A New Generation of Mixed-Integer Programming Codes

23 March 2002 CPAIOR ‘02 1

A New Generationof

Mixed-Integer Programming Codes

Mary Fenelon, Zongao Gu, Javier Lafuente, Ed Rothberg, Roland WunderlingILOG, Inc

Robert E. Bixbyand


Outline

• LP– Overview– Computational results

• MIP– Examples– Historical view– Features– Computational results– One more example -- future


LP


A linear program (LP) is an optimization problem of the form

LP

uxl

bAxToSubject

xcMaximize T


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)

LP


Progress: 1988 – Present

• Algorithms– Simplex algorithms 960x– Simplex + barrier algorithms 2360x

• Machines– Simplex algorithms 800x– Barrier algorithms 13000x

LP

Total: Over 2000000x


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

LP

Key: Ratio > 1 means denominator better


MIP


A mixed-integer program (MIP) is an optimization problem of the form

LP

integerallorsome j

T

x

uxl

bAxtoSubject

xcMaximize


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.

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

MIP


Example 2 (cont.): Avoid structures like

Maximize x + y + zSubject To 2 x + 2 y 1 z = 0 x free y free x,y integer

MIP

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!


• 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

Example 3: A typical situation – Supply-chain scheduling

MIP


Integer optimal solution (0.0001/0): Objective = 1.5091900536e+05Current MIP best bound = 1.5090391809e+05 (gap = 15.0873)Solution time = 3465.73 sec. Iterations = 7885711 Nodes = 489870 (2268)

CPLEX 5.0:

Implied bound cuts applied: 55Flow cuts applied: 200

Integer optimal solution (0.0001/1e-06): Objective = 1.5091904146e+05Current MIP best bound = 1.5090843265e+05 (gap = 10.6088, 0.01%)Solution time = 1.53 sec. Iterations = 3187 Nodes = 58 (2)

CPLEX 6.5:

Supply-chain scheduling (continued): Solving the fixed model

MIP

Original model: Now solves in 2 hours(20% improvement in solution quality)


Computational History:1950 –1998

• 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


1998…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

• 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


Gomory 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) 1

MIP


Computing Gomory Mixed Cuts

1. Make a an ordered list of “sufficiently” fractional variables.

2. Take the first 100. Compute corresponding tableau rows. Reject if coeff. range too big.

3. Add to LP.4. Repeat twice.5. Computed only at root. Slack cuts

purged at end of root computation.

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%)

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

MIP


Computational Results III: 78 Models

CPLEX 5.0 not solvableCPLEX new solvable < 1000 seconds

• No 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

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

possibilities1. Limit traffic at each node2. Limit # of arcs in and out of nodes3. Limit # of jumps4. Symmetry constraint5. 2-line constraint6. Security constraint

• 10 minute solve time limit

MIP


CPLEX solve times (France Telecom):

GUB cover cuts applied: 328Cover cuts applied: 1290Gomory fractional cuts applied: 2

Integer optimal solution: Objective = 1.6461200000e+05Solution time = 525181.51 sec. Iterations = 469805329 Nodes = 3403990

10 Minutes: 34% gap

CPLEX 8.0: Default

GUB cover cuts applied: 803Cover cuts applied: 807Gomory fractional cuts applied: 12

Integer optimal, tolerance (0.0001/1e-06) : Objective = 1.6461200000e+05Current 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

CPLEX 8.0: Tuned with “mip emphasis” (4 processors)

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

MIP

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A New Generation of Mixed-Integer Programming Codes

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