/*
Returns the Nesterov-Todd scaling W at points s and z, and stores the
scaled variable in lmbda.
W * z = W^{-T} * s = lmbda.
W is a MatrixSet with entries:
- W['dnl']: positive vector
- W['dnli']: componentwise inverse of W['dnl']
- W['d']: positive vector
- W['di']: componentwise inverse of W['d']
- W['v']: lists of 2nd order cone vectors with unit hyperbolic norms
- W['beta']: list of positive numbers
- W['r']: list of square matrices
- W['rti']: list of square matrices. rti[k] is the inverse transpose
of r[k].
*/
func computeScaling(s, z, lmbda *matrix.FloatMatrix, dims *DimensionSet, mnl int) (W *FloatMatrixSet, err error) {
/*DEBUGGED*/
err = nil
W = FloatSetNew("dnl", "dnli", "d", "di", "v", "beta", "r", "rti")
// For the nonlinear block:
//
// W['dnl'] = sqrt( s[:mnl] ./ z[:mnl] )
// W['dnli'] = sqrt( z[:mnl] ./ s[:mnl] )
// lambda[:mnl] = sqrt( s[:mnl] .* z[:mnl] )
var stmp, ztmp, lmd *matrix.FloatMatrix
if mnl > 0 {
stmp = matrix.FloatVector(s.FloatArray()[:mnl])
ztmp = matrix.FloatVector(z.FloatArray()[:mnl])
dnl := stmp.Div(ztmp)
dnl.Apply(dnl, math.Sqrt)
dnli := dnl.Copy()
dnli.Apply(dnli, func(a float64) float64 { return 1.0 / a })
W.Set("dnl", dnl)
W.Set("dnli", dnli)
lmd = stmp.Mul(ztmp)
lmd.Apply(lmd, math.Sqrt)
lmbda.SetIndexes(matrix.MakeIndexSet(0, mnl, 1), lmd.FloatArray())
} else {
mnl = 0
}
// For the 'l' block:
//
// W['d'] = sqrt( sk ./ zk )
// W['di'] = sqrt( zk ./ sk )
// lambdak = sqrt( sk .* zk )
//
// where sk and zk are the first dims['l'] entries of s and z.
// lambda_k is stored in the first dims['l'] positions of lmbda.
m := dims.At("l")[0]
td := s.FloatArray()
stmp = matrix.FloatVector(td[mnl : mnl+m])
zd := z.FloatArray()
//fmt.Printf("zdata=%v\n", zd[mnl:mnl+m])
ztmp = matrix.FloatVector(zd[mnl : mnl+m])
d := stmp.Div(ztmp)
d.Apply(d, math.Sqrt)
di := d.Copy()
di.Apply(di, func(a float64) float64 { return 1.0 / a })
//fmt.Printf("d:\n%v\n", d)
//fmt.Printf("di:\n%v\n", di)
W.Set("d", d)
W.Set("di", di)
lmd = stmp.Mul(ztmp)
lmd.Apply(lmd, math.Sqrt)
// lmd has indexes mnl:mnl+m and length of m
lmbda.SetIndexes(matrix.MakeIndexSet(mnl, mnl+m, 1), lmd.FloatArray())
//fmt.Printf("after l:\n%v\n", lmbda)
/*
For the 'q' blocks, compute lists 'v', 'beta'.
The vector v[k] has unit hyperbolic norm:
(sqrt( v[k]' * J * v[k] ) = 1 with J = [1, 0; 0, -I]).
beta[k] is a positive scalar.
The hyperbolic Householder matrix H = 2*v[k]*v[k]' - J
defined by v[k] satisfies
(beta[k] * H) * zk = (beta[k] * H) \ sk = lambda_k
where sk = s[indq[k]:indq[k+1]], zk = z[indq[k]:indq[k+1]].
lambda_k is stored in lmbda[indq[k]:indq[k+1]].
