func kktChol2(G *matrix.FloatMatrix, dims *sets.DimensionSet, A *matrix.FloatMatrix, mnl int) (kktFactor, error) {
if len(dims.At("q")) > 0 || len(dims.At("s")) > 0 {
return nil, errors.New("'chol2' solver only for problems with no second-order or " +
"semidefinite cone constraints")
}
p, n := A.Size()
ml := dims.At("l")[0]
F := &chol2Data{firstcall: true, singular: false, A: A, G: G, dims: dims}
factor := func(W *sets.FloatMatrixSet, H, Df *matrix.FloatMatrix) (KKTFunc, error) {
var err error = nil
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
if F.firstcall {
F.Gs = matrix.FloatZeros(F.G.Size())
if mnl > 0 {
F.Dfs = matrix.FloatZeros(Df.Size())
}
F.S = matrix.FloatZeros(n, n)
F.K = matrix.FloatZeros(p, p)
checkpnt.AddMatrixVar("Gs", F.Gs)
checkpnt.AddMatrixVar("Dfs", F.Dfs)
checkpnt.AddMatrixVar("S", F.S)
checkpnt.AddMatrixVar("K", F.K)
}
if mnl > 0 {
dnli := matrix.FloatZeros(mnl, mnl)
dnli.SetIndexesFromArray(W.At("dnli")[0].FloatArray(), matrix.DiagonalIndexes(dnli)...)
blas.GemmFloat(dnli, Df, F.Dfs, 1.0, 0.0)
}
checkpnt.Check("02factor_chol2", minor)
di := matrix.FloatZeros(ml, ml)
di.SetIndexesFromArray(W.At("di")[0].FloatArray(), matrix.DiagonalIndexes(di)...)
err = blas.GemmFloat(di, G, F.Gs, 1.0, 0.0)
checkpnt.Check("06factor_chol2", minor)
if F.firstcall {
blas.SyrkFloat(F.Gs, F.S, 1.0, 0.0, la.OptTrans)
if mnl > 0 {
blas.SyrkFloat(F.Dfs, F.S, 1.0, 1.0, la.OptTrans)
}
if H != nil {
F.S.Plus(H)
}
checkpnt.Check("10factor_chol2", minor)
err = lapack.Potrf(F.S)
if err != nil {
err = nil // reset error
F.singular = true
// original code recreates F.S as dense if it is sparse and
// A is dense, we don't do it as currently no sparse matrices
//F.S = matrix.FloatZeros(n, n)
//checkpnt.AddMatrixVar("S", F.S)
blas.SyrkFloat(F.Gs, F.S, 1.0, 0.0, la.OptTrans)
if mnl > 0 {
blas.SyrkFloat(F.Dfs, F.S, 1.0, 1.0, la.OptTrans)
}
checkpnt.Check("14factor_chol2", minor)
blas.SyrkFloat(F.A, F.S, 1.0, 1.0, la.OptTrans)
if H != nil {
F.S.Plus(H)
}
lapack.Potrf(F.S)
}
F.firstcall = false
checkpnt.Check("20factor_chol2", minor)
} else {
blas.SyrkFloat(F.Gs, F.S, 1.0, 0.0, la.OptTrans)
if mnl > 0 {
blas.SyrkFloat(F.Dfs, F.S, 1.0, 1.0, la.OptTrans)
}
if H != nil {
F.S.Plus(H)
}
checkpnt.Check("40factor_chol2", minor)
if F.singular {
blas.SyrkFloat(F.A, F.S, 1.0, 1.0, la.OptTrans)
}
lapack.Potrf(F.S)
checkpnt.Check("50factor_chol2", minor)
}
// Asct := L^{-1}*A'. Factor K = Asct'*Asct.
Asct := F.A.Transpose()
blas.TrsmFloat(F.S, Asct, 1.0)
blas.SyrkFloat(Asct, F.K, 1.0, 0.0, la.OptTrans)
lapack.Potrf(F.K)
checkpnt.Check("90factor_chol2", minor)
solve := func(x, y, z *matrix.FloatMatrix) (err error) {
// Solve
//
// [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
// [ A 0 0 ] * [ uy ] = [ by ]
// [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
//
// and return ux, uy, W*uz.
//
// If not F['singular']:
//
// K*uy = A * S^{-1} * ( bx + GG'*W^{-1}*W^{-T}*bz ) - by
// S*ux = bx + GG'*W^{-1}*W^{-T}*bz - A'*uy
// W*uz = W^{-T} * ( GG*ux - bz ).
//
// If F['singular']:
//
// K*uy = A * S^{-1} * ( bx + GG'*W^{-1}*W^{-T}*bz + A'*by )
// - by
// S*ux = bx + GG'*W^{-1}*W^{-T}*bz + A'*by - A'*y.
// W*uz = W^{-T} * ( GG*ux - bz ).
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
// z := W^{-1} * z = W^{-1} * bz
scale(z, W, true, true)
checkpnt.Check("10solve_chol2", minor)
// If not F['singular']:
// x := L^{-1} * P * (x + GGs'*z)
// = L^{-1} * P * (x + GG'*W^{-1}*W^{-T}*bz)
//
// If F['singular']:
// x := L^{-1} * P * (x + GGs'*z + A'*y))
// = L^{-1} * P * (x + GG'*W^{-1}*W^{-T}*bz + A'*y)
if mnl > 0 {
blas.GemvFloat(F.Dfs, z, x, 1.0, 1.0, la.OptTrans)
}
blas.GemvFloat(F.Gs, z, x, 1.0, 1.0, la.OptTrans, &la.IOpt{"offsetx", mnl})
//checkpnt.Check("20solve_chol2", minor)
if F.singular {
blas.GemvFloat(F.A, y, x, 1.0, 1.0, la.OptTrans)
}
checkpnt.Check("30solve_chol2", minor)
blas.TrsvFloat(F.S, x)
//checkpnt.Check("50solve_chol2", minor)
// y := K^{-1} * (Asc*x - y)
// = K^{-1} * (A * S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz) - by)
// (if not F['singular'])
// = K^{-1} * (A * S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz +
// A'*by) - by)
// (if F['singular']).
blas.GemvFloat(Asct, x, y, 1.0, -1.0, la.OptTrans)
//checkpnt.Check("55solve_chol2", minor)
lapack.Potrs(F.K, y)
//checkpnt.Check("60solve_chol2", minor)
// x := P' * L^{-T} * (x - Asc'*y)
// = S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz - A'*y)
// (if not F['singular'])
// = S^{-1} * (bx + GG'*W^{-1}*W^{-T}*bz + A'*by - A'*y)
// (if F['singular'])
blas.GemvFloat(Asct, y, x, -1.0, 1.0)
blas.TrsvFloat(F.S, x, la.OptTrans)
checkpnt.Check("70solve_chol2", minor)
// W*z := GGs*x - z = W^{-T} * (GG*x - bz)
if mnl > 0 {
blas.GemvFloat(F.Dfs, x, z, 1.0, -1.0)
}
blas.GemvFloat(F.Gs, x, z, 1.0, -1.0, &la.IOpt{"offsety", mnl})
checkpnt.Check("90solve_chol2", minor)
return nil
}
return solve, err
}
return factor, 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)
}
// Solution of KKT equations by reduction to a 2 x 2 system, a QR
// factorization to eliminate the equality constraints, and a dense
// Cholesky factorization of order n-p.
//
// Computes the QR factorization
//
// A' = [Q1, Q2] * [R; 0]
//
// and returns a function that (1) computes the Cholesky factorization
//
// Q_2^T * (H + GG^T * W^{-1} * W^{-T} * GG) * Q2 = L * L^T,
//
// given H, Df, W, where GG = [Df; G], and (2) returns a function for
// solving
//
// [ H A' GG' ] [ ux ] [ bx ]
// [ A 0 0 ] * [ uy ] = [ by ].
// [ GG 0 -W'*W ] [ uz ] [ bz ]
//
// H is n x n, A is p x n, Df is mnl x n, G is N x n where
// N = dims['l'] + sum(dims['q']) + sum( k**2 for k in dims['s'] ).
//
func kktChol(G *matrix.FloatMatrix, dims *sets.DimensionSet, A *matrix.FloatMatrix, mnl int) (kktFactor, error) {
p, n := A.Size()
cdim := mnl + dims.Sum("l", "q") + dims.SumSquared("s")
cdim_pckd := mnl + dims.Sum("l", "q") + dims.SumPacked("s")
QA := A.Transpose()
tauA := matrix.FloatZeros(p, 1)
lapack.Geqrf(QA, tauA)
Gs := matrix.FloatZeros(cdim, n)
K := matrix.FloatZeros(n, n)
bzp := matrix.FloatZeros(cdim_pckd, 1)
yy := matrix.FloatZeros(p, 1)
checkpnt.AddMatrixVar("tauA", tauA)
checkpnt.AddMatrixVar("Gs", Gs)
checkpnt.AddMatrixVar("K", K)
factor := func(W *sets.FloatMatrixSet, H, Df *matrix.FloatMatrix) (KKTFunc, error) {
// Compute
//
// K = [Q1, Q2]' * (H + GG' * W^{-1} * W^{-T} * GG) * [Q1, Q2]
//
// and take the Cholesky factorization of the 2,2 block
//
// Q_2' * (H + GG^T * W^{-1} * W^{-T} * GG) * Q2.
var err error = nil
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
// Gs = W^{-T} * GG in packed storage.
