I discover this behavior of Gurobi, which I think is not exactly expected.
According to Gurobi’s doc, if NonConvex is set to 0, then Gurobi should ERROR when a nonconvex quadratic constraint is present.
julia> import JuMP, Gurobi; GRB_ENV = Gurobi.Env();
julia> using LinearAlgebra
julia> A = [0.6500000000000001 0.4330127018922193; 0.4330127018922193 0.24999999999999994];
julia> eigen(A)
Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}
values:
2-element Vector{Float64}:
-0.02696960070847279
0.9269696007084729
julia> model = JuMP.Model(() -> Gurobi.Optimizer(GRB_ENV));
julia> JuMP.@variable(model, x[1:2]);
julia> JuMP.@variable(model, z);
julia> JuMP.set_attribute(model, "Nonconvex", 0) # I explicitly set this
Set parameter NonConvex to value 0
julia> JuMP.@constraint(model, z ≥ x' * A * x) # nonconvex
-0.6500000000000001 x[1]² - 0.8660254037844386 x[2]*x[1] - 0.24999999999999994 x[2]² + z >= 0
julia> JuMP.optimize!(model) # 🔴 I think Gurobi should ERROR here, but it didn't
Set parameter NonConvex to value 0
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
Non-default parameters:
NonConvex 0
Optimize a model with 0 rows, 3 columns and 0 nonzeros
Model fingerprint: 0x36f7eae6
Model has 1 quadratic constraint
Coefficient statistics:
Matrix range [0e+00, 0e+00]
QMatrix range [2e-01, 9e-01]
QLMatrix range [1e+00, 1e+00]
Objective range [0e+00, 0e+00]
Bounds range [0e+00, 0e+00]
RHS range [0e+00, 0e+00]
Presolve removed 0 rows and 3 columns
Presolve time: 0.00s
Presolve: All rows and columns removed
Barrier solved model in 0 iterations and 0.00 seconds (0.00 work units)
Optimal objective 0.00000000e+00
User-callback calls 28, time in user-callback 0.00 sec
julia> JuMP.@objective(model, Min, z);
julia> JuMP.optimize!(model) # By comparison, this is expected
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
Non-default parameters:
NonConvex 0
Optimize a model with 0 rows, 3 columns and 0 nonzeros
Model fingerprint: 0x4893b5bf
Model has 1 quadratic constraint
Coefficient statistics:
Matrix range [0e+00, 0e+00]
QMatrix range [2e-01, 9e-01]
QLMatrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+00]
Bounds range [0e+00, 0e+00]
RHS range [0e+00, 0e+00]
User-callback calls 19, time in user-callback 0.00 sec
ERROR: Gurobi Error 10020: Objective Q not PSD (diagonal adjustment of 7.7e-02 would be required). Set NonConvex parameter to -1 or 2 to solve model.