Handling Challenging Models
Mario Ruthmair
Gurobi Live, Barcelona, 2023
Agenda
- What makes a model challenging?
- How to identify sources of difficulty?
- What can we do?
What makes a model challenging?
- Returns infeasible
- Takes long to solve because:
- Hard to find any or good feasible solutions
- Hard to improve dual bound
- Model is large
- Numerical issues \(\rightarrow\) presentation Mastering Numerical Challenges
Infeasible Model
Optimize a model with 177 rows, 548 columns and 1714 nonzeros
Model fingerprint: 0x0ff2e72c
Variable types: 0 continuous, 548 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+04]
Objective range [5e+00, 1e+04]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+04]
Presolve removed 12 rows and 2 columns
Presolve time: 0.00s
Explored 0 nodes (0 simplex iterations) in 0.00 seconds (0.00 work units)
Thread count was 1 (of 8 available processors)
Solution count 0
Model is infeasible
Best objective -, best bound -, gap -Why infeasible?
Irreducible Inconsistent Subsystem
- Compute an Irreducible Inconsistent Subsystem (IIS) with
computeIIS() - IIS \(=\) “minimal” subset of constraints and variable bounds that conflict
- Useful to find logic errors in model building
- Practical limitations:
- Computational time can be huge, especially for MIPs
- IIS can be large and difficult to interpret
Why infeasible?
Computing Irreducible Inconsistent Subsystem (IIS)...
Constraints | Bounds | Runtime
Min Max Guess | Min Max Guess |
--------------------------------------------------------------------------
0 177 - 0 1096 - 0s
2 2 2 2 2 2 0s
IIS computed: 2 constraints, 2 bounds
IIS runtime: 0.01 seconds (0.00 work units)IIS in LP file format (default bounds \(x \ge 0\)):
Minimize
Subject To
c79: 93 x1 - 189 x2 + 10 x3 >= 12
c156: x1 - x2 + x3 <= 0
Bounds
x2 free
Generals
x1 x2 x3
EndWhy infeasible?
Feasibility Relaxation
- Find a solution with minimum constraint violation with
feasRelax() - Artificial slack variables show constraint violation
- Useful to find input data errors
- Usually quite fast
Why infeasible?
Feasibility Relaxation
Optimize a model with 1273 rows, 1821 columns and 4083 nonzeros
Model fingerprint: 0xe02d128a
Variable types: 1273 continuous, 548 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+04]
Objective range [1e+00, 1e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+00, 1e+04]
Found heuristic solution: objective 1768.0000000
Presolve added 0 rows and 36 columns
Presolve removed 593 rows and 0 columns
Presolve time: 0.00s
Presolved: 680 rows, 1857 columns, 3096 nonzeros
Variable types: 124 continuous, 1733 integer (548 binary)
Found heuristic solution: objective 1306.0000000
Root relaxation: objective 1.269841e-01, 304 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 0.12698 0 59 1306.00000 0.12698 100% - 0s
H 0 0 1168.0000000 0.12698 100% - 0s
H 0 0 1142.0000000 0.12698 100% - 0s
H 0 0 866.2222222 0.12698 100% - 0s
H 0 0 586.9999998 0.12698 100% - 0s
0 0 1.00000 0 74 587.00000 1.00000 100% - 0s
H 0 0 400.9999998 1.00000 100% - 0s
H 0 0 80.0000000 1.00000 98.7% - 0s
0 0 1.00000 0 72 80.00000 1.00000 98.7% - 0s
H 0 0 79.0000000 1.00000 98.7% - 0s
0 0 1.00000 0 70 79.00000 1.00000 98.7% - 0s
H 0 0 10.0000000 1.00000 90.0% - 0s
H 0 0 5.0000000 1.00000 80.0% - 0s
0 0 1.00000 0 16 5.00000 1.00000 80.0% - 0s
0 0 1.00000 0 16 5.00000 1.00000 80.0% - 0s
0 0 1.00000 0 16 5.00000 1.00000 80.0% - 0s
0 2 1.00000 0 16 5.00000 1.00000 80.0% - 0s
H 36 40 4.0000000 1.00000 75.0% 3.1 0s
H 79 161 2.0000000 1.00000 50.0% 2.4 0s
H 117 161 1.0000000 1.00000 0.00% 2.2 0s
Cutting planes:
Gomory: 20
MIR: 10
Explored 172 nodes (1178 simplex iterations) in 0.31 seconds (0.15 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 1 2 4 ... 866.222
Optimal solution found (tolerance 1.00e-04)
Best objective 1.000000000000e+00, best bound 1.000000000000e+00, gap 0.0000%Why infeasible?