*/
ind := mnl + dims.At("l")[0]
var beta *matrix.FloatMatrix
for _, k := range dims.At("q") {
W.Append("v", matrix.FloatZeros(k, 1))
}
beta = matrix.FloatZeros(len(dims.At("q")), 1)
W.Set("beta", beta)
vset := W.At("v")
for k, m := range dims.At("q") {
v := vset[k]
// a = sqrt( sk' * J * sk ) where J = [1, 0; 0, -I]
aa := jnrm2(s, m, ind)
// b = sqrt( zk' * J * zk )
bb := jnrm2(z, m, ind)
// beta[k] = ( a / b )**1/2
beta.SetIndex(k, math.Sqrt(aa/bb))
// c = sqrt( (sk/a)' * (zk/b) + 1 ) / sqrt(2)
c0 := blas.DotFloat(s, z, &la_.IOpt{"n", m},
&la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind})
cc := math.Sqrt((c0/aa/bb + 1.0) / 2.0)
// vk = 1/(2*c) * ( (sk/a) + J * (zk/b) )
blas.CopyFloat(z, v, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m})
blas.ScalFloat(v, -1.0/bb)
v.SetIndex(0, -1.0*v.GetIndex(0))
blas.AxpyFloat(s, v, 1.0/aa, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m})
blas.ScalFloat(v, 1.0/2.0/cc)
// v[k] = 1/sqrt(2*(vk0 + 1)) * ( vk + e ), e = [1; 0]
v.SetIndex(0, v.GetIndex(0)+1.0)
blas.ScalFloat(v, (1.0 / math.Sqrt(2.0*v.GetIndex(0))))
/*
To get the scaled variable lambda_k
d = sk0/a + zk0/b + 2*c
lambda_k = [ c;
(c + zk0/b)/d * sk1/a + (c + sk0/a)/d * zk1/b ]
lambda_k *= sqrt(a * b)
*/
lmbda.SetIndex(ind, cc)
dd := 2*cc + s.GetIndex(ind)/aa + z.GetIndex(ind)/bb
blas.CopyFloat(s, lmbda, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1},
&la_.IOpt{"n", m - 1})
zz := (cc + z.GetIndex(ind)/bb) / dd / aa
ss := (cc + s.GetIndex(ind)/aa) / dd / bb
blas.ScalFloat(lmbda, zz, &la_.IOpt{"offset", ind + 1}, &la_.IOpt{"n", m - 1})
blas.AxpyFloat(z, lmbda, ss, &la_.IOpt{"offsetx", ind + 1},
&la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1})
blas.ScalFloat(lmbda, math.Sqrt(aa*bb), &la_.IOpt{"offset", ind}, &la_.IOpt{"n", m})
ind += m
//fmt.Printf("after q[%d]:\n%v\n", k, lmbda)
}
/*
For the 's' blocks: compute two lists 'r' and 'rti'.
r[k]' * sk^{-1} * r[k] = diag(lambda_k)^{-1}
r[k]' * zk * r[k] = diag(lambda_k)
where sk and zk are the entries inds[k] : inds[k+1] of
s and z, reshaped into symmetric matrices.
rti[k] is the inverse of r[k]', so
rti[k]' * sk * rti[k] = diag(lambda_k)^{-1}
rti[k]' * zk^{-1} * rti[k] = diag(lambda_k).
The vectors lambda_k are stored in
lmbda[ dims['l'] + sum(dims['q']) : -1 ]
*/
for _, k := range dims.At("s") {
W.Append("r", matrix.FloatZeros(k, k))
W.Append("rti", matrix.FloatZeros(k, k))
}
maxs := maxdim(dims.At("s"))
work := matrix.FloatZeros(maxs*maxs, 1)
Ls := matrix.FloatZeros(maxs*maxs, 1)
Lz := matrix.FloatZeros(maxs*maxs, 1)
ind2 := ind
for k, m := range dims.At("s") {
r := W.At("r")[k]
rti := W.At("rti")[k]
// Factor sk = Ls*Ls'; store Ls in ds[inds[k]:inds[k+1]].
blas.CopyFloat(s, Ls, &la_.IOpt{"offsetx", ind2}, &la_.IOpt{"n", m * m})
lapack.PotrfFloat(Ls, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m})
// Factor zs[k] = Lz*Lz'; store Lz in dz[inds[k]:inds[k+1]].
blas.CopyFloat(z, Lz, &la_.IOpt{"offsetx", ind2}, &la_.IOpt{"n", m * m})
lapack.PotrfFloat(Lz, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m})
// SVD Lz'*Ls = U*diag(lambda_k)*V'. Keep U in work.
