In this tutorial you will learn how to create and use QuadraticModels.

\begin{aligned} \min \quad & \tfrac{1}{2}x^T H x + c^T x + c0 \\ & lcon \leq Ax \leq ucon \\ & \ell \leq x \leq u. \end{aligned}

H should be a lower triangular matrix. QuadraticModels can work with different input matrix types:

H = [
6.0 2.0 1.0
2.0 5.0 2.0
1.0 2.0 4.0
]
c = [-8.0; -3; -3]
A = [
1.0 0.0 1.0
0.0 2.0 1.0
]
lcon = [-2.0; 0.]
ucon = [10.0; 20.0]
l = [0.0; 0; 0]
u = [Inf; Inf; Inf]
using LinearAlgebra, SparseArrays, QuadraticModels, SparseMatricesCOO, LinearOperators
HCOO = SparseMatrixCOO(tril(H))
ACOO = SparseMatrixCOO(A)
HCSC = sparse(tril(H))
ACSC = sparse(A)
qmCSC = QuadraticModel(c, HCSC, A = ACSC, lcon = lcon, ucon = ucon, lvar = l, uvar = u,
c0 = 0.0, name = "QM_CSC")
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, SparseArrays.Spars
eMatrixCSC{Float64, Int64}, SparseArrays.SparseMatrixCSC{Float64, Int64}}
Problem name: QM_CSC
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (  0.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: ( 33.33% spa
rsity)   4

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
qmCOO = QuadraticModel(c, HCOO, A = ACOO, lcon = lcon, ucon = ucon, lvar = l, uvar = u,
c0 = 0.0, name = "QM_COO")
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, SparseMatricesCOO.
SparseMatrixCOO{Float64, Int64}, SparseMatricesCOO.SparseMatrixCOO{Float64,
Int64}}
Problem name: QM_COO
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (  0.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: ( 33.33% spa
rsity)   4

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
qmDense = QuadraticModel(c, tril(H), A = A, lcon = lcon, ucon = ucon, lvar = l, uvar = u,
c0 = 0.0, name = "QM_Dense")
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, Matrix{Float64}, M
atrix{Float64}}
Problem name: QM_Dense
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (  0.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: (  0.00% spa
rsity)   6

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
qmLinop = QuadraticModel(c, LinearOperator(Symmetric(H)), A = LinearOperator(A),
lcon = lcon, ucon = ucon, lvar = l, uvar = u,
c0 = 0.0, name = "QM_Linop")
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, LinearOperators.Li
nearOperator{Float64, Int64, LinearOperators.var"#5#8"{LinearAlgebra.Symmet
ric{Float64, Matrix{Float64}}}, LinearOperators.var"#6#9"{LinearAlgebra.Sym
metric{Float64, Matrix{Float64}}}, LinearOperators.var"#7#10"{LinearAlgebra
.Symmetric{Float64, Matrix{Float64}}}, Vector{Float64}}, LinearOperators.Li
nearOperator{Float64, Int64, LinearOperators.var"#5#8"{Matrix{Float64}}, Li
nearOperators.var"#6#9"{Matrix{Float64}}, LinearOperators.var"#7#10"{Matrix
{Float64}}, Vector{Float64}}}
Problem name: QM_Linop
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (100.00% sparsity)   0               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: (100.00% spa
rsity)   0

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0

You can also create a COO QuadraticModel directly from the coordinates (without using SparseMatricesCOO.jl):

Hrows, Hcols, Hvals = findnz(sparse(tril(H)))
Arows, Acols, Avals = findnz(sparse(A))
qmCOO2 = QuadraticModel(c, Hrows, Hcols, Hvals, Arows = Arows, Acols = Acols, Avals = Avals,
lcon = lcon, ucon = ucon, lvar = l, uvar = u,
c0 = 0.0, name = "QM_COO2")
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, SparseMatricesCOO.
SparseMatrixCOO{Float64, Int64}, SparseMatricesCOO.SparseMatrixCOO{Float64,
Int64}}
Problem name: QM_COO2
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (  0.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: ( 33.33% spa
rsity)   4

