问题
This is a part of the engine-log output that I get from a small-scale mixed integer linear optimization problem that I solved in CPLEX 12.7.0
Nodes Cuts/
Node Left Objective IInf Best Integer Best Bound ItCnt Gap
0 0 280.0338 78 280.0338 72
0 0 428.8558 28 Cuts: 89 137
0 0 429.5221 34 Cuts: 2 142
0 0 429.7745 34 MIRcuts: 2 143
* 0+ 0 460.9166 429.7745 6.76%
0 2 429.7745 34 460.9166 429.8666 143 6.74%
Elapsed time = 0.49 sec. (31.07 ticks, tree = 0.01 MB, solutions = 1)
* 35 8 integral 0 438.1448 435.6381 211 0.57%
Cover cuts applied: 17
Implied bound cuts applied: 10
Flow cuts applied: 11
Mixed integer rounding cuts applied: 9
Gomory fractional cuts applied: 24
Root node processing (before b&c):
Real time = 0.45 sec. (31.09 ticks)
Sequential b&c:
Real time = 0.08 sec. (20.80 ticks)
------------
Total (root+branch&cut) = 0.53 sec. (51.89 ticks)
What I understand from this, is that the best integer solution (for the objective function) found has the value of 438.1448, whereas the relaxed solution (non integer values) has the value of 435.6381 as best bound solution.
( 438.1448 / 435.6381 ) - 1 = 0.57% GAP
Does this mean that the solution still has that small gap, however it is proven to be the optimal solution? I had the (maybe wrong) idea that optimality is proven by a 0% gap.
I'm not sure how to interpret it correctly. Thanks for your help in advance.
回答1:
Yes you are right. The optimality is proven if the upper bound and the lower bound evaluate the same value, i.e. CPLEX could prove an optimality gap of 0%.
Since CPLEX stops with a solution that has a gap of 0.57%, I would assume that you configured an MIP-gap <1%. If you are interested in a solution with proven optimal, you should change the MIPGap parameter to zero. See also here.
回答2:
Your understanding of the best bound isn't 100% correct. You can think of the best bound as the best objective value an integer solution could potentially have, based on information the solver has discovered so far. In your case there might actually be a better solution than the one you found, but if there is, it won't have an objective value better than 435.6381.
A more technical definition of the best bound is the best relaxed-but-region-constrained solution for any region that has not yet been eliminated from the search space. Solvers like CPLEX search for an optimal solution by splitting the search space into sub-regions and then ruling out sub-regions that can't possibly contain the optimal integer-feasible solution. These sub-regions get split into sub-sub-regions, and so on. Within each region, the original problem is modified to force variables to fall within the region. The relaxed solution to this modified problem is the best bound for the region. The best of these region-specific best bounds is the best bound for the problem as a whole.
The best bound changes as regions are ruled out. If the best bound does not equal the best solution, then by definition, there is still at least one region other than the region holding the current incumbent that could potentially hold a better solution. Exploring one of these regions might uncover an even better solution than your current incumbent, or it might lead to the region being ruled out. You don't know which until the region is explored. Only when the best solution equals the best bound do you know for sure that there isn't a better solution hiding in a remaining region.
来源：https://stackoverflow.com/questions/43136862/interpretation-of-gap-in-cplex