Feasibility Relaxation
Relation to IIS?
Minimize
Subject To
c79: 93 x1 - 189 x2 + 10 x3 >= 12
c156: x1 - x2 + x3 <= 0
Bounds
x2 free
Generals
x1 x2 x3
End- Solution values from feasibility relaxation:
x1 = 1, x2 = 0, x3 = 0 - Value of constraint slack variable:
ArtN_c156 = 1
How to identify sources of difficulty?
Solver log for model glass4
Optimize a model with 396 rows, 322 columns and 1815 nonzeros
Model fingerprint: 0x18b19fdf
Variable types: 20 continuous, 302 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 8e+06]
Objective range [1e+00, 1e+06]
Bounds range [1e+00, 8e+02]
RHS range [1e+00, 8e+06]
Presolve removed 6 rows and 6 columns
Presolve time: 0.01s
Presolved: 390 rows, 316 columns, 1803 nonzeros
Variable types: 19 continuous, 297 integer (297 binary)
Found heuristic solution: objective 3.133356e+09
Root relaxation: objective 8.000024e+08, 72 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 8.0000e+08 0 72 3.1334e+09 8.0000e+08 74.5% - 0s
H 0 0 2.200019e+09 8.0000e+08 63.6% - 0s
H 0 0 2.000016e+09 8.0000e+08 60.0% - 0s
0 0 8.0000e+08 0 72 2.0000e+09 8.0000e+08 60.0% - 0s
H 0 0 1.930017e+09 8.0000e+08 58.5% - 0s
0 0 8.0000e+08 0 72 1.9300e+09 8.0000e+08 58.5% - 0s
0 0 8.0000e+08 0 83 1.9300e+09 8.0000e+08 58.5% - 0s
0 0 8.0000e+08 0 78 1.9300e+09 8.0000e+08 58.5% - 0s
0 2 8.0000e+08 0 78 1.9300e+09 8.0000e+08 58.5% - 0s
H 95 111 1.816682e+09 8.0000e+08 56.0% 7.6 0s
H 414 410 1.750015e+09 8.0000e+08 54.3% 4.5 0s
H 1334 991 1.733348e+09 8.0000e+08 53.8% 5.5 0s
H 1367 965 1.700014e+09 8.0000e+08 52.9% 5.4 0s
*18952 13298 62 1.700014e+09 8.0000e+08 52.9% 4.3 2s
*21945 13292 48 1.700014e+09 8.0001e+08 52.9% 4.4 2s
*21946 13278 48 1.700014e+09 8.0001e+08 52.9% 4.4 2s
H23596 14591 1.700013e+09 8.0001e+08 52.9% 4.4 2s
H27260 14584 1.688903e+09 8.0001e+08 52.6% 4.3 2s
H30543 15189 1.644458e+09 8.6667e+08 47.3% 4.2 3s
H30547 14430 1.644458e+09 8.6667e+08 47.3% 4.2 3s
H30550 13710 1.644458e+09 8.6667e+08 47.3% 4.2 4s
H30555 13027 1.600013e+09 8.6667e+08 45.8% 4.2 4s
31146 13383 1.0500e+09 62 42 1.6000e+09 9.0000e+08 43.8% 4.4 5s
95341 33758 1.6000e+09 76 13 1.6000e+09 1.0000e+09 37.5% 6.0 10s
187457 77051 1.5000e+09 60 37 1.6000e+09 1.0000e+09 37.5% 5.9 15s
*263218 109630 73 1.600013e+09 1.0000e+09 37.5% 5.7 19s
266480 111130 infeasible 55 1.6000e+09 1.0000e+09 37.5% 5.7 20s
*270267 80393 71 1.550013e+09 1.0000e+09 35.5% 5.7 20s
H271848 77647 1.533346e+09 1.0000e+09 34.8% 5.7 20s
*292350 79627 76 1.500013e+09 1.0000e+09 33.3% 5.7 22s
H293401 79715 1.500013e+09 1.0000e+09 33.3% 5.7 22s
H293417 57180 1.475014e+09 1.0000e+09 32.2% 5.7 22s
H299029 55344 1.460014e+09 1.0000e+09 31.5% 5.7 23s
312999 55932 1.4600e+09 52 35 1.4600e+09 1.0072e+09 31.0% 5.7 25s
*331032 52143 75 1.400013e+09 1.1000e+09 21.4% 5.9 27s
350453 53694 infeasible 54 1.4000e+09 1.1000e+09 21.4% 6.0 30s
384990 51565 1.4000e+09 50 28 1.4000e+09 1.2000e+09 14.3% 6.2 35s
413646 58415 1.3000e+09 68 27 1.4000e+09 1.2000e+09 14.3% 6.3 40s
H413897 11468 1.200013e+09 1.2000e+09 0.00% 6.3 40s
Cutting planes:
Learned: 1
Gomory: 10
Cover: 3
Implied bound: 11
Projected implied bound: 1
MIR: 12
Flow cover: 43
RLT: 2
Relax-and-lift: 10