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0

Some functions work best with SparseMatricesCOO. You can convert your QuadraticModel with

T = Float64
S = Vector{T}
qmCOO3 = convert(QuadraticModel{T, S, SparseMatrixCOO{T, Int}, SparseMatrixCOO{T, Int}},
qmDense)

qmCOO4 = convert(QuadraticModel{T, S, SparseMatrixCOO{T, Int}, SparseMatrixCOO{T, Int}},
qmCSC)
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, SparseMatricesCOO.
SparseMatrixCOO{Float64, Int64}, SparseMatricesCOO.SparseMatrixCOO{Float64,
Int64}}
Problem name: QM_CSC
All variables: ████████████████████ 3      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████████████ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0              low/upp: ████████████
████████ 2
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: (  0.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: ( 33.33% spa
rsity)   4

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0

## Use the NLPModels.jl API

You can use the API from NLPModels.jl with QuadraticModels. Here are some examples:

using NLPModels, Random
Random.seed!(0)

x = rand(3)
grad(qmCOO, x)
3-element Vector{Float64}:
-4.56657066867931
-0.12159022189239677
0.9913545484469992
hess(qmCOO, x)
3×3 LinearAlgebra.Symmetric{Float64, SparseArrays.SparseMatrixCSC{Float64,
Int64}}:
6.0  2.0  1.0
2.0  5.0  2.0
1.0  2.0  4.0

It is possible to convert the model to a QuadraticModel with linear inequality constraints to equality constraints and bounds using SlackModel

using NLPModelsModifiers
qmSlack = SlackModel(qmCOO)
QuadraticModels.QuadraticModel{Float64, Vector{Float64}, SparseMatricesCOO.
SparseMatrixCOO{Float64, Int64}, SparseMatricesCOO.SparseMatrixCOO{Float64,
Int64}}
Problem name: QM_COO
All variables: ████████████████████ 5      All constraints: ████████████
████████ 2
free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 free: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
lower: ████████████⋅⋅⋅⋅⋅⋅⋅⋅ 3                lower: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                upper: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
low/upp: ████████⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 2              low/upp: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
fixed: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                fixed: ████████████
████████ 2
infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               infeas: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzh: ( 60.00% sparsity)   6               linear: ████████████
████████ 2
nonlinear: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0
nnzj: ( 40.00% spa
rsity)   6

Counters:
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 cons: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
cons_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0             cons_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                 jcon: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
⋅⋅⋅⋅⋅⋅⋅⋅ 0              jac_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jac_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                jprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0            jprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jtprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0           jtprod_lin: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jtprod_nln: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0                 hess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0                hprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0
jhess: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ 0               jhprod: ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅⋅⋅ 0

You can read directly MPS or SIF files using QPSReader.jl

using QPSReader
qmCOO4 = QuadraticModel(qps)
Error: SystemError: opening file "AFIRO.SIF": No such file or directory

## Solving

You can use RipQP.jl to solve QuadraticModels:

using RipQP
stats = ripqp(qmCOO)
println(stats)
Generic Execution stats
status: first-order stationary
objective value: -5.6956521641208
primal feasibility: 3.885780586188048e-16
dual feasibility: 5.334399588718952e-12
solution: [1.2608695586860905  2.502461850391081e-8  0.43478259786166773]
multipliers: [4.206802886688845e-12  1.141648742192826e-10]
multipliers_L: [2.3271161304045762e-11  0.39130443799558634  6.3653700416
39302e-11]
multipliers_U: [0.0  0.0  0.0]
iterations: 32
elapsed time: 10.904875993728638
solver specific:
nvar_slack: 5
pdd: 1.4918291270615607e-9
absolute_iter_cnt: 8