Obtaining Suboptimal Solutions with Benders

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diabolik
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Posts: 14
Joined: 1 year ago

Obtaining Suboptimal Solutions with Benders

Post by diabolik »

Hello everyone,

I have the following Benders code for a maximization problem on GAMS. It happens that sometimes GAMS give an suboptimal solution even though it should find the exact solution, since no heuristics exist and Benders tolerance (BdTol= Upper Bound - Lower Bound) is 0. I ensure that both the subproblem and the master problem are solved to optimality and optcr and optca values are taken as 0, too. I realized it as follows: for a dataset I solved the problem without enforcing any variable to take any value, and it provided an objective value. Then, I enforced a variable to take a certain value, and it provided a higher objective value. Shouldn't GAMS find the highest possible value automatically as it is a maximization problem? Do you think there is something wrong with the code?

Code: Select all

Sets
i   /1*100/
j   /1*10/;

Parameters
p /4/
m /1/
BdTol /0/;

Table a(i,j) DATA;

Table b(i,j) DATA;

variables
q 'cut set variable';

binary variables
y(j)
x(i,j);

positive variables
w(j);

*---------------------------------------------------------------------
* Benders Decomposition Initialization
*---------------------------------------------------------------------
display "--------------------- BENDERS ALGORITHM ----------------------------";

scalar UB 'upperbound' /INF/;
scalar LB 'lowerbound' /-INF/;

*Initial values for the master problem variables are given to initiate the algorithm and solving the subproblem.

x.l(i,j)=1;
y.l(j)=1;
q.l=0;

*---------------------------------------------------------------------
* Benders Subproblem
*---------------------------------------------------------------------
variables
z;

equations
subobj          'objective'
subconstr  'dual constraint'
subconstr2;

subobj..z=e=-sum((i,j),b(i,j)*x.l(i,j)*w(j));
subconstr..sum(j,w(j))=e=m;
subconstr2(j)..w(j)=l=y.l(j);
w.up(j)=1;

model subproblem /subobj, subconstr, subconstr2/;

*---------------------------------------------------------------------
* Benders Relaxed Master Problem
*---------------------------------------------------------------------
set iter /iter1*iter1000000/;

set cutset(iter) 'dynamic set';
cutset(iter)=no;

variable z0 'relaxed master objective variable'
q 'cut set variable';

parameters
cutcoeff(iter,i,j);

Equations
master_obj
cut(iter)           'Benders cut for optimal subproblem'
master_constr1
master_constr2
master_constr3;

master_obj..z0=e=sum((i,j),a(i,j)*x(i,j))+q;
cut(cutset)..q=l=-sum((i,j),cutcoeff(cutset,i,j)*x(i,j));
master_constr1(i)..sum(j,x(i,j))=l=1;
master_constr2..sum(j,y(j))=e=p;
master_constr3(i,j)..x(i,j)=l=y(j);

model master /master_obj,cut,master_constr1,master_constr2,master_constr3/;

*---------------------------------------------------------------------
* Benders Algorithm
*---------------------------------------------------------------------
loop(iter$((abs(UB-LB)> BdTol)),
         option lp=CPLEX, optcr=0, optca=0, reslim=1000000, iterlim=1000000;
         solve subproblem maximizing z using lp;
         abort$(subproblem.modelstat>=2) "Subproblem not solved to optimality";
         UB = min(UB, sum((i,j),a(i,j)*x.l(i,j))+q.l);
         cutset(iter) = yes;
         cutcoeff(iter,i,j) = b(i,j)*w.l(j);
         option mip=CPLEX, optcr=0, optca=0, reslim=1000000, iterlim=1000000;
         solve master maximizing z0 using mip;
         abort$(master.modelstat>=2) "Masterproblem not solved to optimality";
         LB = max(LB, sum((i,j),a(i,j)*x.l(i,j))+z.l);
         display x.l, y.l, z0.l, w.l, UB, LB;
);
diabolik
User
User
Posts: 14
Joined: 1 year ago

Re: Obtaining Suboptimal Solutions with Benders

Post by diabolik »

What really interesting is, it finds suboptimal solutions only for some instances, not all the time.