Explored 413963 nodes (2629055 simplex iterations) in 40.32 seconds (23.11 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 1.20001e+09 1.40001e+09 1.46001e+09 ... 1.60001e+09
Optimal solution found (tolerance 1.00e-04)
Best objective 1.200012600000e+09, best bound 1.200006650000e+09, gap 0.0005%Solution progress
Bad solution, good bound
- Bound is strong and converges quickly
- Solutions improve slowly \(\rightarrow\)
- Solver parameters focusing on heuristics
- Enlarge feasible region
- Custom heuristics
Good solution, bad bound
- Solution improves quickly due to internal heuristics (or user solutions)
- Bound is weak and increases slowly \(\rightarrow\)
- Solver parameters focusing on bound
- More cuts
- More preprocessing
- Reformulate model
Tailing off
- Solution and bound improve quickly but take long to close the gap
- Think about termination criteria:
- How precise is data?
- How precise is model?
Gurobi LogTools
- Python package to parse and analyze solver logs
pip install gurobi-logtools
import gurobi_logtools as glt
import plotly.graph_objects as go
results = glt.parse(["*.log"])
log = results.progress("nodelog")
fig = go.Figure()
fig.add_trace(go.Scatter(x=log["Time"], y=log["Incumbent"], name="Upper Bound"))
fig.add_trace(go.Scatter(x=log["Time"], y=log["BestBd"], name="Lower Bound"))
fig.update_xaxes(title_text="Time")
fig.update_yaxes(title_text="Objective value")
fig.show()Gurobi LogTools
Example: glass4
Tune Solver Parameters
Example: glass4
Set parameter Heuristics to value 0.5
Set parameter Cuts to value 0
...
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 8.0000e+08 0 72 3.1334e+09 8.0000e+08 74.5% - 0s
H 0 0 2.200019e+09 8.0000e+08 63.6% - 0s
H 0 0 2.000016e+09 8.0000e+08 60.0% - 0s
0 0 8.0000e+08 0 72 2.0000e+09 8.0000e+08 60.0% - 0s
0 2 8.0000e+08 0 72 2.0000e+09 8.0000e+08 60.0% - 0s
H 171 176 2.000016e+09 8.0000e+08 60.0% 2.2 0s
H 173 176 1.966684e+09 8.0000e+08 59.3% 2.2 0s
H 203 315 1.966684e+09 8.0000e+08 59.3% 2.2 0s
H 321 445 1.900017e+09 8.0000e+08 57.9% 2.3 0s
H 395 445 1.855571e+09 8.0000e+08 56.9% 2.3 0s
H 598 578 1.800015e+09 8.0000e+08 55.6% 2.4 0s
H 680 921 1.794460e+09 8.0000e+08 55.4% 2.6 0s
* 1478 1114 47 1.750016e+09 8.0000e+08 54.3% 3.0 0s
* 1488 1067 47 1.750016e+09 8.0000e+08 54.3% 3.0 0s
* 2065 1434 62 1.733351e+09 8.0000e+08 53.8% 3.2 0s
H 3993 2572 1.733350e+09 8.0000e+08 53.8% 3.0 0s
H 4189 2540 1.700017e+09 8.0000e+08 52.9% 3.0 0s
H 4967 3141 1.700014e+09 8.0000e+08 52.9% 3.0 1s
H 5379 3014 1.510013e+09 8.0000e+08 47.0% 3.1 1s
H 5578 2957 1.500013e+09 8.0000e+08 46.7% 3.1 1s
H 5816 3122 1.500013e+09 8.0000e+08 46.7% 3.2 1s
H 5818 2900 1.475013e+09 8.0000e+08 45.8% 3.2 1s
H20108 11407 1.475012e+09 8.0000e+08 45.8% 4.2 4s
20707 11751 1.0000e+09 44 40 1.4750e+09 8.0000e+08 45.8% 4.5 5s
38874 21313 1.0500e+09 56 33 1.4750e+09 8.0000e+08 45.8% 4.8 10s
*43317 22917 61 1.466682e+09 8.0000e+08 45.5% 5.5 14s
44236 23753 1.1000e+09 39 44 1.4667e+09 8.0000e+08 45.5% 5.7 15s
H44555 22869 1.400014e+09 8.0000e+08 42.9% 5.7 15s
H46604 23057 1.375015e+09 8.0000e+08 41.8% 6.0 15s
*53603 23808 37 1.366680e+09 8.0000e+08 41.5% 5.7 16s
H54046 20392 1.200013e+09 8.0000e+08 33.3% 5.8 17s
Explored 62989 nodes (369883 simplex iterations) in 18.50 seconds (12.03 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 1.20001e+09 1.36668e+09 1.37501e+09 ... 1.51001e+09
