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0w<4!d!dl 0wg4%d%d@z[ 0Zlp\ poO ʚ;1h8ʚ;<4ddddl|- 0X0___PPT10 pp___PPT9/ 0zar2? -O =Fx3A New Generation of Mixed-Integer Programming Codes*43,0 Outline$xLP Overview Computational results MIP Examples Historical view Features Computational results One more example -- futureLSS}LP"<H B A linear program (LP) is an optimization problem of the form dC   $) $$4What s the biggest change?$<1988  One algorithm for LP Primal simplex (Dantzig, 1947) Today  Three algorithms for LP Primal simplex Dual simplex (Lemke, 1954) Barrier (Karmarkar, 1984)` DD,,[ 0Progress: 1988  Present$Algorithms Simplex algorithms 960x Simplex + barrier algorithms 2360x Machines Simplex algorithms 800x Barrier algorithms 13000x ? ?  MAlgorithm Comparison&$$ MIP"<$ J A mixed-integer program (MIP) is an optimization problem of the form dK   $) $$hExample 1: LP still can be HARD SGM: Schedule Generation Model 157323 rows, 182812 columns, 6348437 nzs:i 3f!)eLP relaxation at root node: Barrier: Solve time estimate 3-6 days. Primal steepest edge: 64,000 seconds Branch-and-bound 368 nodes enumerated, infeasibility reduced by 3x. Time: 2 weeks. Currently  solved by decomposition.tNC%NC#AExample 2: MIP really is HARD& 3f (Example 2 (cont.): Avoid structures likeF)  pExample 3: A typical situation  Supply-chain scheduling29 3f.2Model description: Weekly model (repeated), daily buckets: Objective to minimize end-of-day inventory. Production (single facility), inventory, shipping (trucks), wholesalers (demand known) Initial modeling phase Simplified prototype + complicating constraints (consecutive day production, min truck constraints) RESULT: Couldn t get good feasible solutions. Decomposition approach Talk to manual schedulers: They first decide on  producibles schedule. Simulate using Constraint Programming. Fixed model: Fix variables and run MIPr:d/q  (   \  CBComputational History: 1950  19986"$|1954 Dantzig, Fulkerson, S. Johnson: 42 city TSP Solved to optimality using cutting planes and solving LPs by hand 1957 Gomory Cutting plane algorithm: A complete solution 1960 Land, Doig, 1965 Dakin B&B 1971 MPSX/370, Benichou et al. 1972 UMPIRE, Forrest, Hirst, Tomlin (Beale) SOS, pseudo-costs, best projection, & 2ZBZ Z.ZZZKZ&Z$B .    ,  & b9 7 Z1972  1998 Good B&B remained the state-of-the-art in commercial codes, in spite of 1973 Padberg 1974 Balas (disjunctive programming) 1983 Crowder, Johnson, Padberg: PIPX, pure 0/1 MIP 1987 Van Roy and Wolsey: MPSARX, mixed 0/1 MIP Grtschel, Padberg, Rinaldi & TSP (120, 666, 2392 city models solved)PU J  tZ;& *DF1998& A new generation of MIP codes$$Linear programming Stable, robust performance Variable/node selection Probing on dives (strong branching) Primal heuristics 8 different tried at root (one new one is local improvement) Retried based upon success Node presolve Fast, incremental bound strengthening$X&$X     & & jPresolve Probing in constraints: xj ( uj) y, y = 0/1 xj ujy (for all j) Cutting planes Gomory, knapsack covers, flow covers, mix-integer rounding, cliques, GUB covers, implied bounds, path cuts, disjunctive cuts New features Extensions of knapsacks Aggregation for flow covers and MIR  ZZ=ZZZ=Z       =l"TGomory Mixed Cut$ >Given y, xj Z+, and y + aijxj = d = d + f, f > 0 Rounding: Where aij = aij + fj, define t = y + (aijxj: fj f) + (aijxj: fj > f) Z Then (fj xj: fj f) + (fj-1)xj: fj > f) = d - t Disjunction: t d (fjxj : fj f) f t d ((1-fj)xj: fj > f) 1-f Combining: ((fj/f)xj: fj f) + ([(1-fj)/(1-f)]xj: fj > f) 1PZ /PZ +PZ =PZ PZ 7PZ PZ ZPZ PZ BPZ                 *  0 " Computing Gomory Mixed Cuts  Make a an ordered list of  sufficiently fractional variables. Take the first 100. Compute corresponding