The runs for a dataset is given below. The upper ones does not involve any enforcing constraints for a variable to take a certain value. The lower one involves a single variable to take a specific value.
image.png
image.png (10.38 KiB) Viewed 964 times
The following is the log output for the one without the enforced variable.

Code: Select all

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :         -12.203660

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:04.895
--- benders3.gms(358) 88 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter1
---   4,107 rows  4,016 columns  9,820 non-zeroes
---   1,010 discrete-columns
--- Executing CPLEX: elapsed 0:00:04.902

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3387 rows and 3387 columns.
Reduced MIP has 720 rows, 629 columns, and 1867 nonzeros.
Reduced MIP has 629 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.00 sec.
Tried aggregator 1 time.
Presolve time =    0.00 sec.
Found feasible solution after 0.00 sec.  Objective = 0.0000
Probing time =    0.00 sec.
Clique table members: 719.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           362.425771
Iteration:   129   Dual objective     =            96.848559
Iteration:   198   Dual objective     =            94.968744
Root relaxation solution time =    0.00 sec.

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

*     0+    0                            0.0000                    249     --- 
*     0     0      integral     0       93.1164       93.1164      249    0.00%
Elapsed real time =   0.00 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.00
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.00 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4107 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           93.116376    (249 iterations, 0 nodes)
Final Solve:            93.116376    (0 iterations)

Best possible:          93.116376
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:04.954
--- benders3.gms(358) 89 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 89 Mb
--- LOOPS iter = iter2
---   12 rows  11 columns  25 non-zeroes
--- Executing CPLEX: elapsed 0:00:04.959

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :          -2.121247

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:04.974
--- benders3.gms(358) 89 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter2
---   4,108 rows  4,016 columns  9,863 non-zeroes
---   1,010 discrete-columns
--- Executing CPLEX: elapsed 0:00:04.982

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3354 rows and 3354 columns.
Reduced MIP has 754 rows, 662 columns, and 2034 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.00 sec.
Tried aggregator 1 time.
Reduced MIP has 754 rows, 662 columns, and 2034 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.09 sec.
Found feasible solution after 0.09 sec.  Objective = -13.6132
Probing time =    0.00 sec.
Clique table members: 751.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           365.039800
Iteration:   141   Dual objective     =            96.728338
Iteration:   211   Dual objective     =            94.098252
Root relaxation solution time =    0.01 sec.

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

*     0+    0                          -13.6132                    246     --- 
*     0     0      integral     0       93.0748       93.0748      246    0.00%
Elapsed real time =   0.11 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.01
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.01 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4108 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           93.074808    (246 iterations, 0 nodes)
Final Solve:            93.074808    (0 iterations)

Best possible:          93.074808
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:05.115
--- benders3.gms(358) 89 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 89 Mb
--- LOOPS iter = iter3
---   12 rows  11 columns  25 non-zeroes
--- Executing CPLEX: elapsed 0:00:05.120

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :          -2.847644

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:05.132
--- benders3.gms(358) 89 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter3
---   4,109 rows  4,016 columns  9,967 non-zeroes
---   1,010 discrete-columns
--- Executing CPLEX: elapsed 0:00:05.141

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3354 rows and 3354 columns.
Reduced MIP has 755 rows, 662 columns, and 2105 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.00 sec.
Tried aggregator 1 time.
Reduced MIP has 755 rows, 662 columns, and 2105 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.00 sec.
Found feasible solution after 0.00 sec.  Objective = -32.0161
Probing time =    0.00 sec.
Clique table members: 751.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           346.636915
Iteration:   142   Dual objective     =            96.564506
Iteration:   214   Dual objective     =            93.644333
Root relaxation solution time =    0.02 sec.