Optimal solution found (tolerance 1.00e-04)
Best objective 1.200012600000e+09, best bound 1.200012600000e+09, gap 0.0000%Geometry of MIP
- 2 integer variables \(x\) and \(y\)
- 5 linear constraints define gray area \(=\) continuous or LP relaxation
- Feasible solutions are integer points inside the gray area
- Solvers heavily rely on LP relaxation, e.g., in bounds, heuristics, branching
Geometry of MIP
- Tighten the LP relaxation by adding constraints \(=\) Cut
- Set of feasible solutions stays the same
- Gray area gets smaller
- Variable bounds are also cuts
Geometry of MIP
- Dark gray area \(=\) ideal formulation / convex hull
- Solver cuts and presolve try to find convex hull
- Why ideal? MIP reduces to LP
Geometry of MIP
- Objective function vector \(c\)
Geometry of MIP
- Potential initial solution
- Optimal integer solution
- Optimal LP relaxation solution
Geometry of MIP
- Potential initial solution
- Optimal integer solution
- Optimal LP relaxation solution
- Gaps are highly dependent on objective function
- Even small root LP gaps can hide bad formulations
Geometry of MIP
Why is this important?
- Integer problems can have many formulations for the same set of solutions, which one should you choose?
- Tighter formulations often lead to better performance, even if the model has more variables and/or constraints
- Art of Modeling
- Identify and understand strength
- Find cuts
- Reformulate
Example: Min-max
\[ \begin{align*} \min \quad t & & \\ x_j & \le t & \forall j=1,\ldots,n & \\ \sum_{j=1}^n x_j & = 1 & \\ x_j & \in \{0,1\} & \forall j=1,\ldots,n & \end{align*} \]
- Optimal MIP solution: \(t=1,~ x_j=1\), for one particular \(j\)
- LP relaxation solution: \(t=\frac{1}{n},~ x_j=\frac{1}{n}\), for ALL \(j\) \(\rightarrow\) bound is weak
- LP and MIP solutions have an inherent disconnect, even worse for growing \(n\)
- In general, try to push up \(t\) as much as possible
- Here, we can add constraint \(\sum_{j=1}^n x_j \le t\)
Example: Facility Location
- Input data:
- \(S\): Set of supermarkets
- \(W\): Set of candidate warehouse locations
- \(f_{w}\): Fixed cost for building warehouse \(w\)
- \(c_{ws}\): Cost of supplying supermarket \(s\) from warehouse \(w\)
- Choose a subset of warehouses to supply all supermarkets, with minimal total building and supply costs
- Decision Variables:
- \(y_{w} \in \{0, 1\}\): Build warehouse \(w\) (\(y_w = 1\)) or not (\(y_w = 0\))
- \(0 \leq x_{ws} \leq 1\): Fraction of demand of supermarket \(s\) served by warehouse \(w\)
Example: Facility Location
Model
- Objective: Minimize total warehouse building and supply costs \[\text{Min} \quad \sum_{w \in W} f_{w} \cdot y_{w} + \sum_{w \in W} \sum_{s \in S} c_{ws} \cdot x_{ws}\]
- Each supermarket has to be fully supplied, i.e., the sum of fractions received from each warehouse must be equal to 1. \[\sum_{w \in W} x_{ws} = 1 \quad \forall s \in S\]
- Warehouse \(w\) has to be opened if at least one supermarket is supplied. \[\sum_{s \in S} x_{ws} \leq |S| \cdot y_{w} \quad \forall w \in W\]
Example: Facility Location
200 warehouse locations, 2000 supermarkets
Optimize a model with 2200 rows, 400200 columns and 800200 nonzeros
Model fingerprint: 0x889c7bde