tableau rows. Reject if coeff. range too big. Add to LP. Repeat twice. Computed only at root. Slack cuts purged at end of root computation.." Z1q$Computational Results I: 964 models0%3f Ran for 100,000 seconds (defaults) CPLEX 5.0: Failed to solve 426 (44%) CPLEX 8.0: Failed to solve 254 (26 %) Among not solved (with CPLEX 8.0) 109 had gap < 10% 65 had no integral solution (7%) With  mip emphasis feasibility : 19 found no feasible solution (2.0%)x#M"3G#M"3A=5Computational Results II: 651 models (all solvable):63f Ran for 100,000 seconds (defaults) Relative speedups: All models (651): 12x CPLEX 5.0 > 1 second (447): 41x CPLEX 5.0 > 10 seconds (362): 87x CPLEX 5.0 > 100 seconds (281): 171x66$  w]Computational Results III: 78 Models CPLEX 5.0 not solvable CPLEX new solvable < 1000 seconds>^3f 9No cuts 33.3x No presolve 7.7x Old variable selection 2.7x CPLEX 5.0 presolve 2.6x Node presolve 1.3x Heuristics 1.1x Dive probing 1.1x      P.  6Example: Network Design (France Telecom  C. Le Pape & L. Perron)<C(*,1!Construct a virtual private networks Determine routes Determine capacities 6 additional constraints: 64 = 26 possibilities Limit traffic at each node Limit # of arcs in and out of nodes Limit # of jumps Symmetry constraint 2-line constraint Security constraint 10 minute solve time limit %Z&Z1Z ZZ%&!  #CPLEX solve times (France Telecom):6$,Faster integral solutions (France Telecom) :6-Constraint Programming Approach Build greedy initial solution.  Sliced based search to improve solution (Goals & propogation) Results compared to CP approach 33 cases CPLEX gives no integral solution 31 remaining: 18 in which CPLEX produces better solutions Now possible in CPLEX Advanced presolve (to use original problem representation) Concert technology (ILOG Solver-style modeling) Implemented local cuts Implemented ILOG Solver-style goals _ f _ f  ,r  /l,?@ADEFHI(  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@ |?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>> ZR (    6( P  T Click to edit Master title style! !  0   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0@  ``  r*23 March 2002  0 `   n*CPAIOR  02      0t "`   Z*B  s *޽h ? ̙33 $Blank Presentation 0 P( )    N@ii h#   n*  F##FFjj  N؈ii  #  p*  F##FFjjd  c $ ?:9  4  Ntii  G  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S   Tii h   n*  F##FFjj   Tii    p*  F##FFjjH  0"j ? ̙3380___PPT10.@Dc (    N%&e&e     X*   N`)&e&e     Z*   T .&e&e    X*   Tx2&e&e    Z* H  0"j ? ̙3380___PPT10.pnj@ yq0 (  x  c $*D<`     H4ې ?( F  @,   H& ? ) ,$D 0 RMary Fenelon, Zongao Gu, Javier Lafuente, Ed Rothberg, Roland Wunderling ILOG, Inc"SRt     H\7 ? 7  YRobert E. Bixby and*H  0޽h ? ̙33 A `*(  r  S P   x  c $Pp  H  0޽h ? ̙33p ,   (   x   c $`0    H   0޽h ? ̙330   $p(  $x $ c $    $ <<@@ XLP3f  $ Hp ? 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I ,(  r  S 4P     c $p` <$ 0    <@@ XLP3f   H ?L  ,$D 0 VTotal: Over 2000000x&$$H  0޽h ? ̙33:2___PPT10.+fDz' = @B D5' = @BA?%,( < +O%,( < +D' =%(D^' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B%slide(fromBottom)*<3<* DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* '%(D' =-6B%slide(fromBottom)*<3<* 'DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'J%(D' =-6B%slide(fromBottom)*<3<*'JD' =%(D^' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*JS%(D' =-6B%slide(fromBottom)*<3<*JSDR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*So%(D' =-6B%slide(fromBottom)*<3<*SoDR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*o%(D' =-6B%slide(fromBottom)*<3<*oD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+p+0+ ++0+ +k  ( w r  S d)   N   <* ?  Size Prim/ Dual/ Bar/ (#rows) #Models Dual Bar Simp > 0 680 1.5 1.1 1.1 > 10000 248 2.0 1.0 1.2 >100000 73 2.1 1.6 0.9n? 2702y0 2Txb)5 ?   <6@@ XLP3f   <; ?p   k'Key: Ratio > 1 means denominator better((33$H  0޽h ? ̙33d   T(  Tl T C `0    H T 0޽h ? ̙330  (p(  (x ( c $ C    ( <D@@ XLP3f  ( HI ? J v  @,  ( HL ? z  @, l ( 0A #?8c?  #H ( 0޽h ? f D w0(  r  S $SP     S Sp<$ 0    <T@@H,$D 0 YMIP3f   HY ? J >, H  0޽h ? ̙33 ___PPT10.+LFD' = @B DG' = @BA?%,( < +O%,( < +D' =%(D^' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-6B%slide(fromBottom)*<3<*DDR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Dj%(D' =-6B%slide(fromBottom)*<3<*DjD' =%(D^' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*j{%(D' =-6B%slide(fromBottom)*<3<*j{DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D' =-6B%slide(fromBottom)*<3<*{DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+8+0+ +D  E `XP (  l  C lo   {  <Xp A Customer Model: 44 cons, 51 vars, 167 nzs, maximization 51 general integer variables (inf. bounds)$y>M  < x"m\ ,$D 0 T Branch-and-Cut: Initial integer solution -2186.0 Initial upper bound -1379.4 & after 120,000 seconds, 32,000,000 B&C nodes, 5.5 Gig tree Integer solution and bound: UNCHANGEDl8?p%?I'   <@@H YMIP3f H  0޽h ? ̙33___PPT10d+h$D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+8+0+ +  TL`(  ~  s *$   h  B̐f|Pd :  Maximize x + y + z Subject To 2 x + 2 y 1 z = 0 x free y free x,y integer.W-)  <@@H,$D 0 YMIP3f   H$ ? f,$D 0 Note: This problem can be solved in several ways Euclidean reduction on the constraint [Presolve] Removing z=0, objective is integral [Presolve] Bounds on variables (==> local cuts) However: Branch-and-bound cannot solve!22)-))  !  >[))*H  0޽h ? ̙33JB___PPT10"+D ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(Dp ' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bdissolve*<3<*2D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2e%(D' =-o6Bdissolve*<3<*2eD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =-o6Bdissolve*<3<*eD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+8+0+ +  skp(    s *0   * `  Htf| z  >   Hp `     <(@@H YMIP3f H  0޽h ? ̙33 F     (  z    P,$D 0y  BLf|  Integer optimal solution (0.0001/0): Objective = 1.5091900536e+05 Current MIP best bound = 1.5090391809e+05 (gap = 15.0873) Solution time = 3465.73 sec. Iterations = 7885711 Nodes = 489870 (2268)    Bhf|g @ CPLEX 5.0: z     ,$D 0  B|f|  8Implied bound cuts applied: 55 Flow cuts applied: 200 Integer optimal solution (0.0001/1e-06): Objective = 1.5091904146e+05 Current MIP best bound = 1.5090843265e+05 (gap = 10.6088, 0.01%) Solution time = 1.53 sec. Iterations = 3187 Nodes = 58 (2)    Bdf|g @ CPLEX 6.5: A  NP ?s,$D 0 =Supply-chain scheduling (continued): Solving the fixed model6>$ 3f    <@@H,$D 0 YMIP3f *  <$ ? 8H,$D 0 LOriginal model: Now solves in 2 hours (20% improvement in solution quality)"M3f>H  0޽h ? ̙33 ~ ___PPT10^ +7D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DE' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*D' =%(D' =%(DE' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+8+0+ +  Y(  l  C    l  C DP p  l  C Pp p    <@@H YMIP3f H  0޽h ? ̙33%  e( ?? l  C  `   r  S    r  S p     <@@H YMIP3f H  0޽h ? ̙33  me,(  ,x , c $X@   x , c $,    , <@@H YMIP3f H , 0޽h ? ̙33  }u (  r  S P<P     c $  "P@08X  <@@@H YMIP3f H  0޽h ? ̙33 ? g_(  r  S KP   x  c $L    <H@@H YMIP3f H  0޽h ? ̙33  sk(  x  c $lTP   ~  s *HW    <X@@H YMIP3f H  0޽h ? ̙33 H )(  x  c $TcP     c $K 6  N ?0@= P ECPLEX 8.0: DefaultF  N  ? 1 LGUB cover cuts applied: 803 Cover cuts applied: 807 Gomory fractional cuts applied: 12 Integer optimal, tolerance (0.0001/1e-06) : Objective = 1.6461200000e+05 Current MIP best bound = 1.6459555512e+05 (gap = 16.4449, 0.01%) Solution time = 9275.43 sec. Iterations = 26528289 Nodes = 241051 (4219) 10 Minutes: 10% gap >6K7 5D  N ?0 1P  hCPLEX 8.0: Tuned with  mip emphasis (4 processors) 5&   <@@H YMIP3f H  0޽h ? ̙33XQ  #(  r  S |P     c $x<$  0    < @@H YMIP3f H  0޽h ? ̙33mNeN___PPT10EN.