/Reinelt F up r £ ¦ n C ^ W[ ¤ m E MIP Oswald/Reinelt Basics of Linear and Discrete Optimization: Simplex – p.29/30 MIP Oswald/Reinelt G MIP Oswald/Reinelt O Basics of Linear and Discrete Optimization: Simplex – p.29/30 ^[
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