Variable types: 400000 continuous, 200 integer (200 binary)
Coefficient statistics:
Matrix range [1e+00, 2e+03]
Objective range [1e+00, 4e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
Found heuristic solution: objective 1657011.0000
Presolve time: 0.89s
Presolved: 2200 rows, 400200 columns, 800200 nonzeros
Variable types: 400000 continuous, 200 integer (200 binary)
Root relaxation: objective 7.496728e+04, 0 iterations, 0.57 seconds (0.48 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
H 0 0 655075.00000 74967.2750 88.6% - 2s
H 0 0 653009.00000 74967.2750 88.5% - 2s
H 0 0 649303.00000 74967.2750 88.5% - 2s
H 0 0 315273.00000 74967.2750 76.2% - 4s
0 0 90077.2092 0 189 315273.000 90077.2092 71.4% - 7s
H 0 0 309486.00000 113880.544 63.2% - 12s
0 0 113880.637 0 199 309486.000 113880.637 63.2% - 12s
0 0 113880.637 0 199 309486.000 113880.637 63.2% - 13s
H 0 0 288245.00000 113880.637 60.5% - 14s
0 0 127672.131 0 198 288245.000 127672.131 55.7% - 24s
0 0 141821.354 0 200 288245.000 141821.354 50.8% - 38s
0 0 141821.894 0 200 288245.000 141821.894 50.8% - 38s
H 0 0 287028.02305 152967.493 46.7% - 54s
H 0 0 286382.72460 152967.493 46.6% - 54s
H 0 0 284520.36419 152967.493 46.2% - 54s
H 0 0 283286.52103 152967.493 46.0% - 54s
H 0 0 279335.29108 152967.493 45.2% - 54s
H 0 0 278389.32228 152967.493 45.1% - 54s
H 0 0 276709.11508 152967.493 44.7% - 54s
H 0 0 274845.59080 152967.493 44.3% - 55s
H 0 0 272863.07200 152967.493 43.9% - 55s
H 0 0 270067.36859 152967.493 43.4% - 55s
H 0 0 269215.59182 152967.493 43.2% - 55s
H 0 0 267058.50246 152967.493 42.7% - 55s
H 0 0 264503.60754 152967.493 42.2% - 55s
H 0 0 262566.00000 152967.493 41.7% - 55s
H 0 0 260417.00000 152967.493 41.3% - 55s
H 0 0 259862.00000 152967.493 41.1% - 55s
0 0 152967.493 0 198 259862.000 152967.493 41.1% - 55s
H 0 0 247309.00000 164695.350 33.4% - 77s
0 0 164695.350 0 200 247309.000 164695.350 33.4% - 77s
0 0 164697.220 0 200 247309.000 164697.220 33.4% - 78s
0 0 174779.528 0 196 247309.000 174779.528 29.3% - 97s
H 0 0 244829.39820 184077.164 24.8% - 147s
H 0 0 243771.44447 184077.164 24.5% - 147s
H 0 0 241969.66983 184077.164 23.9% - 147s
H 0 0 241173.06143 184077.164 23.7% - 147s
H 0 0 238695.59670 184077.164 22.9% - 148s
H 0 0 238519.36803 184077.164 22.8% - 148s
H 0 0 235385.31861 184077.164 21.8% - 148s
H 0 0 233305.45931 184077.164 21.1% - 148s
H 0 0 230649.58224 184077.164 20.2% - 148s
H 0 0 226799.00000 184077.164 18.8% - 148s
0 0 184077.164 0 196 226799.000 184077.164 18.8% - 148s
0 0 184114.003 0 196 226799.000 184114.003 18.8% - 149s
0 0 192114.343 0 194 226799.000 192114.343 15.3% - 176s
H 0 0 219877.00000 195184.674 11.2% - 219s
0 0 195184.674 0 196 219877.000 195184.674 11.2% - 219s
0 0 195926.373 0 196 219877.000 195926.373 10.9% - 222s
0 0 196165.395 0 196 219877.000 196165.395 10.8% - 223s
0 0 196225.826 0 196 219877.000 196225.826 10.8% - 225s
0 0 196225.826 0 196 219877.000 196225.826 10.8% - 227s
0 2 196225.826 0 196 219877.000 196225.826 10.8% - 235s
23 28 196769.371 7 189 219877.000 196769.371 10.5% 25.9 240s
63 68 197474.225 17 181 219877.000 197225.191 10.3% 19.4 245s
98 107 198364.408 25 173 219877.000 197225.191 10.3% 18.5 250s
143 155 199324.970 36 162 219877.000 197225.191 10.3% 20.1 255s