+ DM' = @B DM' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ?%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* ?D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* ?D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<* ?D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<* ?D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*?D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*?D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*?D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*?D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D"' =%(DQ"' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*VD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*VD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*VD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*VD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*VD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*VD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*VD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*VD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*+8+0+ +% 0 Hu(  HX H C :9    H S  G   wcFeatures discussion Mining the backlog of computational ideas from the theory Get good defaults!!H H 0"j ? ̙33} 0 $,(  $^ $S :9    $c $ G   " H $ 0"j ? ̙33 0 P1(  X  C :9     S l G   35 where get stuck because of LPH  0"j ? ̙33x 0 @H(  X  C :9     S  G   J6Overview: This is about MIP. After being constant for may years, commercial MIP technology has changed fundamentally in the last two+ years. It has happened largely through exploiting a backlog of theoretical contributions. We implemented everything we could find. Of course, all of this builds on LP.H  0"j ? ̙33 0 p:(  X  C :9     S  G   <Begin with foundation: LP. We will do quickly, because that s not the emphasis. Then MIP. Look at some anecdotal info. Why: (1) set stage, (2) averages don t tell the whole story (the landscape is bumpy), unlike LP (where they tell a good story). H  0"j ? ̙33 0 QI (  X  C :9   I  S  G   1000 machine improvement would mean 20 minutes to do thest 368 nodes. There may have been a feasible solution in ~1hour. With 10^6 this would be reduced to a few seconds.5sH  0"j ? ̙33 0 tl0(  X  C :9   l  S  G   This is one of those where variables  chase each other. There is a really simple example, with a fixed variable, that illustrates that same problem: Maximize x + y + z Subject To 2 x + 2 y <= 1 Bounds x free y free z = 0 Generals x y EndH  0"j ? ̙33 0 @:(  X  C :9     S d  G   <(I guess 4.0 can t solve? Mention that we have several of these these days  Where one method is used in a first stage, and another in the second stage. And it can be both ways  Col. generation, Samsung. Two phenomena: Can do more day-of-operation stuff with MIP. And ??H  0"j ? ̙33w 0 `5(  X  C :9     S t G   7#What about turning off cuts in 5.0?H  0"j ? ̙33\ 0 (  X  C :9     S D G   14 days/2000000 =~ 0.6 seconds Two things to note here: for simplex methods (still dominant method): algorithm > machine overall speedup is hugeH  0"j ? ̙33 xUJA=3QX,:V*!FHbL dK., U+mmzgg6,!FlԜd޹3g7/W324! |Lrs;}*|uu?؂E_ &(QYqg*߼[ټ}˧&ӄ}d{# o~_c 9P)ϩ:WߤϢy i~}1WO{!?|}mK-,^۟d>SVf&t-o̅93qVAWoqdh#xVMkQ=Dۉra.Қʔ$c u+/hn֭:J%g8y_޼cw׾` p0Jgq.',5.ҶGifg Hמaɖ)NvSOq; u=#~:/eSDyx"~sϸulMS_@~֥C&XOq\w'Wk4Fܲ{rr֎MBoC0:ݿŸp8u8:".pqlg%|-tU<0JJIס^@c{!B*An!ٶ [:R 8rEcN)9h; -$Y`YZmmofT~pn ʁ!o&Zô{^C4Cuz^(+|{x*rfꬪf1Yo0Z:ex&JBZtD-{c;S"?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    Root EntrydO)PicturesCurrent UserSummaryInformation((PowerPoint Document(DocumentSummaryInformation8