I developed a linear mathematical programming model in Visual Studio (C++) and solved the problem using Cplex (12.7.1). However I noticed some strange behavior of Cplex. For some problem instances, Cplex provides a feasible (non-optimal solution), that could be easily improved by removing slack on certain constraints. A simplified example of the mathematical model is as follows:
Minimize A
Subject to
cX – dY <= A
dY – cX <= A
X, Y binary, A continuous, c,d parameters
Given the values of X and Y in the provided feasible (non-optimal) solution, there is slack on both constraints. The continuous A variable could be easily reduced given the values of decision variables X and Y (i.e., by removing the slack of at least on of the two constraints). I understand that Cplex provides a solution that is feasible given the constraints of the problem. However when branching and solving the simplex in a branch to create a feasible solution, why does this simplex' calculation result in these two non-binding constraints? What can I do to ensure Cplex always provides at least a solution in which one of these two constraints is binding?
None of these tries solved the problem.
int nozones = 2;
int notrucks = 100;
int notimeslots = 24;
IloEnv env;
IloModel model(env);
IloExpr objective(env);
IloExpr constraint(env);
NumVar3Matrix X(env, notimeslots);
for (i = 0; i < notimeslots; i++)
{
X[i] = NumVarMatrix(env, notrucks);
for (l = 0; l < notrucks; l++)
{
X[i][l] = IloNumVarArray(env, nozones);
for (k = 0; k < nozones; k++)
{
X[i][l][k] = IloNumVar(env, 0, 1, ILOINT);
}
}
}
NumVar3Matrix A(env, nozones);
for (k = 0; k < nozones; k++)
{
A[k] = NumVarMatrix(env, notimeslots);
for (int i0 = 0; i0 < notimeslots; i0++)
{
A[k][i0] = IloNumVarArray(env, notimeslots);
for (int i1 = 0; i1 < notimeslots; i1++)
{
A[k][i0][i1] = IloNumVar(env, 0, 9999, ILOFLOAT);
}
}
}
//objective function
for (int k0 = 0; k0 < nozones; k0++)
{
for (int i0 = 0; i0 < notimeslots; i0++)
{
for (int i1 = 0; i1 < notimeslots; i1++)
{
if (i0 > i1)
{
double denominator = (PP.mean[k0] * (double)(notimeslots*notimeslots)); //parameter
objective += A[k0][i0][i1] / denominator;
}
}
}
}
model.add(IloMinimize(env, objective));
//Constraints
for (int k0 = 0; k0 < nozones; k0++)
{
for (int i0 = 0; i0 < notimeslots; i0++)
{
for (int i1 = 0; i1 < notimeslots; i1++)
{
if (i0 > i1)
{
for (int l0 = 0; l0 < notrucks; l0++)
{
constraint += c[k0][l0] * X[i0][l0][k0];
constraint -= d[k0][l0] * X[i1][l0][k0];
}
constraint -= A[k0][i0][i1];
model.add(constraint <= 0);
constraint.clear();
for (int l0 = 0; l0 < notrucks; l0++)
{
constraint -= c[k0][l0] * X[i0][l0][k0];
constraint += d[k0][l0] * X[i1][l0][k0];
}
constraint -= A[k0][i0][i1];
model.add(constraint <= 0);
constraint.clear();
}
}
}
}
Please find the log below:
CPXPARAM_TimeLimit 10
CPXPARAM_Threads 3
CPXPARAM_MIP_Tolerances_MIPGap 9.9999999999999995e-08
CPXPARAM_MIP_Strategy_CallbackReducedLP 0
Tried aggregator 2 times.
MIP Presolve eliminated 412 rows and 384 columns.
MIP Presolve modified 537 coefficients.
Aggregator did 21 substitutions.
Reduced MIP has 595 rows, 475 columns, and 10901 nonzeros.
Reduced MIP has 203 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.09 sec. (8.97 ticks)
Found incumbent of value 1254245.248934 after 0.11 sec. (10.55 ticks)
Probing time = 0.00 sec. (0.39 ticks)
Tried aggregator 1 time.
Reduced MIP has 595 rows, 475 columns, and 10901 nonzeros.
Reduced MIP has 203 binaries, 272 generals, 0 SOSs, and 0 indicators.
Presolve time = 0.03 sec. (4.47 ticks)
Probing time = 0.00 sec. (0.55 ticks)
Clique table members: 51.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: deterministic, using up to 3 threads.