183 194 200229.907 47 151 219877.000 197225.191 10.3% 21.8 260s
223 233 201567.765 56 142 219877.000 197225.191 10.3% 23.1 266s
259 271 202726.045 64 134 219877.000 197225.191 10.3% 25.0 271s
298 316 204081.770 72 127 219877.000 197225.191 10.3% 27.1 276s
330 345 206740.858 79 120 219877.000 197225.191 10.3% 28.3 280s
358 380 206797.231 84 115 219877.000 197225.191 10.3% 29.6 285s
404 439 208523.098 94 104 219877.000 197225.191 10.3% 30.7 290s
473 503 210330.037 107 91 219877.000 197225.191 10.3% 30.4 296s
533 541 214736.349 123 72 219877.000 197225.191 10.3% 32.7 302s
561 574 215748.307 129 65 219877.000 197225.191 10.3% 33.9 305s
644 631 cutoff 168 219877.000 197231.207 10.3% 33.5 312s
672 651 197448.104 8 188 219877.000 197448.104 10.2% 33.5 315s
718 697 199094.241 22 176 219877.000 197475.343 10.2% 32.9 321s
H 733 688 219526.00000 197475.343 10.0% 32.8 321s
738 710 200028.714 28 170 219526.000 197475.343 10.0% 32.8 325s
H 760 714 218836.00000 197475.343 9.76% 33.0 329s
H 771 713 218823.00000 197475.343 9.76% 33.3 329s
776 734 201927.105 38 159 218823.000 197475.343 9.76% 33.4 333s
797 763 202687.326 42 155 218823.000 197475.343 9.76% 33.8 337s
826 791 203413.089 47 150 218823.000 197475.343 9.76% 33.5 341s
854 816 203917.562 51 147 218823.000 197475.343 9.76% 33.6 345s
879 839 204990.022 60 138 218823.000 197475.343 9.76% 34.0 350s
902 876 207260.562 66 132 218823.000 197475.343 9.76% 34.6 355s
939 911 208445.648 79 119 218823.000 197475.343 9.76% 34.7 360s
974 952 209719.263 88 110 218823.000 197475.343 9.76% 35.0 365s
1015 999 211292.539 95 102 218823.000 197475.343 9.76% 35.4 372s
1063 1000 213064.979 75 196 218823.000 197475.343 9.76% 36.1 399s
* 1064 950 12 215114.00000 215114.000 0.00% 36.0 407s
Explored 1064 nodes (62733 simplex iterations) in 407.36 seconds (209.74 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 215114 218823 218836 ... 238519
Optimal solution found (tolerance 1.00e-04)
Best objective 2.151140000000e+05, best bound 2.151140000000e+05, gap 0.0000%Example: Facility Location
Cuts
- Warehouse \(w\) has to be opened if at least one supermarket is supplied. \[\sum_{s \in S} x_{ws} \leq |S| \cdot y_{w} \quad \forall w \in W\]
- In most solutions, the left-hand side will be much smaller than \(|S| \cdot y_w\)
- In the LP relaxation, due to the large Big-M value \(|S|\) and since we minimize \(f_w \cdot y_w\), the value for \(y_w\) can be set to a small fractional value
- Constraint Disaggregation: If any single supermarket is supplied by a warehouse, the warehouse has to be opened. \[x_{ws} \leq y_{w} \quad \forall w \in W, \forall s \in S\]
Example: Facility Location
Optimize a model with 402000 rows, 400200 columns and 1200000 nonzeros
Model fingerprint: 0xab88c055
Variable types: 400000 continuous, 200 integer (200 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e+00, 4e+03]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
Found heuristic solution: objective 1657011.0000
Presolve time: 2.35s
Presolved: 402000 rows, 400200 columns, 1200000 nonzeros
Variable types: 400000 continuous, 200 integer (200 binary)
Deterministic concurrent LP optimizer: primal simplex, dual simplex, and barrier
Showing barrier log only...
