Sometimes a commercial solver is really better...

Solving a model with an integer valued objective, with 500 binary variables, gave some interesting results.

• Cplex. Optimal solution \(z=60\) found in about 8 seconds. Using 4 threads on an old laptop. Log is below.
• CBC. Hit timelimit of 2 hours. Objective = 62 (non-optimal). Also using 4 threads on the same machine. The strange thing is that with CBC the best possible bound is not changing at all. Not by a millimeter. See the highlighted numbers in the log below. CBC is a very good solver, but sometimes I see things like this.
• Gurobi and Xpress solve this problem very fast. Ran on NEOS via an MPS file. 
• SCIP has also problems with this model. Ran on NEOS via an MPS file.

All runs were with default settings. It looks like the commercial solvers do a much better job than the open source and academic codes. Sometimes you get what you pay for.

Cplex Log

Tried aggregator 2 times.
MIP Presolve eliminated 4 rows and 5 columns.
MIP Presolve modified 3 coefficients.
Aggregator did 500 substitutions.
Reduced MIP has 999 rows, 1499 columns, and 2497 nonzeros.
Reduced MIP has 500 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.02 sec. (7.38 ticks)
Found incumbent of value 500.000000 after 0.03 sec. (8.85 ticks)
Probing time = 0.00 sec. (0.20 ticks)
Tried aggregator 1 time.
Reduced MIP has 999 rows, 1499 columns, and 2497 nonzeros.
Reduced MIP has 500 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.00 sec. (5.26 ticks)
Probing time = 0.00 sec. (0.19 ticks)
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: deterministic, using up to 4 threads.
Parallel mode: deterministic, using up to 3 threads for concurrent optimization.
Tried aggregator 1 time.
LP Presolve eliminated 999 rows and 1499 columns.
All rows and columns eliminated.
Presolve time = 0.02 sec. (0.71 ticks)
Initializing dual steep norms . . .
Root relaxation solution time = 0.02 sec. (1.15 ticks)

        Nodes                                         Cuts/
   Node  Left     Objective  IInf  Best Integer    Best Bound    ItCnt     Gap

*     0+    0                          500.0000        0.0000           100.00%
Found incumbent of value 500.000000 after 0.06 sec. (16.78 ticks)
      0     0       55.2664   490      500.0000       55.2664        0   88.95%
*     0+    0                           75.0000       55.2664            26.31%
Found incumbent of value 75.000000 after 0.08 sec. (21.35 ticks)
      0     0       55.2825   353       75.0000     Cuts: 349      222   26.29%
      0     0       55.4160   311       75.0000     Cuts: 349      455   26.11%
      0     0       55.4160   275       75.0000     Cuts: 349      775   26.11%
*     0+    0                           72.0000       55.4160            23.03%
Found incumbent of value 72.000000 after 0.59 sec. (116.10 ticks)
      0     0       55.4160   182       72.0000     Cuts: 349     1032   23.03%
      0     0       55.4160   141       72.0000     Cuts: 349     1208   23.03%
      0     0       55.4160   121       72.0000     Cuts: 192     1336   23.03%
      0     0       55.4160   105       72.0000     Cuts: 128     1410   23.03%
      0     0       55.4160    98       72.0000      Cuts: 85     1472   23.03%
      0     0       55.4160    61       72.0000      Cuts: 74     1494   23.03%
      0     0       55.4160    67       72.0000     Cuts: 160     1571   23.03%
      0     2       55.4160    56       72.0000       55.4160     1571   23.03%
Elapsed time = 1.33 sec. (273.54 ticks, tree = 0.01 MB, solutions = 3)
   1239   854       55.4160    59       72.0000       55.4160     3845   23.03%
                                                     Cuts: 38                  
*  1471+ 1230                           71.0000       55.4160            21.95%
                                                     Cuts: 16                  
Found incumbent of value 71.000000 after 3.39 sec. (827.52 ticks)
*  1474+ 1230                           70.0000       55.4160            20.83%
Found incumbent of value 70.000000 after 3.39 sec. (828.63 ticks)
*  1481+ 1230                           69.0000       55.4160            19.69%
Found incumbent of value 69.000000 after 3.41 sec. (831.79 ticks)
*  1490+ 1230                           68.0000       55.4160            18.51%
Found incumbent of value 68.000000 after 3.44 sec. (834.87 ticks)
*  1731+ 1033                           67.0000       58.7491            12.31%
Found incumbent of value 67.000000 after 6.63 sec. (1824.17 ticks)
*  1731+  688                           65.0000       59.0933             9.09%
Found incumbent of value 65.000000 after 7.25 sec. (2079.31 ticks)
*  1731+  458                           62.0000       60.0000             3.23%
Found incumbent of value 62.000000 after 7.47 sec. (2162.68 ticks)
*  1731+  305                           61.0000       60.0000             1.64%
Found incumbent of value 61.000000 after 8.20 sec. (2495.45 ticks)
*  1731+    0                           60.0000       60.0000             0.00%
Found incumbent of value 60.000000 after 8.58 sec. (2603.40 ticks)
*  1731     0      integral     0       60.0000       60.0000    10141    0.00%
Found incumbent of value 60.000000 after 8.61 sec. (2604.50 ticks)

Cover cuts applied:  150
Implied bound cuts applied:  14
Flow cuts applied:  37
Mixed integer rounding cuts applied:  249
Gomory fractional cuts applied:  26

