Benchmark and profile the performance of IP solvers in our using case

[GiggleLiu]

• Gurobi
• HiGHS
• SCIP

[ArrogantGao]

I benchmarked some different solvers supporting JuMP.jl, both IP and LP, see https://github.com/ArrogantGao/JuMPSolver_benchmarks

I used a bottle neck case for solving the MIS problem, see https://arrogantgao.github.io/OptimalBranching.jl/dev/generated/rule_discovery/#Bottleneck-case, which take about 15000 variables and 71 constraints.
The results are summarized as below.

Integer programming:

Solver	CX	Time
SCIP	1.0932155031	0.62172113
HiGHS	1.0932155031	1.04424529
GLPK	1.0932155031	0.18660210

and Cbc crash directly

Linear programming:

Solver	cx	Time
Ipopt	1.1213026185	1.194681631
ECOS	1.1240009475	6.343032757
Cbc	1.0932155031	0.069755896
GLPK	1.0932155031	0.182092119
HiGHS	1.0932155031	0.126678814
MadNLP	1.1202363410	3.143519999
SCIP	1.0985375825	0.582486811
SDPA	1.1231308607	2.564197325
Tulip	1.1187268938	0.306318450

Hypatia, Loraine and DSDP are slow in this case and take more than 100s to solve.
COSMO, QSQP, ProxSDP and SCS failed to solve the problem.

[ArrogantGao]

I added some benchmark for the commerical solver CPLEX, it is indeed much faster then the non commerical ones, about 10 time faster than HiGHS.

CPLEX does not allow me to share the results, one can run the code themselves to verify.