Root barrier log...
Ordering time: 0.18s
Barrier statistics:
Dense cols : 200
AA' NZ : 8.000e+05
Factor NZ : 1.645e+06 (roughly 340 MB of memory)
Factor Ops : 9.667e+07 (less than 1 second per iteration)
Threads : 2
Objective Residual
Iter Primal Dual Primal Dual Compl Time
0 2.21451667e+07 7.45226501e+05 2.00e+01 2.66e+02 7.12e+01 5s
1 2.83097541e+06 -7.72382509e+05 2.48e+00 7.53e+01 7.12e+00 6s
2 6.94893623e+05 -1.44052241e+05 6.66e-15 1.63e-01 7.08e-01 6s
3 6.19409426e+05 3.44940542e+04 9.10e-15 1.61e-01 4.92e-01 7s
4 5.64068664e+05 9.87373612e+04 1.38e-14 1.04e-01 3.91e-01 7s
5 5.49580322e+05 1.16302608e+05 1.33e-14 6.42e-02 3.64e-01 8s
6 5.25675690e+05 1.29417444e+05 1.60e-14 6.21e-02 3.33e-01 8s
7 4.94701173e+05 1.38858058e+05 7.55e-15 6.10e-02 2.99e-01 9s
8 4.45389485e+05 1.76495697e+05 7.55e-15 8.92e-03 2.26e-01 10s
9 4.01222789e+05 1.97074403e+05 1.34e-14 1.80e-02 1.71e-01 10s
10 3.72653325e+05 2.05428659e+05 1.43e-14 1.57e-02 1.40e-01 11s
11 3.45938327e+05 2.09878961e+05 1.19e-14 1.31e-02 1.14e-01 12s
12 3.15836621e+05 2.11984365e+05 4.77e-15 9.89e-03 8.72e-02 13s
13 2.95957845e+05 2.13062637e+05 7.55e-15 7.99e-03 6.97e-02 13s
14 2.79458149e+05 2.13924452e+05 2.55e-14 5.93e-03 5.51e-02 14s
Barrier performed 14 iterations in 13.93 seconds (4.14 work units)
Barrier solve interrupted - model solved by another algorithm
Concurrent spin time: 1.23s (can be avoided by choosing Method=3)
Solved with dual simplex
Root relaxation: objective 2.151140e+05, 13089 iterations, 11.13 seconds (2.90 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
* 0 0 0 215114.00000 215114.000 0.00% - 14s
Explored 1 nodes (13089 simplex iterations) in 14.05 seconds (4.26 work units)
Thread count was 8 (of 8 available processors)
Solution count 2: 215114 1.65701e+06
Optimal solution found (tolerance 1.00e-04)
Best objective 2.151140000000e+05, best bound 2.151140000000e+05, gap 0.0000%Solver Heuristics
- General-purpose \(=\) work for models from many different areas
- Solver includes more than 30 different heuristics
- Main concept:
- Fix or round values for (some) integer variables based on
- Relaxation solutions
- Available integer solutions
- Solve sub-MIP for the rest
- Fix or round values for (some) integer variables based on
- Sometimes relaxations are hard to solve or do not give much information \(\rightarrow\) NoRel heuristic
- Often works well for large binary models
- Activate via parameter
NoRelHeurTimeorNoRelHeurWork
NoRel Heuristic
Example: glass4
Set parameter Heuristics to value 0.5
Set parameter NoRelHeurTime to value 5
Set parameter Cuts to value 0
...
