func TestConeLp(t *testing.T) {
gdata := [][]float64{
[]float64{16., 7., 24., -8., 8., -1., 0., -1., 0., 0., 7.,
-5., 1., -5., 1., -7., 1., -7., -4.},
[]float64{-14., 2., 7., -13., -18., 3., 0., 0., -1., 0., 3.,
13., -6., 13., 12., -10., -6., -10., -28.},
[]float64{5., 0., -15., 12., -6., 17., 0., 0., 0., -1., 9.,
6., -6., 6., -7., -7., -6., -7., -11.}}
hdata := []float64{-3., 5., 12., -2., -14., -13., 10., 0., 0., 0., 68.,
-30., -19., -30., 99., 23., -19., 23., 10.}
// these reference values obtained from running cvxopt conelp.py example
xref := []float64{-1.22091525026262993, 0.09663323966626469, 3.57750155386611057}
sref := []float64{
0.00000172588537019, 13.35314040819201864,
94.28805677232460880, -53.44110853283719109,
18.97172963929198275, -75.32834138499130461,
10.00000013568614321, -1.22091525026262993,
0.09663323966626476, 3.57750155386611146,
44.05899318373081286, -58.82581769017131990,
4.26572401145687596, -58.82581769017131990,
124.10382738701650851, 40.46243652188705653,
4.26572401145687596, 40.46243652188705653,
47.17458693781828316}
zref := []float64{
0.09299833991484617, 0.00000001060210894,
0.23532251654806322, 0.13337937743566930,
-0.04734875722474355, 0.18800192060450249,
0.00000001245876667, 0.00000000007816348,
-0.00000000039584268, -0.00000000183463577,
0.12558704894101563, 0.08777794737598217,
-0.08664401207348003, 0.08777794737598217,
0.06135161787371416, -0.06055906182304811,
-0.08664401207348003, -0.06055906182304811,
0.05977675078191153}
c := matrix.FloatVector([]float64{-6., -4., -5.})
G := matrix.FloatMatrixFromTable(gdata)
h := matrix.FloatVector(hdata)
dims := sets.DSetNew("l", "q", "s")
dims.Set("l", []int{2})
dims.Set("q", []int{4, 4})
dims.Set("s", []int{3})
var solopts SolverOptions
solopts.MaxIter = 30
solopts.ShowProgress = false
sol, err := ConeLp(c, G, h, nil, nil, dims, &solopts, nil, nil)
if err == nil {
fail := false
x := sol.Result.At("x")[0]
s := sol.Result.At("s")[0]
z := sol.Result.At("z")[0]
t.Logf("Optimal\n")
t.Logf("x=\n%v\n", x.ToString("%.9f"))
t.Logf("s=\n%v\n", s.ToString("%.9f"))
t.Logf("z=\n%v\n", z.ToString("%.9f"))
xe, _ := nrmError(matrix.FloatVector(xref), x)
if xe > TOL {
t.Logf("x differs [%.3e] from exepted too much.", xe)
fail = true
}
se, _ := nrmError(matrix.FloatVector(sref), s)
if se > TOL {
t.Logf("s differs [%.3e] from exepted too much.", se)
fail = true
}
ze, _ := nrmError(matrix.FloatVector(zref), z)
if ze > TOL {
t.Logf("z differs [%.3e] from exepted too much.", ze)
fail = true
}
if fail {
t.Fail()
}
} else {
t.Logf("status: %s\n", err)
t.Fail()
}
}
func qcl1(A, b *matrix.FloatMatrix) (*cvx.Solution, error) {
// Returns the solution u, z of
//
// (primal) minimize || u ||_1
// subject to || A * u - b ||_2 <= 1
//
// (dual) maximize b^T z - ||z||_2
// subject to || A'*z ||_inf <= 1.