Root relaxation solution time = 0.05 sec. (15.41 ticks)
Nodes Cuts/
Node Left Objective IInf Best Integer Best Bound ItCnt Gap
* 0+ 0 1254245.2489 13879.8564 98.89%
* 0+ 0 1225612.3997 13879.8564 98.87%
* 0+ 0 1217588.5782 13879.8564 98.86%
* 0+ 0 1209564.7566 13879.8564 98.85%
* 0+ 0 1201540.9350 13879.8564 98.84%
* 0+ 0 1193517.1135 13879.8564 98.84%
* 0+ 0 1185493.2919 13879.8564 98.83%
* 0+ 0 1177589.9029 13879.8564 98.82%
0 0 334862.8273 139 1177589.9029 334862.8273 387 71.56%
* 0+ 0 920044.8009 334862.8273 63.60%
0 0 335605.5047 162 920044.8009 Cuts: 248 516 63.52%
* 0+ 0 732802.2256 335605.5047 54.20%
* 0+ 0 669710.6005 335605.5047 49.89%
0 0 336504.5144 153 669710.6005 Cuts: 248 617 49.75%
0 0 338357.1160 172 669710.6005 Cuts: 248 705 49.48%
0 0 338950.0580 178 669710.6005 Cuts: 248 796 49.39%
0 0 339315.6848 189 669710.6005 Cuts: 248 900 49.33%
0 0 339447.9616 193 669710.6005 Cuts: 248 977 49.31%
0 0 339663.6342 203 669710.6005 Cuts: 228 1091 49.28%
0 0 339870.9021 205 669710.6005 Cuts: 210 1154 49.25%
* 0+ 0 531348.6042 339870.9021 36.04%
0 0 340009.1008 207 531348.6042 Cuts: 241 1225 35.87%
0 0 340855.1873 202 531348.6042 Cuts: 231 1318 35.85%
0 0 341229.8328 202 531348.6042 Cuts: 248 1424 35.78%
0 0 341409.5769 200 531348.6042 Cuts: 248 1502 35.75%
0 0 341615.2848 286 531348.6042 Cuts: 248 1568 35.71%
0 0 341704.8400 300 531348.6042 Cuts: 225 1626 35.69%
0 0 341805.5681 222 531348.6042 Cuts: 191 1687 35.67%
* 0+ 0 489513.3319 341805.5681 30.17%
0 0 341834.6048 218 489513.3319 Cuts: 169 1739 30.17%
0 0 341900.1390 228 489513.3319 Cuts: 205 1788 30.16%
0 0 341945.8278 211 489513.3319 Cuts: 197 1855 30.15%
* 0+ 0 489468.1697 341945.8278 30.14%
0 2 341945.8278 202 489468.1697 341945.8278 1855 30.14%
Elapsed time = 5.53 sec. (446.68 ticks, tree = 0.01 MB, solutions = 14)
* 199+ 154 484741.1904 341968.3817 29.45%
263 222 342462.1403 198 484741.1904 341968.3817 12287 29.45%
* 550+ 420 461678.3486 341993.1725 25.92%
555 403 411858.3790 117 461678.3486 341993.1725 21480 25.92%
* 566+ 319 439985.4277 341993.1725 22.27%
660 321 350009.7742 289 439985.4277 341993.1725 16141 22.27%
* 670+ 427 438464.9662 342020.7550 22.00%
Flow cuts applied: 15
Mixed integer rounding cuts applied: 65
Zero-half cuts applied: 6
Gomory fractional cuts applied: 15
Root node processing (before b&c):
Real time = 5.53 sec. (446.21 ticks)
Parallel b&c, 3 threads:
Real time = 4.50 sec. (1093.39 ticks)
Sync time (average) = 0.59 sec.
Wait time (average) = 0.04 sec.
------------
Total (root+branch&cut) = 10.03 sec. (1539.61 ticks)
The expected result is that in all feasible solutions that Cplex provides, for all pairs of constraints that at least on of them is binding (without slack).
I assume CPLEX aborted due to hitting your time limit, hence the solution was not proven to be an optimum. Is this correct ?
This is not a bug. CPLEX does not make such guarantees for a user terminated run. CPLEX stops as soon as possible, when a solution satisfying the user requests/settings is found.
To get the behavior your are looking for, then in the C API you can use :
to solve the fixed problem. Since the resulting problem is a pure LP, you can now call :
And as mentioned in the link you can use solveFixed() for higher level API's.
Daniel also answered your cross-post here :
Please reply at the IBM developer forum if anything is not clear, thank you.