Root node processing (before b&c):
  Real time             =    1.31 sec. (273.18 ticks)
Parallel b&c, 4 threads:
  Real time             =    7.30 sec. (2331.56 ticks)
  Sync time (average)   =    0.25 sec.
  Wait time (average)   =    0.01 sec.
                          ------------
Total (root+branch&cut) =    8.61 sec. (2604.75 ticks)
MIP status(101): integer optimal solution

CBC Log

Calling CBC main solution routine...
Integer solution of 74 found by feasibility pump after 0 iterations and 0 nodes (2.78 seconds)
Integer solution of 72 found by RINS after 0 iterations and 0 nodes (2.96 seconds)
128 added rows had average density of 31.601563
At root node, 128 cuts changed objective from 55.26645 to 55.416026 in 10 passes
Cut generator 0 (Probing) - 367 row cuts average 2.1 elements, 0 column cuts (12 active)  in 0.022 seconds - new frequency is 1
Cut generator 1 (Gomory) - 598 row cuts average 26.1 elements, 0 column cuts (0 active)  in 0.090 seconds - new frequency is 1
Cut generator 2 (Knapsack) - 14 row cuts average 11.9 elements, 0 column cuts (0 active)  in 0.037 seconds - new frequency is -100
Cut generator 3 (Clique) - 0 row cuts average 0.0 elements, 0 column cuts (0 active)  in 0.003 seconds - new frequency is -100
Cut generator 4 (MixedIntegerRounding2) - 368 row cuts average 11.2 elements, 0 column cuts (0 active)  in 0.023 seconds - new frequency is 1
Cut generator 5 (FlowCover) - 394 row cuts average 2.8 elements, 0 column cuts (0 active)  in 0.041 seconds - new frequency is 1
Cut generator 6 (TwoMirCuts) - 598 row cuts average 35.5 elements, 0 column cuts (0 active)  in 0.072 seconds - new frequency is -100
After 0 nodes, 1 on tree, 72 best solution, best possible 55.416026 (3.94 seconds)
Integer solution of 70 found by heuristic after 4138 iterations and 57 nodes (14.24 seconds)
Integer solution of 69 found by heuristic after 12712 iterations and 287 nodes (23.32 seconds)
Integer solution of 66 found by heuristic after 18149 iterations and 487 nodes (25.94 seconds)
After 1005 nodes, 555 on tree, 66 best solution, best possible 55.416026 (31.73 seconds)
Integer solution of 65 found by heuristic after 32591 iterations and 1015 nodes (33.93 seconds)
Integer solution of 64 found by heuristic after 43252 iterations and 1405 nodes (39.79 seconds)
Integer solution of 63 found by heuristic after 52409 iterations and 1805 nodes (44.62 seconds)
After 2017 nodes, 1104 on tree, 63 best solution, best possible 55.416026 (45.77 seconds)
After 3045 nodes, 1653 on tree, 63 best solution, best possible 55.416026 (52.29 seconds)
After 4099 nodes, 2212 on tree, 63 best solution, best possible 55.416026 (55.02 seconds)
After 5163 nodes, 2769 on tree, 63 best solution, best possible 55.416026 (57.08 seconds)

. . .

After 131221 nodes, 37229 on tree, 63 best solution, best possible 55.416026 (610.94 seconds)
After 132282 nodes, 37230 on tree, 63 best solution, best possible 55.416026 (613.18 seconds)
After 133298 nodes, 37231 on tree, 63 best solution, best possible 55.416026 (615.50 seconds)
After 134320 nodes, 37318 on tree, 63 best solution, best possible 55.416026 (621.40 seconds)
Integer solution of 62 found by heuristic after 2839018 iterations and 134502 nodes (621.99 seconds)
After 135334 nodes, 37422 on tree, 62 best solution, best possible 55.416026 (627.07 seconds)
After 136352 nodes, 37407 on tree, 62 best solution, best possible 55.416026 (631.83 seconds)
After 137391 nodes, 37395 on tree, 62 best solution, best possible 55.416026 (637.74 seconds)
After 138400 nodes, 37385 on tree, 62 best solution, best possible 55.416026 (642.25 seconds)

 . . .

After 1566320 nodes, 37408 on tree, 62 best solution, best possible 55.416026 (7177.35 seconds)
After 1567325 nodes, 37412 on tree, 62 best solution, best possible 55.416026 (7182.33 seconds)
After 1568336 nodes, 37421 on tree, 62 best solution, best possible 55.416026 (7187.07 seconds)
After 1569358 nodes, 37402 on tree, 62 best solution, best possible 55.416026 (7192.93 seconds)
After 1570410 nodes, 37400 on tree, 62 best solution, best possible 55.416026 (7199.07 seconds)
Thread 0 used 30010 times,  waiting to start 2696,  0 locks, 0 locked, 0 waiting for locks
Thread 1 used 30010 times,  waiting to start 2480,  0 locks, 0 locked, 0 waiting for locks
Thread 2 used 30010 times,  waiting to start 2083,  0 locks, 0 locked, 0 waiting for locks
Thread 3 used 30010 times,  waiting to start 1770,  0 locks, 0 locked, 0 waiting for locks
Main thread 6737 waiting for threads,  0 locks, 0 locked, 0 waiting for locks
Exiting on maximum time
Partial search - best objective 62 (best possible 55.416026), took 19832056 iterations and 1570713 nodes (7202.49 seconds)
Strong branching done 3540964 times (6897 iterations), fathomed 137838 nodes and fixed 779663 variables
Maximum depth 198, 2057372 variables fixed on reduced cost

Time limit reached. Have feasible solution.
MIP solution:    6.200000e+01   (1570713 nodes, 7202.62 CPU seconds, 7202.62 wall clock seconds)