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

*     0+    0                          -32.0161                    252     --- 
*     0     0      integral     0       92.6192       92.6192      252    0.00%
Elapsed real time =   0.09 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.09
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.09 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4109 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.01 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           92.619154    (252 iterations, 0 nodes)
Final Solve:            92.619154    (0 iterations)

Best possible:          92.619154
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:05.283
--- benders3.gms(358) 89 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 89 Mb
--- LOOPS iter = iter4
---   12 rows  11 columns  24 non-zeroes
--- Executing CPLEX: elapsed 0:00:05.290

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :           0.000000

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:05.304
--- benders3.gms(358) 89 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter4
---   4,110 rows  4,016 columns  10,038 non-zeroes
---   1,010 discrete-columns
--- Executing CPLEX: elapsed 0:00:05.311

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3355 rows and 3354 columns.
Reduced MIP has 755 rows, 662 columns, and 2105 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.02 sec.
Tried aggregator 1 time.
Reduced MIP has 755 rows, 662 columns, and 2105 nonzeros.
Reduced MIP has 661 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.09 sec.
Found feasible solution after 0.09 sec.  Objective = -32.0161
Probing time =    0.00 sec.
Clique table members: 751.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           346.636915
Iteration:   142   Dual objective     =            96.564506
Iteration:   214   Dual objective     =            93.644333
Root relaxation solution time =    0.00 sec.

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

*     0+    0                          -32.0161                    252     --- 
*     0     0      integral     0       92.6192       92.6192      252    0.00%
Elapsed real time =   0.09 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.02
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.02 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4110 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           92.619154    (252 iterations, 0 nodes)
Final Solve:            92.619154    (0 iterations)

Best possible:          92.619154
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:05.434
--- benders3.gms(358) 89 Mb
*** Status: Normal completion
--- Job benders3.gms Stop 12/10/22 14:37:01 elapsed 0:00:05.481
The following is the log output for the one with the enforced variable.

Code: Select all

--- benders3.gms(151) 86 Mb
--- Starting execution: elapsed 0:00:04.400
--- benders3.gms(358) 88 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 88 Mb
--- LOOPS iter = iter1
---   12 rows  11 columns  31 non-zeroes
--- Executing CPLEX: elapsed 0:00:04.406

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :         -12.203660

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:04.417
--- benders3.gms(358) 88 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter1
---   4,107 rows  4,016 columns  9,820 non-zeroes
---   1,009 discrete-columns
--- Executing CPLEX: elapsed 0:00:04.425

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3547 rows and 3500 columns.
Reduced MIP has 560 rows, 516 columns, and 1458 nonzeros.
Reduced MIP has 516 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.00 sec.
Tried aggregator 2 times.
MIP Presolve eliminated 1 rows and 1 columns.
Aggregator did 1 substitutions.
Reduced MIP has 558 rows, 514 columns, and 1454 nonzeros.
Reduced MIP has 514 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.01 sec.
Found feasible solution after 0.01 sec.  Objective = 25.1590
Probing time =    0.00 sec.
Clique table members: 557.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           213.565458
Iteration:    77   Dual objective     =            96.476340
Iteration:   150   Dual objective     =            93.814763
Root relaxation solution time =    0.00 sec.

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

*     0+    0                           25.1590                    193     --- 
*     0     0      integral     0       93.1164       93.1164      193    0.00%
Elapsed real time =   0.01 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.00
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.00 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4107 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           93.116376    (193 iterations, 0 nodes)
Final Solve:            93.116376    (0 iterations)

Best possible:          93.116376
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:04.551
--- benders3.gms(358) 89 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 89 Mb
--- LOOPS iter = iter2
---   12 rows  11 columns  25 non-zeroes
--- Executing CPLEX: elapsed 0:00:04.555

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :          -2.121247

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:04.568
--- benders3.gms(358) 89 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter2
---   4,108 rows  4,016 columns  9,863 non-zeroes
---   1,009 discrete-columns
--- Executing CPLEX: elapsed 0:00:04.576

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3460 rows and 3413 columns.
Reduced MIP has 648 rows, 603 columns, and 1781 nonzeros.
Reduced MIP has 602 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.00 sec.
Tried aggregator 1 time.
Reduced MIP has 648 rows, 603 columns, and 1781 nonzeros.
Reduced MIP has 602 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.00 sec.
Found feasible solution after 0.00 sec.  Objective = -2.6132
Probing time =    0.00 sec.
Clique table members: 645.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           339.579984
Iteration:   130   Dual objective     =            96.654157
Iteration:   208   Dual objective     =            93.001859
Root relaxation solution time =    0.09 sec.

