func syrkTest(t *testing.T, C, A *matrix.FloatMatrix, flags Flags, vlen, nb int) bool {
//var B0 *matrix.FloatMatrix
P := A.Cols()
S := 0
E := C.Rows()
C0 := C.Copy()
trans := linalg.OptNoTrans
if flags&TRANSA != 0 {
trans = linalg.OptTrans
P = A.Rows()
}
uplo := linalg.OptUpper
if flags&LOWER != 0 {
uplo = linalg.OptLower
}
blas.SyrkFloat(A, C0, 1.0, 1.0, uplo, trans)
if A.Rows() < 8 {
//t.Logf("..A\n%v\n", A)
t.Logf(" BLAS C0:\n%v\n", C0)
}
Ar := A.FloatArray()
Cr := C.FloatArray()
DSymmRankBlk(Cr, Ar, 1.0, 1.0, flags, C.LeadingIndex(), A.LeadingIndex(),
P, S, E, vlen, nb)
result := C0.AllClose(C)
t.Logf(" C0 == C: %v\n", result)
if A.Rows() < 8 {
t.Logf(" DMRank C:\n%v\n", C)
}
return result
}
func (gp *gpConvexProg) F2(x, z *matrix.FloatMatrix) (f, Df, H *matrix.FloatMatrix, err error) {
err = nil
f = matrix.FloatZeros(gp.mnl+1, 1)
Df = matrix.FloatZeros(gp.mnl+1, gp.n)
H = matrix.FloatZeros(gp.n, gp.n)
y := gp.g.Copy()
Fsc := matrix.FloatZeros(gp.maxK, gp.n)
blas.GemvFloat(gp.F, x, y, 1.0, 1.0)
//fmt.Printf("y=\n%v\n", y.ToString("%.3f"))
for i, s := range gp.ind {
start := s[0]
stop := s[1]
// yi := exp(yi) = exp(Fi*x+gi)
ymax := maxvec(y.FloatArray()[start:stop])
ynew := matrix.Exp(matrix.FloatVector(y.FloatArray()[start:stop]).Add(-ymax))
y.SetIndexesFromArray(ynew.FloatArray(), matrix.Indexes(start, stop)...)
// fi = log sum yi = log sum exp(Fi*x+gi)
ysum := blas.AsumFloat(y, &la.IOpt{"n", stop - start}, &la.IOpt{"offset", start})
f.SetIndex(i, ymax+math.Log(ysum))
blas.ScalFloat(y, 1.0/ysum, &la.IOpt{"n", stop - start}, &la.IOpt{"offset", start})
blas.GemvFloat(gp.F, y, Df, 1.0, 0.0, la.OptTrans, &la.IOpt{"m", stop - start},
&la.IOpt{"incy", gp.mnl + 1}, &la.IOpt{"offseta", start},
&la.IOpt{"offsetx", start}, &la.IOpt{"offsety", i})
Fsc.SetSubMatrix(0, 0, gp.F.GetSubMatrix(start, 0, stop-start))
for k := start; k < stop; k++ {
blas.AxpyFloat(Df, Fsc, -1.0, &la.IOpt{"n", gp.n},
&la.IOpt{"incx", gp.mnl + 1}, &la.IOpt{"incy", Fsc.Rows()},
&la.IOpt{"offsetx", i}, &la.IOpt{"offsety", k - start})
blas.ScalFloat(Fsc, math.Sqrt(y.GetIndex(k)),
&la.IOpt{"inc", Fsc.Rows()}, &la.IOpt{"offset", k - start})
}
// H += z[i]*Hi = z[i] *Fisc' * Fisc
blas.SyrkFloat(Fsc, H, z.GetIndex(i), 1.0, la.OptTrans,
&la.IOpt{"k", stop - start})
}
return
}
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
}
// Computes analytic center of A*x <= b with A m by n of rank n.
// We assume that b > 0 and the feasible set is bounded.
func acent(A, b *matrix.FloatMatrix, niters int) (x *matrix.FloatMatrix, ntdecrs []float64, err error) {
err = nil
if niters <= 0 {
niters = MAXITERS
}
ntdecrs = make([]float64, 0, niters)
if A.Rows() != b.Rows() {
return nil, nil, errors.New("A.Rows() != b.Rows()")
}
m, n := A.Size()
x = matrix.FloatZeros(n, 1)
H := matrix.FloatZeros(n, n)
// Helper m*n matrix
Dmn := matrix.FloatZeros(m, n)
for i := 0; i < niters; i++ {
// Gradient is g = A^T * (1.0/(b - A*x)). d = 1.0/(b - A*x)
// d is m*1 matrix, g is n*1 matrix
d := matrix.Minus(b, matrix.Times(A, x)).Inv()
g := matrix.Times(A.Transpose(), d)
// Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
// in the original python code expression d[:,n*[0]] creates
// a m*n matrix where each column is copy of column 0.
// We do it here manually.
for i := 0; i < n; i++ {
Dmn.SetColumn(i, d)
}
// Function mul creates element wise product of matrices.
Asc := matrix.Mul(Dmn, A)
blas.SyrkFloat(Asc, H, 1.0, 0.0, linalg.OptTrans)
// Newton step is v = H^-1 * g.
v := matrix.Scale(g, -1.0)
PosvFloat(H, v)
// Directional derivative and Newton decrement.
lam := blas.DotFloat(g, v)
ntdecrs = append(ntdecrs, math.Sqrt(-lam))
if ntdecrs[len(ntdecrs)-1] < TOL {
return x, ntdecrs, err
}
// Backtracking line search.
// y = d .* A*v
y := matrix.Mul(d, matrix.Times(A, v))
step := 1.0
for 1-step*y.Max() < 0 {
step *= BETA
}
search:
for {
// t = -step*y + 1 [e.g. t = 1 - step*y]
t := matrix.Scale(y, -step).Add(1.0)
// ts = sum(log(1-step*y))
ts := t.Log().Sum()
if -ts < ALPHA*step*lam {
break search
}
step *= BETA
}
v.Scale(step)
x.Plus(v)
}
// no solution !!
err = errors.New(fmt.Sprintf("Iteration %d exhausted\n", niters))
return x, ntdecrs, err
}
// 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
}