Hi All,

I’m now trying to solve a large Variational Inequality problem. And I solve it by two ways, using PATH solver solve it directly and use Dantzig-Wolfe method to solve it in a decomposition way. And when I implement the second way in GAMS, I found the time to solve it is relatively short, but the execution time for the master problem will be much slower, which cause the DW method much solver compared to PATH solver in total. I am wondering if there is a method to solve this problem？

I attach the solver part of the log file:

The first is from PATH solver:

```
MODEL STATISTICS
BLOCKS OF EQUATIONS 3 SINGLE EQUATIONS 21,839
BLOCKS OF VARIABLES 5 SINGLE VARIABLES 62,774
NON ZERO ELEMENTS 184,155 NON LINEAR N-Z 81,840
DERIVATIVE POOL 20 CONSTANT POOL 46
CODE LENGTH 368,310
---- 395 Solve Fini gasEnEquil 0.073 1.634 SECS 31 MB 184155
GENERATION TIME = 0.800 SECONDS 31 MB 26.1.0 rf2b37b9 DEX-DEG
EXECUTION TIME = 1.634 SECONDS 31 MB 26.1.0 rf2b37b9 DEX-DEG
---- 395 GAMS Fini 0.029 0.029 SECS 31 MB
---- 1 InitE 0.001 0.001 SECS 14 MB
---- 1 ExecInit 0.000 0.001 SECS 14 MB
---- 395 Solve Alg gasEnEquil 0.000 0.001 SECS 14 MB
GAMS 26.1.0 rf2b37b9 Released Feb 2, 2019 DEX-DEG x86 64bit/Mac OS X 04/26/19 12:59:06 Page 1857
G e n e r a l A l g e b r a i c M o d e l i n g S y s t e m
Solution Report SOLVE gasEnEquil Using EMP From line 395
S O L V E S U M M A R Y
MODEL gasEnEquil
TYPE EMP
SOLVER JAMS FROM LINE 395
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 2 Locally Optimal
RESOURCE USAGE, LIMIT 2375.177 1000.000
ITERATION COUNT, LIMIT 26615 2000000000
EVALUATION ERRORS 0 0
JAMS 1.0 26.1.0 rf2b37b9 Released Feb 02, 2019 DEG x86 64bit/Mac OS X
JAMS - Solver for Extended Mathematical Programs (EMP)
------------------------------------------------------
--- Using Option File
Reading parameter(s) from "/Users/jiajieshen/Dropbox/optimization/FProject_1/MyTest_v3/simpleModel/simpleModel_test/DWgas_mP_v2_l_LS2/jams.opt"
>> *filename gasEnEnquil.gms
>> *dict gasEnEnquil.txt
>> subsolver path
>> subsolveropt 1
Finished reading from "/Users/jiajieshen/Dropbox/optimization/FProject_1/MyTest_v3/simpleModel/simpleModel_test/DWgas_mP_v2_l_LS2/jams.opt"
--- EMP Summary
Logical Constraints = 0
Disjunctions = 0
Adjusted Constraint = 0
Flipped Constraints = 0
Dual Variable Maps = 0
Dual Equation Maps = 0
VI Functions = 1364
QVI Parameters = 0
Equilibrium Agent = 16
Bilevel Followers = 0
--- The model /Users/jiajieshen/Dropbox/optimization/FProject_1/MyTest_v3/simpleModel/simpleModel_test/DWgas_mP_v2_l_LS2/225a/emp.dat will be solved by GAMS
---
--- Returning from GAMS step
83204 row/cols, 265860 non-zeros, 0.00% dense.
```

and one iteration of my master problem will be:

```
MODEL STATISTICS
BLOCKS OF EQUATIONS 3 SINGLE EQUATIONS 1,468
BLOCKS OF VARIABLES 3 SINGLE VARIABLES 1,468
NON ZERO ELEMENTS 282,391 NON LINEAR N-Z 0
DERIVATIVE POOL 20 CONSTANT POOL 16
CODE LENGTH 0
GENERATION TIME = 0.294 SECONDS 607 MB 26.1.0 rf2b37b9 DEX-DEG
EXECUTION TIME = 131.764 SECONDS 607 MB 26.1.0 rf2b37b9 DEX-DEG
L O O P S FOR/WHILE 102
GAMS 26.1.0 rf2b37b9 Released Feb 2, 2019 DEX-DEG x86 64bit/Mac OS X 04/26/19 12:40:01 Page 234037
the main file for Dantzig-Wolfe algorithm for equilibrium
Solution Report SOLVE gasEnMaster Using MCP From line 767
S O L V E S U M M A R Y
MODEL gasEnMaster
TYPE MCP
SOLVER PATH FROM LINE 767
**** SOLVER STATUS 1 Normal Completion
**** MODEL STATUS 1 Optimal
RESOURCE USAGE, LIMIT 0.735 1000.000
ITERATION COUNT, LIMIT 618 2000000000
EVALUATION ERRORS 0 0
1468 row/cols, 282391 non-zeros, 13.10% dense.
```

You can see that my master problem size is smaller than the original problem, only the nonzero part is more than the original problem: (282,391 vs 184,155), however, the execution time is much larger(131.764 seconds vs 1.634 seconds) and the memory is much larger(607 MB vs 31 MB). And the master problem solving time is much faster (Resource usage: 0.735 vs 2375.177).

So I think the execution time can be improved in some way. Does anyone has experience on that?

Thanks