【发布时间】：2019-08-27 03:26:54
【问题描述】：
我在 Visual Studio (C++) 中开发了一个线性数学规划模型，并使用 Cplex (12.7.1) 解决了这个问题。但是我注意到 Cplex 的一些奇怪行为。对于某些问题实例，Cplex 提供了一个可行的（非最佳解决方案），可以通过消除某些约束的松弛来轻松改进。数学模型的简化示例如下：
最小化 A
服从
cX - dY
dY - cX
X、Y二进制、A连续、c、d参数
在所提供的可行（非最优）解决方案中给定 X 和 Y 的值，这两个约束都存在松弛。给定决策变量 X 和 Y 的值（即，通过消除两个约束中的至少一个的松弛），可以轻松地减少连续变量 A。我知道 Cplex 提供了一个在问题的约束下可行的解决方案。但是，当在分支中对单纯形进行分支求解以创建可行解时，为什么这个单纯形的计算会导致这两个非约束约束？我可以做些什么来确保 Cplex 始终至少提供一个解决方案，其中绑定了这两个约束之一？
- 我尝试包含没有松弛的解决方案，以测试预期的解决方案是否被 Cplex 识别为可行的解决方案（即，为了测试用 C++ 编程的数学模型是否没有错误）；
- 我尝试增加 Cplex 的容差 (IloCplex::Param::MIP::Tolerances::MIPGap)；
- 我尝试切换 Cplex 的动态搜索 (IloCplex::Param::MIP::Strategy::Search)。
这些尝试都没有解决问题。
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();
}
}
}
}
请在下面找到日志：
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)
预期的结果是，在 Cplex 提供的所有可行解决方案中，对于至少有一个约束的所有约束对（没有松弛）。
【问题讨论】：
-
提供完整的引擎日志确实有助于找出可能出现的问题。
标签： c++ optimization mathematical-optimization cplex