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A New Generation of Mixed-Integer Programming Codes

A New Generation of Mixed-Integer Programming Codes. Robert E. Bixby and. Mary Fenelon, Zongao Gu, Javier Lafuente, Ed Rothberg, Roland Wunderling ILOG, Inc. Outline. LP Overview Computational results MIP Examples Historical view Features Computational results

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A New Generation of Mixed-Integer Programming Codes

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  1. A New GenerationofMixed-Integer Programming Codes Robert E. Bixby and Mary Fenelon, Zongao Gu, Javier Lafuente, Ed Rothberg, Roland Wunderling ILOG, Inc CPAIOR ‘02

  2. Outline • LP • Overview • Computational results • MIP • Examples • Historical view • Features • Computational results • One more example -- future CPAIOR ‘02

  3. LP CPAIOR ‘02

  4. LP A linear program (LP) is an optimization problem of the form CPAIOR ‘02

  5. LP What’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) CPAIOR ‘02

  6. LP Progress: 1988 – Present • Algorithms • Simplex algorithms 960x • Simplex + barrier algorithms 2360x • Machines • Simplex algorithms 800x • Barrier algorithms 13000x Total: Over 2000000x CPAIOR ‘02

  7. LP Algorithm Comparison 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 0.9 Key: Ratio > 1 means denominator better CPAIOR ‘02

  8. MIP CPAIOR ‘02

  9. LP A mixed-integer program (MIP) is an optimization problem of the form CPAIOR ‘02

  10. MIP Example 1: LP still can be HARDSGM: Schedule Generation Model157323 rows, 182812 columns, 6348437 nzs • LP 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. CPAIOR ‘02

  11. MIP Example 2: MIP really is HARD A Customer Model: 44 cons, 51 vars, 167 nzs, maximization 51 general integer variables (inf. bounds) • 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: UNCHANGED CPAIOR ‘02

  12. MIP Example 2 (cont.):Avoid structures like Maximize x + y + z Subject To 2 x + 2 y  1 z = 0 x free y free x,y integer • 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! CPAIOR ‘02

  13. MIP Example 3:A typical situation – Supply-chain scheduling • Model 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 MIP CPAIOR ‘02

  14. CPLEX 5.0: 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) CPLEX 6.5: Implied 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) MIP Supply-chain scheduling (continued): Solving the fixed model Original model: Now solves in 2 hours (20% improvement in solution quality) CPAIOR ‘02

  15. 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, … 1972 – 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 Grötschel, Padberg, Rinaldi …TSP (120, 666, 2392 city models solved) MIP Computational History:1950 –1998 CPAIOR ‘02

  16. 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 Presolve 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 MIP 1998…A new generation of MIP codes CPAIOR ‘02

  17. MIP Gomory Mixed Cut • Given y, xjZ+, and y +  aijxj = d = d + f, f > 0 • Rounding: Where aij = aij + fj, define t = y + (aijxj: fj  f) + (aijxj: 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)  1 CPAIOR ‘02

  18. MIP 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. CPAIOR ‘02

  19. MIP Computational Results I:964 models • 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%) CPAIOR ‘02

  20. MIP Computational Results II:651 models (all solvable) • 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): 171x CPAIOR ‘02

  21. MIP Computational Results III:78 ModelsCPLEX 5.0 not solvableCPLEX new solvable < 1000 seconds • No cuts33.3x • No presolve7.7x • Old variable selection2.7x • CPLEX 5.0 presolve2.6x • Node presolve1.3x • Heuristics 1.1x • Dive probing1.1x CPAIOR ‘02

  22. MIP Example:Network Design (France Telecom – C. Le Pape & L. Perron) • 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 CPAIOR ‘02

  23. MIP CPLEX solve times (France Telecom): CPLEX 8.0: Default GUB cover cuts applied: 328 Cover cuts applied: 1290 Gomory fractional cuts applied: 2 Integer optimal solution: Objective = 1.6461200000e+05 Solution time = 525181.51 sec. Iterations = 469805329 Nodes = 3403990 10 Minutes: 34% gap CPLEX 8.0: Tuned with “mip emphasis” (4 processors) GUB 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 CPAIOR ‘02

  24. MIP Faster integral solutions (France Telecom) : • 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 CPAIOR ‘02

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