It has been a while that CPLEX can solve nonconvex quadratic programming problems. But I am wondering if that has been considered in JuMP.

The reason I am asking is that I coded the following:

```
Q=[-50.0 0.0 0.0 0.0 0.0 42.0
0.0 -50.0 0.0 0.0 0.0 44.0
0.0 0.0 -50.0 0.0 0.0 45.0
0.0 0.0 0.0 -50.0 0.0 47.0
0.0 0.0 0.0 0.0 -50.0 47.5
42.0 44.0 45.0 47.0 47.5 -0.0];
A_x=[ 1.0 0.0 0.0 0.0 0.0 0.0
0.0 1.0 0.0 0.0 0.0 0.0
0.0 0.0 1.0 0.0 0.0 0.0
0.0 0.0 0.0 1.0 0.0 0.0
0.0 0.0 0.0 0.0 1.0 0.0
0.0 0.0 0.0 0.0 0.0 1.0
0.0 0.0 0.0 0.0 0.0 -1.0
20.0 12.0 11.0 7.0 4.0 0.0];
b_x=[1.0
1.0
1.0
1.0
1.0
0.5
-0.5
40.0]
n=6;
model = Model(CPLEX.Optimizer)
@variable(model, x[1:n]>=0)
@objective(model, Min, x'*Q*x)
con = @constraint(model, A_x*x .<= b_x)
sol_cplex=optimize!(model)
```

in Julia 1.5.3 using JuMP 0.21.5 and CPLEX 12.10, and got the following error:

```
ERROR: LoadError: CPLEX Error 5002: %s is not convex.
Stacktrace:
[1] error(::String) at .\error.jl:33
[2] _check_ret(::CPLEX.Env, ::Int32) at C:\Users\20176914\.julia\packages\CPLEX\uUjlQ\src\MOI\MOI_wrapper.jl:126
[3] _check_ret at C:\Users\20176914\.julia\packages\CPLEX\uUjlQ\src\MOI\MOI_wrapper.jl:243 [inlined]
[4] _optimize!(::CPLEX.Optimizer) at C:\Users\20176914\.julia\packages\CPLEX\uUjlQ\src\MOI\MOI_wrapper.jl:2431
[5] optimize!(::CPLEX.Optimizer) at C:\Users\20176914\.julia\packages\CPLEX\uUjlQ\src\MOI\MOI_wrapper.jl:2488
[6] optimize!(::MathOptInterface.Bridges.LazyBridgeOptimizer{CPLEX.Optimizer}) at C:\Users\20176914\.julia\packages\MathOptInterface\ZJFKw\src\Bridges\bridge_optimizer.jl:264
[7] optimize!(::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.AbstractOptimizer,MathOptInterface.Utilities.UniversalFallback{MathOptInterface.Utilities.Model{Float64}}}) at C:\Users\20176914\.julia\packages\MathOptInterface\ZJFKw\src\Utilities\cachingoptimizer.jl:215
[8] optimize!(::Model, ::Nothing; bridge_constraints::Bool, ignore_optimize_hook::Bool, kwargs::Base.Iterators.Pairs{Union{},Union{},Tuple{},NamedTuple{(),Tuple{}}}) at C:\Users\20176914\.julia\packages\JuMP\qhoVb\src\optimizer_interface.jl:130
[9] optimize! at C:\Users\20176914\.julia\packages\JuMP\qhoVb\src\optimizer_interface.jl:106 [inlined] (repeats 2 times)
[10] top-level scope at C:\Users\20176914\surfdrive\scientific\University\Codes\MATLAB\Bilinearprog\readLP.jl:46
[11] include(::String) at .\client.jl:457
[12] top-level scope at REPL[2]:1
in expression starting at C:\Users\20176914\surfdrive\scientific\University\Codes\MATLAB\Bilinearprog\readLP.jl:46
```

I know from past experience that there are different functions to make the distinction between convex and non-convex. So, I am now wondering if it is only possible to solve convex QPs in JuMP using CPLEX.

Does anyone have any experience with that?