Clarify error message in quad_to_soc bridge

[ASaragga]

Running the model

model = Model(COSMO.Optimizer)  # or model = Model(SCS.Optimizer)
@variable(model, w[1:6]>= 0)
@constraint(model, sum(w) == 1)
@constraint(model, w' * pcov * w == 0.038^2)
@objective(model, Max, pmean' * w)
optimize!(model)
optimal_solution = value.(w)

with:

pmean = [0.006719140992780818, 0.011577810667787833, 0.01344208119264699, 0.005535719592233162, 0.026922967360723037, 0.00934581295816873]

pcov = [0.018717403832096687 0.005706355121276367 0.006459876807928641 0.0012524618528797894 0.002563861720279623 0.0007671488407868487; 0.005706355121276367 0.006526570285541143 0.004384136499235087 0.0011480135148245983 0.0018044216713121253 0.001154428875412326; 0.006459876807928641 0.004384136499235087 0.0054284300755523115 0.0006596834879959167 0.0016581295100441041 0.0007133949662164625; 0.0012524618528797894 0.0011480135148245983 0.0006596834879959167 0.0026543564006455294 0.00048713871877351897 0.0011206346404939; 0.002563861720279623 0.0018044216713121253 0.0016581295100441041 0.00048713871877351897 0.0023567945429189227 0.00043958950944804554; 0.0007671488407868487 0.001154428875412326 0.0007133949662164625 0.0011206346404939 0.00043958950944804554 0.0020143379751173373]

one gets the following error,

ERROR: LoadError: The optimizer supports second-order cone constraints and not quadratic constraints but you entered a quadratic constraint of type: `MathOptInterface.ScalarQuadraticFunction{Float64}`-in-`MathOptInterface.GreaterThan{Float64}`. A bridge attempted to transform the quadratic constraint to a second order cone constraint but the constraint is not strongly convex, i.e., the symmetric matrix of quadratic coefficients is not positive definite. Convex constraints that are not strongly convex, i.e. the matrix is positive semidefinite but not positive definite, are not supported yet.

However the matrix pcov is positive definite (all eigen values > 0) and the constraint is therefore strongly convex.

[odow]

A quadratic equality constraint is non-convex.

The error message is a bit confusing, because we probably tried two rewrite the equality constraint into two inequalities, and then bridge those.

You should formulate a second-order-cone constraint directly: https://jump.dev/JuMP.jl/stable/constraints/#Quadratic-constraints, https://en.wikipedia.org/wiki/Second-order_cone_programming

[odow]

Note that this works:

model = Model(SCS.Optimizer)
@variable(model, w[1:6]>= 0)
@constraint(model, sum(w) == 1)
@constraint(model, w' * pcov * w <= 0.038^2)
@objective(model, Max, pmean' * w)
optimize!(model)
optimal_solution = value.(w)

I'll transfer this to MOI where the error message could be improved (to state that this could come from a different constraint).