for i := 0; i < m; i++ {
blas.ScalFloat(Ls, 0.0, &la_.IOpt{"offset", i * m}, &la_.IOpt{"n", i})
}
blas.CopyFloat(Ls, work, &la_.IOpt{"n", m * m})
blas.TrmmFloat(Lz, work, 1.0, la_.OptTransA, &la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m},
&la_.IOpt{"n", m}, &la_.IOpt{"m", m})
lapack.GesvdFloat(work, lmbda, nil, nil,
la_.OptJobuO, &la_.IOpt{"lda", m}, &la_.IOpt{"offsetS", ind},
&la_.IOpt{"n", m}, &la_.IOpt{"m", m})
// r = Lz^{-T} * U
blas.CopyFloat(work, r, &la_.IOpt{"n", m * m})
blas.TrsmFloat(Lz, r, 1.0, la_.OptTransA,
&la_.IOpt{"lda", m}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m})
// rti = Lz * U
blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m})
blas.TrmmFloat(Lz, rti, 1.0,
&la_.IOpt{"lda", m}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m})
// r := r * diag(sqrt(lambda_k))
// rti := rti * diag(1 ./ sqrt(lambda_k))
for i := 0; i < m; i++ {
a := math.Sqrt(lmbda.GetIndex(ind + i))
blas.ScalFloat(r, a, &la_.IOpt{"offset", m * i}, &la_.IOpt{"n", m})
blas.ScalFloat(rti, 1.0/a, &la_.IOpt{"offset", m * i}, &la_.IOpt{"n", m})
}
ind += m
ind2 += m * m
}
return
}
func updateScaling(W *FloatMatrixSet, lmbda, s, z *matrix.FloatMatrix) (err error) {
err = nil
var stmp, ztmp *matrix.FloatMatrix
/*
Nonlinear and 'l' blocks
d := d .* sqrt( s ./ z )
lmbda := lmbda .* sqrt(s) .* sqrt(z)
*/
mnl := 0
dnlset := W.At("dnl")
dnliset := W.At("dnli")
dset := W.At("d")
diset := W.At("di")
beta := W.At("beta")[0]
if dnlset != nil && dnlset[0].NumElements() > 0 {
mnl = dnlset[0].NumElements()
}
ml := dset[0].NumElements()
m := mnl + ml
//fmt.Printf("ml=%d, mnl=%d, m=%d'n", ml, mnl, m)
stmp = matrix.FloatVector(s.FloatArray()[:m])
stmp.Apply(stmp, math.Sqrt)
s.SetIndexes(matrix.MakeIndexSet(0, m, 1), stmp.FloatArray())
ztmp = matrix.FloatVector(z.FloatArray()[:m])
ztmp.Apply(ztmp, math.Sqrt)
z.SetIndexes(matrix.MakeIndexSet(0, m, 1), ztmp.FloatArray())
// d := d .* s .* z
if len(dnlset) > 0 {
blas.TbmvFloat(s, dnlset[0], &la_.IOpt{"n", mnl}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
blas.TbsvFloat(z, dnlset[0], &la_.IOpt{"n", mnl}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
dnliset[0].Apply(dnlset[0], func(a float64) float64 { return 1.0 / a })
}
blas.TbmvFloat(s, dset[0], &la_.IOpt{"n", ml},
&la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}, &la_.IOpt{"offseta", mnl})
blas.TbsvFloat(z, dset[0], &la_.IOpt{"n", ml},
&la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}, &la_.IOpt{"offseta", mnl})
diset[0].Apply(dset[0], func(a float64) float64 { return 1.0 / a })
// lmbda := s .* z
blas.CopyFloat(s, lmbda, &la_.IOpt{"n", m})
blas.TbmvFloat(z, lmbda, &la_.IOpt{"n", m}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
//fmt.Printf("-- end of l:\nz=\n%v\nlmbda=\n%v\n", z.ConvertToString(), lmbda.ConvertToString())
//fmt.Printf("W[d]=\n%v\n", dset[0].ConvertToString())
//fmt.Printf("W[di]=\n%v\n", diset[0].ConvertToString())
// 'q' blocks.
// Let st and zt be the new variables in the old scaling:
//
// st = s_k, zt = z_k
//
// and a = sqrt(st' * J * st), b = sqrt(zt' * J * zt).
//
// 1. Compute the hyperbolic Householder transformation 2*q*q' - J
// that maps st/a to zt/b.
//
// c = sqrt( (1 + st'*zt/(a*b)) / 2 )
// q = (st/a + J*zt/b) / (2*c).
//
// The new scaling point is
//
// wk := betak * sqrt(a/b) * (2*v[k]*v[k]' - J) * q
//
// with betak = W['beta'][k].
//
// 3. The scaled variable:
//
// lambda_k0 = sqrt(a*b) * c
// lambda_k1 = sqrt(a*b) * ( (2vk*vk' - J) * (-d*q + u/2) )_1
//
// where
//
// u = st/a - J*zt/b
// d = ( vk0 * (vk'*u) + u0/2 ) / (2*vk0 *(vk'*q) - q0 + 1).
//
// 4. Update scaling
//
// v[k] := wk^1/2
// = 1 / sqrt(2*(wk0 + 1)) * (wk + e).
// beta[k] *= sqrt(a/b)
ind := m
for k, v := range W.At("v") {
m = v.NumElements()
// ln = sqrt( lambda_k' * J * lambda_k ) !! NOT USED!!
jnrm2(lmbda, m, ind) // ?? NOT USED ??
// a = sqrt( sk' * J * sk ) = sqrt( st' * J * st )
// s := s / a = st / a
aa := jnrm2(s, m, ind)
blas.ScalFloat(s, 1.0/aa, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
// b = sqrt( zk' * J * zk ) = sqrt( zt' * J * zt )
// z := z / a = zt / b
bb := jnrm2(z, m, ind)
blas.ScalFloat(z, 1.0/bb, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
// c = sqrt( ( 1 + (st'*zt) / (a*b) ) / 2 )
cc := blas.DotFloat(s, z, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind},
&la_.IOpt{"n", m})
cc = math.Sqrt((1.0 + cc) / 2.0)
// vs = v' * st / a
vs := blas.DotFloat(v, s, &la_.IOpt{"offsety", ind}, &la_.IOpt{"n", m})
// vz = v' * J *zt / b
vz := jdot(v, z, m, 0, ind)
// vq = v' * q where q = (st/a + J * zt/b) / (2 * c)
vq := (vs + vz) / 2.0 / cc
// vq = v' * q where q = (st/a + J * zt/b) / (2 * c)
vu := vs - vz
// lambda_k0 = c
lmbda.SetIndex(ind, cc)
// wk0 = 2 * vk0 * (vk' * q) - q0
wk0 := 2.0*v.GetIndex(0)*vq - (s.GetIndex(ind)+z.GetIndex(ind))/2.0/cc
// d = (v[0] * (vk' * u) - u0/2) / (wk0 + 1)
dd := (v.GetIndex(0)*vu - s.GetIndex(ind)/2.0 + z.GetIndex(ind)/2.0) / (wk0 + 1.0)
// lambda_k1 = 2 * v_k1 * vk' * (-d*q + u/2) - d*q1 + u1/2
blas.CopyFloat(v, lmbda, &la_.IOpt{"offsetx", 1}, &la_.IOpt{"offsety", ind + 1},
&la_.IOpt{"n", m - 1})
blas.ScalFloat(lmbda, (2.0 * (-dd*vq + 0.5*vu)),
&la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1})
blas.AxpyFloat(s, lmbda, 0.5*(1.0-dd/cc),
&la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1})
blas.AxpyFloat(z, lmbda, 0.5*(1.0+dd/cc),
&la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1})
// Scale so that sqrt(lambda_k' * J * lambda_k) = sqrt(aa*bb).
blas.ScalFloat(lmbda, math.Sqrt(aa*bb), &la_.IOpt{"offset", ind}, &la_.IOpt{"n", m})
// v := (2*v*v' - J) * q
// = 2 * (v'*q) * v' - (J* st/a + zt/b) / (2*c)
blas.ScalFloat(v, 2.0*vq)
v.SetIndex(0, v.GetIndex(0)-(s.GetIndex(ind)/2.0/cc))
blas.AxpyFloat(s, v, 0.5/cc, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", 1},
&la_.IOpt{"n", m - 1})
blas.AxpyFloat(z, v, -0.5/cc, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m})
// v := v^{1/2} = 1/sqrt(2 * (v0 + 1)) * (v + e)
v0 := v.GetIndex(0) + 1.0
v.SetIndex(0, v0)
blas.ScalFloat(v, 1.0/math.Sqrt(2.0*v0))
// beta[k] *= ( aa / bb )**1/2
bk := beta.GetIndex(k)
beta.SetIndex(k, bk*math.Sqrt(aa/bb))
ind += m
}
//fmt.Printf("-- end of q:\nz=\n%v\nlmbda=\n%v\n", z.ConvertToString(), lmbda.ConvertToString())
//fmt.Printf("beta=\n%v\n", beta.ConvertToString())
// 's' blocks
//
// Let st, zt be the updated variables in the old scaling:
//
// st = Ls * Ls', zt = Lz * Lz'.
//
// where Ls and Lz are the 's' components of s, z.
//
// 1. SVD Lz'*Ls = Uk * lambda_k^+ * Vk'.
//
// 2. New scaling is
//
// r[k] := r[k] * Ls * Vk * diag(lambda_k^+)^{-1/2}
// rti[k] := r[k] * Lz * Uk * diag(lambda_k^+)^{-1/2}.
//
maxr := 0
for _, m := range W.At("r") {
if m.Rows() > maxr {
maxr = m.Rows()
}
}
work := matrix.FloatZeros(maxr*maxr, 1)
vlensum := 0
for _, m := range W.At("v") {
vlensum += m.NumElements()
}
ind = mnl + ml + vlensum
ind2 := ind
ind3 := 0
rset := W.At("r")
rtiset := W.At("rti")
for k, _ := range rset {
r := rset[k]
rti := rtiset[k]
m = r.Rows()
//fmt.Printf("m=%d, r=\n%v\nrti=\n%v\n", m, r.ConvertToString(), rti.ConvertToString())
// r := r*sk = r*Ls
blas.GemmFloat(r, s, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m},
&la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m},
&la_.IOpt{"offsetb", ind2})
//fmt.Printf("1 work=\n%v\n", work.ConvertToString())
blas.CopyFloat(work, r, &la_.IOpt{"n", m * m})
// rti := rti*zk = rti*Lz
blas.GemmFloat(rti, z, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m},
&la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m},
&la_.IOpt{"offsetb", ind2})
//fmt.Printf("2 work=\n%v\n", work.ConvertToString())
blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m})
// SVD Lz'*Ls = U * lmbds^+ * V'; store U in sk and V' in zk. '
blas.GemmFloat(z, s, work, 1.0, 0.0, la_.OptTransA, &la_.IOpt{"m", m},
&la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m},
&la_.IOpt{"ldc", m}, &la_.IOpt{"offseta", ind2}, &la_.IOpt{"offsetb", ind2})
//fmt.Printf("3 work=\n%v\n", work.ConvertToString())
// U = s, Vt = z
lapack.GesvdFloat(work, lmbda, s, z, la_.OptJobuAll, la_.OptJobvtAll,
&la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m}, &la_.IOpt{"ldu", m},
&la_.IOpt{"ldvt", m}, &la_.IOpt{"offsets", ind}, &la_.IOpt{"offsetu", ind2},
&la_.IOpt{"offsetvt", ind2})
// r := r*V
blas.GemmFloat(r, z, work, 1.0, 0.0, la_.OptTransB, &la_.IOpt{"m", m},
&la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m},
&la_.IOpt{"offsetb", ind2})
//fmt.Printf("4 work=\n%v\n", work.ConvertToString())
blas.CopyFloat(work, r, &la_.IOpt{"n", m * m})
// rti := rti*U
blas.GemmFloat(rti, s, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m},
&la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m},
&la_.IOpt{"offsetb", ind2})
//fmt.Printf("5 work=\n%v\n", work.ConvertToString())
blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m})
for i := 0; i < m; i++ {
a := 1.0 / math.Sqrt(lmbda.GetIndex(ind+i))
blas.ScalFloat(r, a, &la_.IOpt{"n", m}, &la_.IOpt{"offset", m * i})
blas.ScalFloat(rti, a, &la_.IOpt{"n", m}, &la_.IOpt{"offset", m * i})
}
ind += m
ind2 += m * m
ind3 += m // !!NOT USED: ind3!!
}
//fmt.Printf("-- end of s:\nz=\n%v\nlmbda=\n%v\n", z.ConvertToString(), lmbda.ConvertToString())
return
}