纸浆挂在不可行的 integer LP 问题上

Question

I have a loop which solves a large number of integer LP problems using PuLP. At some point it comes across something like 2a + 2b = 1 . This is obviously infeasible, and yet PuLP just hangs on this input.

I've tried solving this using CBC and it instantly returns the correct result -- infeasible or unbounded. So it's not an issue with CBC.

Here's code that reproduces the problem:

from pulp import *

a = LpVariable('a', cat=LpInteger)
b = LpVariable('b', cat=LpInteger)

prob = LpProblem()
prob += 2*a + 2*b == 1

prob.solve(solver=PULP_CBC_CMD())
print(prob.status)

Answer 1

The problem does not come from PuLP (PuLP is a modeling library) but from the CBC solver.

Examples of outputs I get:

...
Cbc0010I After 35000 nodes, 30979 on tree, 1e+50 best solution, best possible 0 (666.85 seconds)
Cbc0010I After 36000 nodes, 31979 on tree, 1e+50 best solution, best possible 0 (691.99 seconds)
Cbc0010I After 37000 nodes, 32979 on tree, 1e+50 best solution, best possible 0 (717.61 seconds)
Cbc0010I After 38000 nodes, 33977 on tree, 1e+50 best solution, best possible 0 (742.93 seconds)
...

1e+50 as a best solution means that a feasible point has not been found in the branching tree (no incumbent).
However, it goes on with branching.

With the other solvers I've tried the answer is immediate.

GLPK

GLPK Integer Optimizer, v4.60
1 row, 3 columns, 2 non-zeros
2 integer variables, none of which are binary
Preprocessing...
1 row, 0 columns, 0 non-zeros
0 integer variables, none of which are binary
Scaling...
 A: min|aij| =  1.000e+00  max|aij| =  1.000e+00  ratio =  1.000e+00
Problem data seem to be well scaled
Solving LP relaxation...
GLPK Simplex Optimizer, v4.60
1 row, 0 columns, 0 non-zeros
~     0: obj =   0.000000000e+00  infeas =  1.000e+00
PROBLEM HAS NO FEASIBLE SOLUTION
Time used:   0.0 secs
Memory used: 0.0 Mb (40416 bytes)

CPLEX

Presolve time = 0.00 sec. (0.00 ticks)

Root node processing (before b&c):
  Real time             =    0.00 sec. (0.00 ticks)
Parallel b&c, 4 threads:
  Real time             =    0.00 sec. (0.00 ticks)
  Sync time (average)   =    0.00 sec.
  Wait time (average)   =    0.00 sec.
                          ------------
Total (root+branch&cut) =    0.00 sec. (0.00 ticks)


MIP - Integer infeasible.
Current MIP best bound =  0.0000000000e+00 (gap is infinite)
Solution time =    0.00 sec.  Iterations = 0  Nodes = 0
Deterministic time = 0.00 ticks  (2.89 ticks/sec)

GUROBI

OBJ: 1 rows, 3 columns, 2 nonzeros
Optimize a model with 1 rows, 3 columns and 2 nonzeros
Variable types: 1 continuous, 2 integer (0 binary)
Coefficient statistics:
  Matrix range     [2e+00, 2e+00]
  Objective range  [1e+00, 1e+00]
  Bounds range     [0e+00, 0e+00]
  RHS range        [1e+00, 1e+00]
Presolve removed 0 rows and 2 columns
Presolve time: 0.00s

Solution count 0

Model is infeasible