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

*     0+    0                           -2.6132                    213     --- 
*     0     0      integral     0       92.8258       92.8258      213    0.00%
Elapsed real time =   0.09 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.09
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.09 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4108 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           92.825775    (213 iterations, 0 nodes)
Final Solve:            92.825775    (0 iterations)

Best possible:          92.825775
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:04.703
--- benders3.gms(358) 89 Mb
--- Generating LP model subproblem
--- benders3.gms(360) 89 Mb
--- LOOPS iter = iter3
---   12 rows  11 columns  24 non-zeroes
--- Executing CPLEX: elapsed 0:00:04.707

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
LP Presolve eliminated 12 rows and 11 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
LP status(1): optimal

Optimal solution found.
Objective :           0.000000

--- Reading solution for model subproblem
--- Executing after solve: elapsed 0:00:04.720
--- benders3.gms(358) 89 Mb
--- Generating MIP model master
--- benders3.gms(367) 89 Mb
--- LOOPS iter = iter3
---   4,109 rows  4,016 columns  9,901 non-zeroes
---   1,009 discrete-columns
--- Executing CPLEX: elapsed 0:00:04.727

IBM ILOG CPLEX   Jul  4, 2012 23.9.5 WEX 36376.36401 WEI x86_64/MS Windows
--- GAMS/Cplex licensed for continuous and discrete problems.
Cplex 12.4.0.1

Reading data...
Starting Cplex...
Tried aggregator 1 time.
MIP Presolve eliminated 3461 rows and 3413 columns.
Reduced MIP has 648 rows, 603 columns, and 1781 nonzeros.
Reduced MIP has 602 binaries, 0 generals, 0 SOSs, and 0 indicators.
Probing time =    0.01 sec.
Tried aggregator 1 time.
Reduced MIP has 648 rows, 603 columns, and 1781 nonzeros.
Reduced MIP has 602 binaries, 0 generals, 0 SOSs, and 0 indicators.
Presolve time =    0.09 sec.
Found feasible solution after 0.09 sec.  Objective = -2.6132
Probing time =    0.00 sec.
Clique table members: 645.
MIP emphasis: balance optimality and feasibility.
MIP search method: dynamic search.
Parallel mode: none, using 1 thread.
Tried aggregator 1 time.
No LP presolve or aggregator reductions.
Presolve time =    0.00 sec.
Initializing dual steep norms . . .

Iteration log . . .
Iteration:     1   Dual objective     =           339.579984
Iteration:   130   Dual objective     =            96.654157
Iteration:   208   Dual objective     =            93.001859
Root relaxation solution time =    0.00 sec.

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

*     0+    0                           -2.6132                    213     --- 
*     0     0      integral     0       92.8258       92.8258      213    0.00%
Elapsed real time =   0.11 sec. (tree size =  0.00 MB, solutions = 2)

Root node processing (before b&c):
  Real time             =    0.02
Sequential b&c:
  Real time             =    0.00
                          -------
Total (root+branch&cut) =    0.02 sec.
MIP status(101): integer optimal solution
Fixing integer variables, and solving final LP...
Tried aggregator 1 time.
LP Presolve eliminated 4109 rows and 4016 columns.
All rows and columns eliminated.
Presolve time =    0.00 sec.
Fixed MIP status(1): optimal

Proven optimal solution.

MIP Solution:           92.825775    (213 iterations, 0 nodes)
Final Solve:            92.825775    (0 iterations)

Best possible:          92.825775
Absolute gap:            0.000000
Relative gap:            0.000000

--- Reading solution for model master
--- Executing after solve: elapsed 0:00:04.861
--- benders3.gms(358) 89 Mb
*** Status: Normal completion
--- Job benders3.gms Stop 12/10/22 14:25:56 elapsed 0:00:04.910
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