//
// Exploits structure, assuming A is m by n with m >= n.
m, n := A.Size()
Fkkt := func(W *sets.FloatMatrixSet) (f cvx.KKTFunc, err error) {
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
err = nil
f = nil
beta := W.At("beta")[0].GetIndex(0)
v := W.At("v")[0]
// As = 2 * v *(v[1:].T * A)
//v_1 := matrix.FloatNew(1, v.NumElements()-1, v.FloatArray()[1:])
v_1 := v.SubMatrix(1, 0).Transpose()
As := matrix.Times(v, matrix.Times(v_1, A)).Scale(2.0)
//As_1 := As.GetSubMatrix(1, 0, m, n)
//As_1.Scale(-1.0)
//As.SetSubMatrix(1, 0, matrix.Minus(As_1, A))
As_1 := As.SubMatrix(1, 0, m, n)
As_1.Scale(-1.0)
As_1.Minus(A)
As.Scale(1.0 / beta)
S := matrix.Times(As.Transpose(), As)
checkpnt.AddMatrixVar("S", S)
d1 := W.At("d")[0].SubMatrix(0, 0, n, 1).Copy()
d2 := W.At("d")[0].SubMatrix(n, 0).Copy()
// D = 4.0 * (d1**2 + d2**2)**-1
d := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2)).Inv().Scale(4.0)
// S[::n+1] += d
S.Diag().Plus(d.Transpose())
err = lapack.Potrf(S)
checkpnt.Check("00-Fkkt", minor)
if err != nil {
return
}
f = func(x, y, z *matrix.FloatMatrix) (err error) {
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
} else {
loopf += 1
minor = loopf
}
checkpnt.Check("00-f", minor)
// -- z := - W**-T * z
// z[:n] = -div( z[:n], d1 )
z_val := z.SubMatrix(0, 0, n, 1)
z_res := matrix.Div(z_val, d1).Scale(-1.0)
z.SubMatrix(0, 0, n, 1).Set(z_res)
// z[n:2*n] = -div( z[n:2*n], d2 )
z_val = z.SubMatrix(n, 0, n, 1)
z_res = matrix.Div(z_val, d2).Scale(-1.0)
z.SubMatrix(n, 0, n, 1).Set(z_res)
// z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
v0_z2n := v.GetIndex(0) * z.GetIndex(2*n)
v1_z2n := blas.DotFloat(v, z, &linalg.IOpt{"offsetx", 1}, &linalg.IOpt{"offsety", 2*n + 1})
z_res = matrix.Scale(v, -2.0*(v0_z2n-v1_z2n))
z.SubMatrix(2*n, 0, z_res.NumElements(), 1).Plus(z_res)
// z[2*n+1:] *= -1.0
z.SubMatrix(2*n+1, 0).Scale(-1.0)
// z[2*n:] /= beta
z.SubMatrix(2*n, 0).Scale(1.0 / beta)
// -- x := x - G' * W**-1 * z
// z_n = z[:n], z_2n = z[n:2*n], z_m = z[-(m+1):],
z_n := z.SubMatrix(0, 0, n, 1)
z_2n := z.SubMatrix(n, 0, n, 1)
z_m := z.SubMatrix(z.NumElements()-(m+1), 0)
// x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
z_res = matrix.Minus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
a_res := matrix.Times(As.Transpose(), z_m)
z_res.Plus(a_res).Scale(-1.0)
x.SubMatrix(0, 0, n, 1).Plus(z_res)
// x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
z_res = matrix.Plus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
x.SubMatrix(n, 0, z_res.NumElements(), 1).Plus(z_res)
checkpnt.Check("15-f", minor)
// Solve for x[:n]:
//
// S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
// w1 = (d1**2 - d2**2), w2 = (d1**2 + d2**2)
w1 := matrix.Minus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))
w2 := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))
// x[:n] += -mul( div(w1, w2), x[n:])
x_n := x.SubMatrix(n, 0)
x_val := matrix.Mul(matrix.Div(w1, w2), x_n).Scale(-1.0)
x.SubMatrix(0, 0, n, 1).Plus(x_val)
checkpnt.Check("25-f", minor)
// Solve for x[n:]:
//
// (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
err = lapack.Potrs(S, x)
if err != nil {
fmt.Printf("Potrs error: %s\n", err)
}
checkpnt.Check("30-f", minor)
// Solve for x[n:]:
//
// (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
// w1 = (d1**-2 - d2**-2), w2 = (d1**-2 + d2**-2)
w1 = matrix.Minus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
w2 = matrix.Plus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
x_n = x.SubMatrix(0, 0, n, 1)
// x[n:] += mul( d1**-2 - d2**-2, x[:n])
x_val = matrix.Mul(w1, x_n)
x.SubMatrix(n, 0, x_val.NumElements(), 1).Plus(x_val)
checkpnt.Check("35-f", minor)
// x[n:] = div( x[n:], d1**-2 + d2**-2)
x_n = x.SubMatrix(n, 0)
x_val = matrix.Div(x_n, w2)
x.SubMatrix(n, 0, x_val.NumElements(), 1).Set(x_val)
checkpnt.Check("40-f", minor)
// x_n = x[:n], x-2n = x[n:2*n]
x_n = x.SubMatrix(0, 0, n, 1)
x_2n := x.SubMatrix(n, 0, n, 1)
// z := z + W^-T * G*x
// z[:n] += div( x[:n] - x[n:2*n], d1)
x_val = matrix.Div(matrix.Minus(x_n, x_2n), d1)
z.SubMatrix(0, 0, n, 1).Plus(x_val)
checkpnt.Check("44-f", minor)
// z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
x_val = matrix.Div(matrix.Plus(x_n, x_2n).Scale(-1.0), d2)
z.SubMatrix(n, 0, n, 1).Plus(x_val)
checkpnt.Check("48-f", minor)
// z[2*n:] += As*x[:n]
x_val = matrix.Times(As, x_n)
z.SubMatrix(2*n, 0, x_val.NumElements(), 1).Plus(x_val)
checkpnt.Check("50-f", minor)
return nil
}
return
}
// matrix(n*[0.0] + n*[1.0])
c := matrix.FloatZeros(2*n, 1)
c.SubMatrix(n, 0).SetIndexes(1.0)
h := matrix.FloatZeros(2*n+m+1, 1)
h.SetIndexes(1.0, 2*n)
// h[2*n+1:] = -b
h.SubMatrix(2*n+1, 0).Plus(b).Scale(-1.0)
G := &matrixFs{A}
dims := sets.DSetNew("l", "q", "s")
dims.Set("l", []int{2 * n})
dims.Set("q", []int{m + 1})
var solopts cvx.SolverOptions
solopts.ShowProgress = true
if maxIter > 0 {
solopts.MaxIter = maxIter
}
if len(solver) > 0 {
solopts.KKTSolverName = solver
}
return cvx.ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, nil, nil)
}
func mcsdp(w *matrix.FloatMatrix) (*Solution, error) {
//
// Returns solution x, z to
//
// (primal) minimize sum(x)
// subject to w + diag(x) >= 0
//
// (dual) maximize -tr(w*z)
// subject to diag(z) = 1
// z >= 0.
//
n := w.Rows()
G := &matrixFs{n}
cngrnc := func(r, x *matrix.FloatMatrix, alpha float64) (err error) {
// Congruence transformation
//
// x := alpha * r'*x*r.
//
// r and x are square matrices.
//
err = nil
// tx = matrix(x, (n,n)) is copying and reshaping
// scale diagonal of x by 1/2, (x is (n,n))
tx := x.Copy()
matrix.Reshape(tx, n, n)
tx.Diag().Scale(0.5)
// a := tril(x)*r
// (python: a = +r is really making a copy of r)
a := r.Copy()
err = blas.TrmmFloat(tx, a, 1.0, linalg.OptLeft)
// x := alpha*(a*r' + r*a')
err = blas.Syr2kFloat(r, a, tx, alpha, 0.0, linalg.OptTrans)
// x[:] = tx[:]
tx.CopyTo(x)
return
}
Fkkt := func(W *sets.FloatMatrixSet) (KKTFunc, error) {
// Solve
// -diag(z) = bx
// -diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
//
// On entry, x and z contain bx and bs.
// On exit, they contain the solution, with z scaled
// (inv(rti)'*z*inv(rti) is returned instead of z).
//
// We first solve
//
// ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
//
// and take z = -rti' * (diag(x) + bs) * rti.
var err error = nil
rti := W.At("rti")[0]
// t = rti*rti' as a nonsymmetric matrix.
t := matrix.FloatZeros(n, n)
err = blas.GemmFloat(rti, rti, t, 1.0, 0.0, linalg.OptTransB)
if err != nil {
return nil, err
}
// Cholesky factorization of tsq = t.*t.
tsq := matrix.Mul(t, t)
err = lapack.Potrf(tsq)
if err != nil {
return nil, err
}
f := func(x, y, z *matrix.FloatMatrix) (err error) {
// tbst := t * zs * t = t * bs * t
tbst := z.Copy()
matrix.Reshape(tbst, n, n)
cngrnc(t, tbst, 1.0)
// x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
diag := tbst.Diag().Transpose()
x.Minus(diag)
// x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
err = lapack.Potrs(tsq, x)
// z := z + diag(x) = bs + diag(x)
// z, x are really column vectors here
z.AddIndexes(matrix.MakeIndexSet(0, n*n, n+1), x.FloatArray())
// z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
cngrnc(rti, z, -1.0)
return nil
}
return f, nil
}
c := matrix.FloatWithValue(n, 1, 1.0)
// initial feasible x: x = 1.0 - min(lmbda(w))
lmbda := matrix.FloatZeros(n, 1)
wp := w.Copy()
lapack.Syevx(wp, lmbda, nil, 0.0, nil, []int{1, 1}, linalg.OptRangeInt)
x0 := matrix.FloatZeros(n, 1).Add(-lmbda.GetAt(0, 0) + 1.0)
s0 := w.Copy()
s0.Diag().Plus(x0.Transpose())
matrix.Reshape(s0, n*n, 1)
// initial feasible z is identity
z0 := matrix.FloatIdentity(n)
matrix.Reshape(z0, n*n, 1)
dims := sets.DSetNew("l", "q", "s")
dims.Set("s", []int{n})
primalstart := sets.FloatSetNew("x", "s")
dualstart := sets.FloatSetNew("z")
primalstart.Set("x", x0)
primalstart.Set("s", s0)
dualstart.Set("z", z0)
var solopts SolverOptions
solopts.MaxIter = 30
solopts.ShowProgress = false
h := w.Copy()
matrix.Reshape(h, h.NumElements(), 1)
return ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, primalstart, dualstart)
}
func TestConeQp(t *testing.T) {
adata := [][]float64{
[]float64{0.3, -0.4, -0.2, -0.4, 1.3},
[]float64{0.6, 1.2, -1.7, 0.3, -0.3},
[]float64{-0.3, 0.0, 0.6, -1.2, -2.0}}
// reference values from cvxopt coneqp.py
xref := []float64{0.72558318685981904, 0.61806264311119252, 0.30253527966423444}
sref := []float64{
0.72558318685981904, 0.61806264311119263,
0.30253527966423449, 1.00000000000041678,
-0.72558318686012169, -0.61806264311145032,
-0.30253527966436067}
zref := []float64{
0.00000003332583626, 0.00000005116586239,
0.00000009993673262, 0.56869648433154019,
0.41264857754144563, 0.35149286573190930,
0.17201618570052318}
A := matrix.FloatMatrixFromTable(adata, matrix.ColumnOrder)
b := matrix.FloatVector([]float64{1.5, 0.0, -1.2, -0.7, 0.0})
_, n := A.Size()
N := n + 1 + n
h := matrix.FloatZeros(N, 1)
h.SetIndex(n, 1.0)
I0 := matrix.FloatDiagonal(n, -1.0)
I1 := matrix.FloatIdentity(n)
G, _ := matrix.FloatMatrixStacked(matrix.StackDown, I0, matrix.FloatZeros(1, n), I1)
At := A.Transpose()
P := matrix.Times(At, A)
q := matrix.Times(At, b).Scale(-1.0)
dims := sets.DSetNew("l", "q", "s")
dims.Set("l", []int{n})
dims.Set("q", []int{n + 1})
var solopts SolverOptions
solopts.MaxIter = 10
solopts.ShowProgress = false
sol, err := ConeQp(P, q, G, h, nil, nil, dims, &solopts, nil)
if err == nil {
fail := false
x := sol.Result.At("x")[0]
s := sol.Result.At("s")[0]
z := sol.Result.At("z")[0]
t.Logf("Optimal\n")
t.Logf("x=\n%v\n", x.ToString("%.9f"))
t.Logf("s=\n%v\n", s.ToString("%.9f"))
t.Logf("z=\n%v\n", z.ToString("%.9f"))
xe, _ := nrmError(matrix.FloatVector(xref), x)
if xe > TOL {
t.Logf("x differs [%.3e] from exepted too much.", xe)
fail = true
}
se, _ := nrmError(matrix.FloatVector(sref), s)
if se > TOL {
t.Logf("s differs [%.3e] from exepted too much.", se)
fail = true
}
ze, _ := nrmError(matrix.FloatVector(zref), z)
if ze > TOL {
t.Logf("z differs [%.3e] from exepted too much.", ze)
fail = true
}
if fail {
t.Fail()
}
}
}