if mnl > 0 {
Gs.SetSubMatrix(0, 0, Df)
}
Gs.SetSubMatrix(mnl, 0, G)
checkpnt.Check("00factor_chol", minor)
scale(Gs, W, true, true)
pack2(Gs, dims, mnl)
//checkpnt.Check("10factor_chol", minor)
// K = [Q1, Q2]' * (H + Gs' * Gs) * [Q1, Q2].
blas.SyrkFloat(Gs, K, 1.0, 0.0, la.OptTrans, &la.IOpt{"k", cdim_pckd})
if H != nil {
K.SetSubMatrix(0, 0, matrix.Plus(H, K.GetSubMatrix(0, 0, H.Rows(), H.Cols())))
}
//checkpnt.Check("20factor_chol", minor)
symm(K, n, 0)
lapack.Ormqr(QA, tauA, K, la.OptLeft, la.OptTrans)
lapack.Ormqr(QA, tauA, K, la.OptRight)
//checkpnt.Check("30factor_chol", minor)
// Cholesky factorization of 2,2 block of K.
lapack.Potrf(K, &la.IOpt{"n", n - p}, &la.IOpt{"offseta", p * (n + 1)})
checkpnt.Check("40factor_chol", minor)
solve := func(x, y, z *matrix.FloatMatrix) (err error) {
// Solve
//
// [ 0 A' GG'*W^{-1} ] [ ux ] [ bx ]
// [ A 0 0 ] * [ uy ] = [ by ]
// [ W^{-T}*GG 0 -I ] [ W*uz ] [ W^{-T}*bz ]
//
// and return ux, uy, W*uz.
//
// On entry, x, y, z contain bx, by, bz. On exit, they contain
// the solution ux, uy, W*uz.
//
// If we change variables ux = Q1*v + Q2*w, the system becomes
//
// [ K11 K12 R ] [ v ] [Q1'*(bx+GG'*W^{-1}*W^{-T}*bz)]
// [ K21 K22 0 ] * [ w ] = [Q2'*(bx+GG'*W^{-1}*W^{-T}*bz)]
// [ R^T 0 0 ] [ uy ] [by ]
//
// W*uz = W^{-T} * ( GG*ux - bz ).
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
// bzp := W^{-T} * bz in packed storage
scale(z, W, true, true)
pack(z, bzp, dims, &la.IOpt{"mnl", mnl})
// x := [Q1, Q2]' * (x + Gs' * bzp)
// = [Q1, Q2]' * (bx + Gs' * W^{-T} * bz)
blas.GemvFloat(Gs, bzp, x, 1.0, 1.0, la.OptTrans, &la.IOpt{"m", cdim_pckd})
lapack.Ormqr(QA, tauA, x, la.OptLeft, la.OptTrans)
// y := x[:p]
// = Q1' * (bx + Gs' * W^{-T} * bz)
blas.Copy(y, yy)
blas.Copy(x, y, &la.IOpt{"n", p})
// x[:p] := v = R^{-T} * by
blas.Copy(yy, x)
lapack.Trtrs(QA, x, la.OptUpper, la.OptTrans, &la.IOpt{"n", p})
// x[p:] := K22^{-1} * (x[p:] - K21*x[:p])
// = K22^{-1} * (Q2' * (bx + Gs' * W^{-T} * bz) - K21*v)
blas.GemvFloat(K, x, x, -1.0, 1.0, &la.IOpt{"m", n - p}, &la.IOpt{"n", p},
&la.IOpt{"offseta", p}, &la.IOpt{"offsety", p})
lapack.Potrs(K, x, &la.IOpt{"n", n - p}, &la.IOpt{"offseta", p * (n + 1)},
&la.IOpt{"offsetb", p})
// y := y - [K11, K12] * x
// = Q1' * (bx + Gs' * W^{-T} * bz) - K11*v - K12*w
blas.GemvFloat(K, x, y, -1.0, 1.0, &la.IOpt{"m", p}, &la.IOpt{"n", n})
// y := R^{-1}*y
// = R^{-1} * (Q1' * (bx + Gs' * W^{-T} * bz) - K11*v
// - K12*w)
lapack.Trtrs(QA, y, la.OptUpper, &la.IOpt{"n", p})
// x := [Q1, Q2] * x
lapack.Ormqr(QA, tauA, x, la.OptLeft)
// bzp := Gs * x - bzp.
// = W^{-T} * ( GG*ux - bz ) in packed storage.
// Unpack and copy to z.
blas.GemvFloat(Gs, x, bzp, 1.0, -1.0, &la.IOpt{"m", cdim_pckd})
unpack(bzp, z, dims, &la.IOpt{"mnl", mnl})
checkpnt.Check("90solve_chol", minor)
return nil
}
return solve, err
}
return factor, nil
}