Optimize a model with 396 rows, 322 columns and 1815 nonzeros
Model fingerprint: 0x18b19fdf
Variable types: 20 continuous, 302 integer (0 binary)
Coefficient statistics:
Matrix range [1e+00, 8e+06]
Objective range [1e+00, 1e+06]
Bounds range [1e+00, 8e+02]
RHS range [1e+00, 8e+06]
Presolve removed 6 rows and 6 columns
Presolve time: 0.00s
Presolved: 390 rows, 316 columns, 1803 nonzeros
Variable types: 19 continuous, 297 integer (297 binary)
Found heuristic solution: objective 3.133356e+09
Starting NoRel heuristic
Found heuristic solution: objective 3.033354e+09
Found heuristic solution: objective 2.633356e+09
Found heuristic solution: objective 2.633356e+09
Found heuristic solution: objective 2.633355e+09
Found heuristic solution: objective 2.533354e+09
Found heuristic solution: objective 2.333352e+09
Found heuristic solution: objective 2.300021e+09
Found heuristic solution: objective 2.200019e+09
Found heuristic solution: objective 2.125017e+09
Found heuristic solution: objective 2.100018e+09
Found heuristic solution: objective 2.000018e+09
Found heuristic solution: objective 2.000017e+09
Found heuristic solution: objective 2.000017e+09
Found heuristic solution: objective 1.983351e+09
Found heuristic solution: objective 1.950017e+09
Found heuristic solution: objective 1.933351e+09
Found heuristic solution: objective 1.933350e+09
Found heuristic solution: objective 1.933350e+09
Found heuristic solution: objective 1.850015e+09
Found heuristic solution: objective 1.800015e+09
Found heuristic solution: objective 1.700015e+09
Found heuristic solution: objective 1.700015e+09
Found heuristic solution: objective 1.700014e+09
Found heuristic solution: objective 1.700014e+09
Found heuristic solution: objective 1.690015e+09
Found heuristic solution: objective 1.650015e+09
Found heuristic solution: objective 1.650014e+09
Found heuristic solution: objective 1.616682e+09
Found heuristic solution: objective 1.600013e+09
Found heuristic solution: objective 1.600013e+09
Found heuristic solution: objective 1.600013e+09
Found heuristic solution: objective 1.500013e+09
Found heuristic solution: objective 1.500013e+09
Found heuristic solution: objective 1.475013e+09
Elapsed time for NoRel heuristic: 5s (best bound 8.00002e+08)
Root simplex log...
Iteration Objective Primal Inf. Dual Inf. Time
0 8.0000240e+08 1.843200e+04 0.000000e+00 5s
72 8.0000240e+08 0.000000e+00 0.000000e+00 5s
Root relaxation: objective 8.000024e+08, 72 iterations, 0.00 seconds (0.00 work units)
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 8.0000e+08 0 72 1.4750e+09 8.0000e+08 45.8% - 5s
0 0 8.0000e+08 0 72 1.4750e+09 8.0000e+08 45.8% - 5s
0 2 8.0000e+08 0 72 1.4750e+09 8.0000e+08 45.8% - 5s
12212 5296 1.1000e+09 47 46 1.4750e+09 8.0000e+08 45.8% 5.9 10s
*12457 5028 66 1.400013e+09 8.0000e+08 42.9% 6.1 10s
H12698 3108 1.200013e+09 8.0000e+08 33.3% 6.2 10s
Explored 17122 nodes (115780 simplex iterations) in 11.99 seconds (6.38 work units)
Thread count was 8 (of 8 available processors)
Solution count 10: 1.20001e+09 1.40001e+09 1.47501e+09 ... 1.65001e+09
Optimal solution found (tolerance 1.00e-04)
Best objective 1.200012600000e+09, best bound 1.200003200000e+09, gap 0.0008%Custom Heuristics
When?
- Solver cannot find any solution
- Solution quality is not sufficient
- Solution improvement too slow
- Model is too large
Custom Heuristics
What can we do?
- Make model easier to be feasible:
- Relax constraints
- Use penalties for hard constraints
- Construct initial solution and hand it over as MIP start:
- Can be complete or partial
- No need for high solution quality
- Improve solutions in callbacks:
- Round LP solutions with problem-specific knowledge
- Improve integer solutions with local-search techniques
- Partition model by time, location, etc.
- Variable attribute
Partition, parameterPartitionPlace - Guides solver heuristics
- Variable attribute
Example: Rolling Horizon MIP Heuristic
Example: Local Search MIP Heuristic
Conclusion
How to handle challenging models?
- Modeling phase
- Clean up input data
- Understand model strength
- Tighten variable bounds
- Reduce Big-M values
- Add problem-specific cuts
- Reformulate model
- Solution phase
- Analyze solver logs \(\rightarrow\) Gurobi Experts
- Run custom heuristics
- Analyze model structure \(\rightarrow\) Gurobi Experts
- Tune solver parameters \(\rightarrow\) Gurobi Experts
©️ Gurobi Optimization