Gurobi's Root relaxation objective in MIP Logging is Not a natural LP relaxation one

Specific Domains Optimization (Mathematical)
1

I’m not sure whether this is reasonable, but it is somewhat misleading.
Compare the 2 lines where I draw 3 orange balls.

julia> import JuMP, Gurobi; GRB_ENV = Gurobi.Env();
Set parameter Username
Set parameter LicenseID to value 2663413
Academic license - for non-commercial use only - expires 2026-05-11

julia> import LinearAlgebra.⋅ as ⋅ # dot product

julia> import Random; Random.seed!(3); # ⚠️ when modify this, U should also modify hyperparam

julia> 

julia> begin
           I = 7; # number of types of demand
           w = rand(2.3:.01:4.8, I); # length of each demand
           S = 2 # number of scenes
           d = [5.0 7.44; 5.0 3.72; 4.0 3.57; 2.0 2.48; 6.0 4.96; 2.0 2.48; 2.0 1.24] # number of demads in 2 scenes
           J = 3; # number of types of raw rolls
           W = 2.57 * [3, 4, 5.]; # unit length of each type of raw roll
           c = [.32 .44; .38 .32; .44 .30]; # (delayed to 2nd stage) cost of producing one unit length of raw roll
       end;

julia> begin
           Rfm = JuMP.Model(() -> Gurobi.Optimizer(GRB_ENV)); # a reformulation which facilitates decomposition
           JuMP.@variable(Rfm, m[1:J]);
           JuMP.@variable(Rfm, 0 ≤ n[1:I, 1:J, 1:S], Int);
               JuMP.@variable(Rfm, 0 ≤ b[1:J, 1:S] ≤ 4, Int);
                   JuMP.@constraint(Rfm, [j = 1:J, s = 1:S], w ⋅ view(n, :, j, s) ≤ W[j]b[j, s]);
           JuMP.@constraint(Rfm, [i = 1:I, s = 1:S], sum(view(n, i, :, s)) ≥ d[i, s]); # 🟡 [β ≥ 0]
           JuMP.@constraint(Rfm, [j = 1:J, s = 1:S], m[j] == b[j, s]); # 🟡 [π]
               JuMP.@objective(Rfm, Min, sum(c[j, s]b[j, s]/S for j = 1:J, s = 1:S));
       end;

julia> JuMP.optimize!(Rfm)
Gurobi Optimizer version 12.0.2 build v12.0.2rc0 (win64 - Windows 11.0 (26100.2))

CPU model: 11th Gen Intel(R) Core(TM) i5-1135G7 @ 2.40GHz, instruction set [SSE2|AVX|AVX2|AVX512]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads

Optimize a model with 26 rows, 51 columns and 102 nonzeros
Model fingerprint: 0x78f1bb14
Variable types: 3 continuous, 48 integer (0 binary)
Coefficient statistics:
  Matrix range     [1e+00, 1e+01]
  Objective range  [1e-01, 2e-01]
  Bounds range     [4e+00, 4e+00]
  RHS range        [1e+00, 7e+00]
Found heuristic solution: objective 4.4000000
Presolve removed 6 rows and 6 columns
Presolve time: 0.00s
Presolved: 20 rows, 45 columns, 90 nonzeros
Variable types: 0 continuous, 45 integer (0 binary)

Root relaxation: objective 3.791803e+00, 🟠🟠🟠

Explored 1 nodes (25 simplex iterations) in 0.00 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)

Solution count 2: 4.02 4.4

Optimal solution found (tolerance 1.00e-04)
Best objective 4.020000000000e+00, best bound 4.020000000000e+00, gap 0.0000%

julia> JuMP.unset_integer.(n); JuMP.unset_integer.(b);

julia> JuMP.optimize!(Rfm)
Gurobi Optimizer version 12.0.2 build v12.0.2rc0 (win64 - Windows 11.0 (26100.2))

Iteration    Objective       Primal Inf.    Dual Inf.      Time
       0    0.0000000e+00   4.119000e+01   0.000000e+00      0s
      28    3.4581323e+00   0.000000e+00   0.000000e+00      0s

Solved in 28 iterations and 0.00 seconds (0.00 work units)
Optimal objective  3.458132296e+00 🟠🟠🟠
2

I think the re-occuring issue is that you have a mental model of how Gurobi works that does not match what it does in practice.

The root relaxation may have inferred additional information about the model from the fact that some variables are integer. It is not as simple as the textbook implementation of “solve the problem with integrality relaxed.”

3

I was writting code to derive the Lagrangian lower bound, ended up finding out that it is outperformed by the “Root relaxation objective” in the MIP Logging, which made me have a false belief as if there was some error in my code—until I discovered the fact in the title.

I think both are meaningful. Is there an attribute that I can retrieve them?
Such that there are in total 3 performance indices:

  1. Lagrangian bound (which is model-dependent)
  2. Gurobi’s Root relaxation bound
  3. Textbook LP relaxation bound.
4

Here’s the documentation for the list of attributes: