/*
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
}
// Solves a pair of primal and dual cone programs
//
// minimize c'*x
// subject to G*x + s = h
// A*x = b
// s >= 0
//
// maximize -h'*z - b'*y
// subject to G'*z + A'*y + c = 0
// z >= 0.
//
// The inequalities are with respect to a cone C defined as the Cartesian
// product of N + M + 1 cones:
//
// C = C_0 x C_1 x .... x C_N x C_{N+1} x ... x C_{N+M}.
//
// The first cone C_0 is the nonnegative orthant of dimension ml.
// The next N cones are second order cones of dimension mq[0], ...,
// mq[N-1]. The second order cone of dimension m is defined as
//
// { (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.
//
// The next M cones are positive semidefinite cones of order ms[0], ...,
// ms[M-1] >= 0.
//
func ConeLp(c, G, h, A, b *matrix.FloatMatrix, dims *DimensionSet, solopts *SolverOptions, primalstart, dualstart *FloatMatrixSet) (sol *Solution, err error) {
err = nil
const EXPON = 3
const STEP = 0.99
sol = &Solution{Unknown,
nil, nil, nil, nil, nil,
0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0}
//var primalstart *FloatMatrixSet = nil
//var dualstart *FloatMatrixSet = nil
var refinement int
if solopts.Refinement > 0 {
refinement = solopts.Refinement
} else {
refinement = 0
if len(dims.At("q")) > 0 || len(dims.At("s")) > 0 {
refinement = 1
}
}
feasTolerance := FEASTOL
absTolerance := ABSTOL
relTolerance := RELTOL
if solopts.FeasTol > 0.0 {
feasTolerance = solopts.FeasTol
}
if solopts.AbsTol > 0.0 {
absTolerance = solopts.AbsTol
}
if solopts.RelTol > 0.0 {
relTolerance = solopts.RelTol
}
solvername := solopts.KKTSolverName
if len(solvername) == 0 {
if dims != nil && (len(dims.At("q")) > 0 || len(dims.At("s")) > 0) {
solvername = "qr"
} else {
solvername = "chol2"
}
}
if c == nil || c.Cols() > 1 {
err = errors.New("'c' must be matrix with 1 column")
return
}
if h == nil || h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = DSetNew("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
if err = checkConeLpDimensions(dims); err != nil {
return
}
cdim := dims.Sum("l", "q") + dims.SumSquared("s")
cdim_diag := dims.Sum("l", "q", "s")
if h.Rows() != cdim {
err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim))
return
}
// Data for kth 'q' constraint are found in rows indq[k]:indq[k+1] of G.
indq := make([]int, 0, 100)
indq = append(indq, dims.At("l")[0])
for _, k := range dims.At("q") {
indq = append(indq, indq[len(indq)-1]+k)
}
// Data for kth 's' constraint are found in rows inds[k]:inds[k+1] of G.
inds := make([]int, 0, 100)
inds = append(inds, indq[len(indq)-1])
for _, k := range dims.At("s") {
inds = append(inds, inds[len(inds)-1]+k*k)
}
if G != nil && !G.SizeMatch(cdim, c.Rows()) {
estr := fmt.Sprintf("'G' must be of size (%d,%d)", cdim, c.Rows())
err = errors.New(estr)
return
}
Gf := func(x, y *matrix.FloatMatrix, alpha, beta float64, opts ...la.Option) error {
return sgemv(G, x, y, alpha, beta, dims, opts...)
}
// Check A and set defaults if it is nil
if A == nil {
// zeros rows reduces Gemv to vector products
A = matrix.FloatZeros(0, c.Rows())
}
if A.Cols() != c.Rows() {
estr := fmt.Sprintf("'A' must have %d columns", c.Rows())
err = errors.New(estr)
return
}
Af := func(x, y *matrix.FloatMatrix, alpha, beta float64, opts ...la.Option) error {
return blas.GemvFloat(A, x, y, alpha, beta, opts...)
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() != A.Rows() {
estr := fmt.Sprintf("'b' must have length %d", A.Rows())
err = errors.New(estr)
return
}
// kktsolver(W) returns a routine for solving 3x3 block KKT system
//
// [ 0 A' G'*W^{-1} ] [ ux ] [ bx ]
// [ A 0 0 ] [ uy ] = [ by ].
// [ G 0 -W' ] [ uz ] [ bz ]
var factor kktFactor
var kktsolver kktFactor = nil
if kktfunc, ok := solvers[solvername]; ok {
// kkt function returns us problem spesific factor function.
factor, err = kktfunc(G, dims, A, 0)
// solver is
kktsolver = func(W *FloatMatrixSet, H, Df *matrix.FloatMatrix) (kktFunc, error) {
return factor(W, nil, nil)
}
} else {
err = errors.New(fmt.Sprintf("solver '%s' not known", solvername))
return
}
// res() evaluates residual in 5x5 block KKT system
//
// [ vx ] [ 0 ] [ 0 A' G' c ] [ ux ]
// [ vy ] [ 0 ] [-A 0 0 b ] [ uy ]
// [ vz ] += [ W'*us ] - [-G 0 0 h ] [ W^{-1}*uz ]
// [ vtau ] [ dg*ukappa ] [-c' -b' -h' 0 ] [ utau/dg ]
//
// vs += lmbda o (dz + ds)
// vkappa += lmbdg * (dtau + dkappa).
ws3 := matrix.FloatZeros(cdim, 1)
wz3 := matrix.FloatZeros(cdim, 1)
//
res := func(ux, uy, uz, utau, us, ukappa, vx, vy, vz, vtau, vs, vkappa *matrix.FloatMatrix, W *FloatMatrixSet, dg float64, lmbda *matrix.FloatMatrix) (err error) {
err = nil
// vx := vx - A'*uy - G'*W^{-1}*uz - c*utau/dg
Af(uy, vx, -1.0, 1.0, la.OptTrans)
//fmt.Printf("post-Af vx=\n%v\n", vx)
blas.Copy(uz, wz3)
scale(wz3, W, false, true)
Gf(wz3, vx, -1.0, 1.0, la.OptTrans)
blas.AxpyFloat(c, vx, -utau.Float()/dg)
// vy := vy + A*ux - b*utau/dg
Af(ux, vy, 1.0, 1.0)
blas.AxpyFloat(b, vy, -utau.Float()/dg)
// vz := vz + G*ux - h*utau/dg + W'*us
Gf(ux, vz, 1.0, 1.0)
blas.AxpyFloat(h, vz, -utau.Float()/dg)
blas.Copy(us, ws3)
scale(ws3, W, true, false)
blas.AxpyFloat(ws3, vz, 1.0)
// vtau := vtau + c'*ux + b'*uy + h'*W^{-1}*uz + dg*ukappa
var vtauplus float64 = dg*ukappa.Float() + blas.DotFloat(c, ux) +
blas.DotFloat(b, uy) + sdot(h, wz3, dims, 0)
vtau.SetValue(vtau.Float() + vtauplus)
// vs := vs + lmbda o (uz + us)
blas.Copy(us, ws3)
blas.AxpyFloat(uz, ws3, 1.0)
sprod(ws3, lmbda, dims, 0, &la.SOpt{"diag", "D"})
blas.AxpyFloat(ws3, vs, 1.0)
// vkappa += vkappa + lmbdag * (utau + ukappa)
lscale := lmbda.GetIndex(lmbda.NumElements() - 1)
var vkplus float64 = lscale * (utau.Float() + ukappa.Float())
vkappa.SetValue(vkappa.Float() + vkplus)
return
}
resx0 := math.Max(1.0, math.Sqrt(blas.DotFloat(c, c)))
resy0 := math.Max(1.0, math.Sqrt(blas.DotFloat(b, b)))
resz0 := math.Max(1.0, snrm2(h, dims, 0))
// select initial points
//fmt.Printf("** initial resx0=%.4f, resy0=%.4f, resz0=%.4f \n", resx0, resy0, resz0)
x := c.Copy()
blas.ScalFloat(x, 0.0)
y := b.Copy()
blas.ScalFloat(y, 0.0)
s := matrix.FloatZeros(cdim, 1)
z := matrix.FloatZeros(cdim, 1)
dx := c.Copy()
dy := b.Copy()
ds := matrix.FloatZeros(cdim, 1)
dz := matrix.FloatZeros(cdim, 1)
// these are singleton matrix
dkappa := matrix.FloatValue(0.0)
dtau := matrix.FloatValue(0.0)
var W *FloatMatrixSet
var f kktFunc
if primalstart == nil || dualstart == nil {
// Factor
//
// [ 0 A' G' ]
// [ A 0 0 ].
// [ G 0 -I ]
//
W = FloatSetNew("d", "di", "v", "beta", "r", "rti")
dd := dims.At("l")[0]
mat := matrix.FloatOnes(dd, 1)
W.Set("d", mat)
mat = matrix.FloatOnes(dd, 1)
W.Set("di", mat)
dq := len(dims.At("q"))
W.Set("beta", matrix.FloatOnes(dq, 1))
for _, n := range dims.At("q") {
vm := matrix.FloatZeros(n, 1)
vm.SetIndex(0, 1.0)
W.Append("v", vm)
}
for _, n := range dims.At("s") {
W.Append("r", matrix.FloatIdentity(n))
W.Append("rti", matrix.FloatIdentity(n))
}
f, err = kktsolver(W, nil, nil)
if err != nil {
fmt.Printf("kktsolver error: %s\n", err)
return
}
}
if primalstart == nil {
// minimize || G * x - h ||^2
// subject to A * x = b
//
// by solving
//
// [ 0 A' G' ] [ x ] [ 0 ]
// [ A 0 0 ] * [ dy ] = [ b ].
// [ G 0 -I ] [ -s ] [ h ]
blas.ScalFloat(x, 0.0)
blas.CopyFloat(y, dy)
blas.CopyFloat(h, s)
err = f(x, dy, s)
if err != nil {
fmt.Printf("f(x,dy,s): %s\n", err)
return
}
blas.ScalFloat(s, -1.0)
//fmt.Printf("** initial s:\n%v\n", s)
} else {
blas.Copy(primalstart.At("x")[0], x)
blas.Copy(primalstart.At("s")[0], s)
}
// ts = min{ t | s + t*e >= 0 }
ts, _ := maxStep(s, dims, 0, nil)
if ts >= 0 && primalstart != nil {
err = errors.New("initial s is not positive")
return
}
if dualstart == nil {
// minimize || z ||^2
// subject to G'*z + A'*y + c = 0
//
// by solving
//
// [ 0 A' G' ] [ dx ] [ -c ]
// [ A 0 0 ] [ y ] = [ 0 ].
// [ G 0 -I ] [ z ] [ 0 ]
blas.Copy(c, dx)
blas.ScalFloat(dx, -1.0)
blas.ScalFloat(y, 0.0)
err = f(dx, y, z)
if err != nil {
fmt.Printf("f(dx,y,z): %s\n", err)
return
}
} else {
if len(dualstart.At("y")) > 0 {
blas.Copy(dualstart.At("y")[0], y)
}
blas.Copy(dualstart.At("z")[0], z)
}
// ts = min{ t | z + t*e >= 0 }
tz, _ := maxStep(z, dims, 0, nil)
if tz >= 0 && dualstart != nil {
err = errors.New("initial z is not positive")
return
}
nrms := snrm2(s, dims, 0)
nrmz := snrm2(z, dims, 0)
gap := 0.0
pcost := 0.0
dcost := 0.0
relgap := 0.0
if primalstart == nil && dualstart == nil {
gap = sdot(s, z, dims, 0)
pcost = blas.DotFloat(c, x)
dcost = -blas.DotFloat(b, y) - sdot(h, z, dims, 0)
if pcost < 0.0 {
relgap = gap / -pcost
} else if dcost > 0.0 {
relgap = gap / dcost
} else {
relgap = math.NaN()
}
if ts <= 0 && tz < 0 &&
(gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance)) {
// Constructed initial points happen to be feasible and optimal
ind := dims.At("l")[0] + dims.Sum("q")
for _, m := range dims.At("s") {
symm(s, m, ind)
symm(z, m, ind)
ind += m * m
}
// rx = A'*y + G'*z + c
rx := c.Copy()
Af(y, rx, 1.0, 1.0, la.OptTrans)
Gf(z, rx, 1.0, 1.0, la.OptTrans)
resx := math.Sqrt(blas.Dot(rx, rx).Float())
// ry = b - A*x
ry := b.Copy()
Af(x, ry, -1.0, -1.0)
resy := math.Sqrt(blas.Dot(ry, ry).Float())
// rz = s + G*x - h
rz := matrix.FloatZeros(cdim, 1)
Gf(x, rz, 1.0, 0.0)
blas.AxpyFloat(s, rz, 1.0)
blas.AxpyFloat(h, rz, -1.0)
resz := snrm2(rz, dims, 0)
pres := math.Max(resy/resy0, resz/resz0)
dres := resx / resx0
cx := blas.Dot(c, x).Float()
by := blas.Dot(b, y).Float()
hz := sdot(h, z, dims, 0)
sol.X = x
sol.Y = y
sol.S = s
sol.Z = z
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", x)
sol.Result.Append("y", y)
sol.Result.Append("s", s)
sol.Result.Append("z", z)
sol.Status = Optimal
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = cx
sol.DualObjective = -(by + hz)
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
return
}
if ts >= -1e-8*math.Max(nrms, 1.0) {
a := 1.0 + ts
is := make([]int, 0)
// indexes s[:dims['l']]
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
// indexes s[indq[:-1]]
is = append(is, indq[:len(indq)-1]...)
// indexes s[ind:ind+m*m:m+1] (diagonal)
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
}
for _, k := range is {
s.SetIndex(k, a+s.GetIndex(k))
}
//fmt.Printf("scaled s=\n%v\n", s.ConvertToString())
}
if tz >= -1e-8*math.Max(nrmz, 1.0) {
a := 1.0 + tz
is := make([]int, 0)
// indexes z[:dims['l']]
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
// indexes z[indq[:-1]]
is = append(is, indq[:len(indq)-1]...)
// indexes z[ind:ind+m*m:m+1] (diagonal)
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
}
for _, k := range is {
z.SetIndex(k, a+z.GetIndex(k))
}
//fmt.Printf("scaled z=\n%v\n", z.ConvertToString())
}
} else if primalstart == nil && dualstart != nil {
if ts >= -1e-8*math.Max(nrms, 1.0) {
a := 1.0 + ts
is := make([]int, 0)
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
is = append(is, indq[:len(indq)-1]...)
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
}
for _, k := range is {
s.SetIndex(k, a+s.GetIndex(k))
}
}
} else if primalstart != nil && dualstart == nil {
if tz >= -1e-8*math.Max(nrmz, 1.0) {
a := 1.0 + tz
is := make([]int, 0)
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
is = append(is, indq[:len(indq)-1]...)
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
}
for _, k := range is {
z.SetIndex(k, a+z.GetIndex(k))
}
}
}
tau := matrix.FloatValue(1.0)
kappa := matrix.FloatValue(1.0)
wkappa3 := matrix.FloatValue(0.0)
rx := c.Copy()
hrx := c.Copy()
ry := b.Copy()
hry := b.Copy()
rz := matrix.FloatZeros(cdim, 1)
hrz := matrix.FloatZeros(cdim, 1)
sigs := matrix.FloatZeros(dims.Sum("s"), 1)
sigz := matrix.FloatZeros(dims.Sum("s"), 1)
lmbda := matrix.FloatZeros(cdim_diag+1, 1)
lmbdasq := matrix.FloatZeros(cdim_diag+1, 1)
gap = sdot(s, z, dims, 0)
var x1, y1, z1 *matrix.FloatMatrix
var dg, dgi float64
var th *matrix.FloatMatrix
var WS fClosure
var f3 kktFunc
//fmt.Printf("preloop x=\n%v\n", x.ConvertToString())
//fmt.Printf("preloop z=\n%v\n", z.ConvertToString())
//fmt.Printf("preloop s=\n%v\n", s.ConvertToString())
for iter := 0; iter < solopts.MaxIter+1; iter++ {
// hrx = -A'*y - G'*z
Af(y, hrx, -1.0, 0.0, la.OptTrans)
Gf(z, hrx, -1.0, 1.0, la.OptTrans)
hresx := math.Sqrt(blas.DotFloat(hrx, hrx))
// rx = hrx - c*tau
// = -A'*y - G'*z - c*tau
blas.Copy(hrx, rx)
err = blas.AxpyFloat(c, rx, -tau.Float())
resx := math.Sqrt(blas.DotFloat(rx, rx)) / tau.Float()
// hry = A*x
Af(x, hry, 1.0, 0.0)
hresy := math.Sqrt(blas.DotFloat(hry, hry))
// ry = hry - b*tau
// = A*x - b*tau
blas.Copy(hry, ry)
blas.AxpyFloat(b, ry, -tau.Float())
resy := math.Sqrt(blas.DotFloat(ry, ry)) / tau.Float()
// hrz = s + G*x
Gf(x, hrz, 1.0, 0.0)
blas.AxpyFloat(s, hrz, 1.0)
hresz := snrm2(hrz, dims, 0)
// rz = hrz - h*tau
// = s + G*x - h*tau
blas.ScalFloat(rz, 0.0)
blas.AxpyFloat(hrz, rz, 1.0)
blas.AxpyFloat(h, rz, -tau.Float())
resz := snrm2(rz, dims, 0) / tau.Float()
// rt = kappa + c'*x + b'*y + h'*z '
cx := blas.DotFloat(c, x)
by := blas.DotFloat(b, y)
hz := sdot(h, z, dims, 0)
rt := kappa.Float() + cx + by + hz
// Statistics for stopping criteria
pcost = cx / tau.Float()
dcost = -(by + hz) / tau.Float()
if pcost < 0.0 {
relgap = gap / -pcost
} else if dcost > 0.0 {
relgap = gap / dcost
} else {
relgap = math.NaN()
}
pres := math.Max(resy/resy0, resz/resz0)
dres := resx / resx0
pinfres := math.NaN()
if hz+by < 0.0 {
pinfres = hresx / resx0 / (-hz - by)
}
dinfres := math.NaN()
if cx < 0.0 {
dinfres = math.Max(hresy/resy0, hresz/resz0) / (-cx)
}
if solopts.ShowProgress {
if iter == 0 {
// show headers of something
fmt.Printf("% 10s% 12s% 10s% 8s% 7s % 5s\n",
"pcost", "dcost", "gap", "pres", "dres", "k/t")
}
// show something
fmt.Printf("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e% 7.0e\n",
iter, pcost, dcost, gap, pres, dres, kappa.GetIndex(0)/tau.GetIndex(0))
}
if (pres <= feasTolerance && dres <= feasTolerance &&
(gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance))) ||
iter == solopts.MaxIter {
// done
blas.ScalFloat(x, 1.0/tau.Float())
blas.ScalFloat(y, 1.0/tau.Float())
blas.ScalFloat(s, 1.0/tau.Float())
blas.ScalFloat(z, 1.0/tau.Float())
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(s, m, ind)
symm(z, m, ind)
ind += m * m
}
ts, _ = maxStep(s, dims, 0, nil)
tz, _ = maxStep(z, dims, 0, nil)
if iter == solopts.MaxIter {
// MaxIterations exceeded
if solopts.ShowProgress {
fmt.Printf("No solution. Max iterations exceeded\n")
}
err = errors.New("No solution. Max iterations exceeded")
sol.X = x
sol.Y = y
sol.S = s
sol.Z = z
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", x)
sol.Result.Append("y", y)
sol.Result.Append("s", s)
sol.Result.Append("z", z)
sol.Status = Unknown
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
sol.PrimalResidualCert = pinfres
sol.DualResidualCert = dinfres
sol.Iterations = iter
return
} else {
// Optimal
if solopts.ShowProgress {
fmt.Printf("Optimal solution.\n")
}
err = nil
sol.X = x
sol.Y = y
sol.S = s
sol.Z = z
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", x)
sol.Result.Append("y", y)
sol.Result.Append("s", s)
sol.Result.Append("z", z)
sol.Status = Optimal
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
sol.PrimalResidualCert = math.NaN()
sol.DualResidualCert = math.NaN()
sol.Iterations = iter
return
}
} else if !math.IsNaN(pinfres) && pinfres <= feasTolerance {
// Primal Infeasible
if solopts.ShowProgress {
fmt.Printf("Primal infeasible.\n")
}
err = errors.New("Primal infeasible")
blas.ScalFloat(y, 1.0/(-hz-by))
blas.ScalFloat(z, 1.0/(-hz-by))
sol.X = nil
sol.Y = nil
sol.S = nil
sol.Z = nil
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(z, m, ind)
ind += m * m
}
tz, _ = maxStep(z, dims, 0, nil)
sol.Status = PrimalInfeasible
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", nil)
sol.Result.Append("y", nil)
sol.Result.Append("s", nil)
sol.Result.Append("z", nil)
sol.Gap = math.NaN()
sol.RelativeGap = math.NaN()
sol.PrimalObjective = math.NaN()
sol.DualObjective = 1.0
sol.PrimalInfeasibility = math.NaN()
sol.DualInfeasibility = math.NaN()
sol.PrimalSlack = math.NaN()
sol.DualSlack = -tz
sol.PrimalResidualCert = pinfres
sol.DualResidualCert = math.NaN()
sol.Iterations = iter
return
} else if !math.IsNaN(dinfres) && dinfres <= feasTolerance {
// Dual Infeasible
if solopts.ShowProgress {
fmt.Printf("Dual infeasible.\n")
}
err = errors.New("Primal infeasible")
blas.ScalFloat(x, 1.0/(-cx))
blas.ScalFloat(s, 1.0/(-cx))
sol.X = nil
sol.Y = nil
sol.S = nil
sol.Z = nil
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(s, m, ind)
ind += m * m
}
ts, _ = maxStep(s, dims, 0, nil)
sol.Status = PrimalInfeasible
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", nil)
sol.Result.Append("y", nil)
sol.Result.Append("s", nil)
sol.Result.Append("z", nil)
sol.Gap = math.NaN()
sol.RelativeGap = math.NaN()
sol.PrimalObjective = 1.0
sol.DualObjective = math.NaN()
sol.PrimalInfeasibility = math.NaN()
sol.DualInfeasibility = math.NaN()
sol.PrimalSlack = -ts
sol.DualSlack = math.NaN()
sol.PrimalResidualCert = math.NaN()
sol.DualResidualCert = dinfres
sol.Iterations = iter
return
}
// Compute initial scaling W:
//
// W * z = W^{-T} * s = lambda
// dg * tau = 1/dg * kappa = lambdag.
if iter == 0 {
W, err = computeScaling(s, z, lmbda, dims, 0)
// dg = sqrt( kappa / tau )
// dgi = sqrt( tau / kappa )
// lambda_g = sqrt( tau * kappa )
//
// lambda_g is stored in the last position of lmbda.
dg = math.Sqrt(kappa.Float() / tau.Float())
dgi = math.Sqrt(float64(tau.Float() / kappa.Float()))
lmbda.SetIndex(-1, math.Sqrt(float64(tau.Float()*kappa.Float())))
}
// lmbdasq := lmbda o lmbda
ssqr(lmbdasq, lmbda, dims, 0)
lmbdasq.SetIndex(-1, lmbda.GetIndex(-1)*lmbda.GetIndex(-1))
// f3(x, y, z) solves
//
// [ 0 A' G' ] [ ux ] [ bx ]
// [ A 0 0 ] [ uy ] = [ by ].
// [ G 0 -W'*W ] [ W^{-1}*uz ] [ bz ]
//
// On entry, x, y, z contain bx, by, bz.
// On exit, they contain ux, uy, uz.
//
// Also solve
//
// [ 0 A' G' ] [ x1 ] [ c ]
// [-A 0 0 ]*[ y1 ] = -dgi * [ b ].
// [-G 0 W'*W ] [ W^{-1}*z1 ] [ h ]
f3, err = kktsolver(W, nil, nil)
if err != nil {
fmt.Printf("kktsolver error=%v\n", err)
return
}
if iter == 0 {
x1 = c.Copy()
y1 = b.Copy()
z1 = matrix.FloatZeros(cdim, 1)
}
blas.Copy(c, x1)
blas.ScalFloat(x1, -1.0)
blas.Copy(b, y1)
blas.Copy(h, z1)
err = f3(x1, y1, z1)
//fmt.Printf("f3 result: x1=\n%v\nf3 result: z1=\n%v\n", x1, z1)
blas.ScalFloat(x1, dgi)
blas.ScalFloat(y1, dgi)
blas.ScalFloat(z1, dgi)
if err != nil {
if iter == 0 && primalstart != nil && dualstart != nil {
err = errors.New("Rank(A) < p or Rank([G; A]) < n")
return
} else {
t_ := 1.0 / tau.Float()
blas.ScalFloat(x, t_)
blas.ScalFloat(y, t_)
blas.ScalFloat(s, t_)
blas.ScalFloat(z, t_)
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(s, m, ind)
symm(z, m, ind)
ind += m * m
}
ts, _ = maxStep(s, dims, 0, nil)
tz, _ = maxStep(z, dims, 0, nil)
err = errors.New("Terminated (singular KKT matrix).")
sol.X = x
sol.Y = y
sol.S = s
sol.Z = z
sol.Result = FloatSetNew("x", "y", "s", "x")
sol.Result.Append("x", x)
sol.Result.Append("y", y)
sol.Result.Append("s", s)
sol.Result.Append("z", z)
sol.Status = Unknown
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
sol.Iterations = iter
return
}
}
// f6_no_ir(x, y, z, tau, s, kappa) solves
//
// [ 0 ] [ 0 A' G' c ] [ ux ] [ bx ]
// [ 0 ] [ -A 0 0 b ] [ uy ] [ by ]
// [ W'*us ] - [ -G 0 0 h ] [ W^{-1}*uz ] = -[ bz ]
// [ dg*ukappa ] [ -c' -b' -h' 0 ] [ utau/dg ] [ btau ]
//
// lmbda o (uz + us) = -bs
// lmbdag * (utau + ukappa) = -bkappa.
//
// On entry, x, y, z, tau, s, kappa contain bx, by, bz, btau,
// bkappa. On exit, they contain ux, uy, uz, utau, ukappa.
// th = W^{-T} * h
if iter == 0 {
th = matrix.FloatZeros(cdim, 1)
}
blas.Copy(h, th)
scale(th, W, true, true)
f6_no_ir := func(x, y, z, tau, s, kappa *matrix.FloatMatrix) (err error) {
// Solve
//
// [ 0 A' G' 0 ] [ ux ]
// [ -A 0 0 b ] [ uy ]
// [ -G 0 W'*W h ] [ W^{-1}*uz ]
// [ -c' -b' -h' k/t ] [ utau/dg ]
//
// [ bx ]
// [ by ]
// = [ bz - W'*(lmbda o\ bs) ]
// [ btau - bkappa/tau ]
//
// us = -lmbda o\ bs - uz
// ukappa = -bkappa/lmbdag - utau.
// First solve
//
// [ 0 A' G' ] [ ux ] [ bx ]
// [ A 0 0 ] [ uy ] = [ -by ]
// [ G 0 -W'*W ] [ W^{-1}*uz ] [ -bz + W'*(lmbda o\ bs) ]
err = nil
// y := -y = -by
blas.ScalFloat(y, -1.0)
// s := -lmbda o\ s = -lmbda o\ bs
err = sinv(s, lmbda, dims, 0)
blas.ScalFloat(s, -1.0)
// z := -(z + W'*s) = -bz + W'*(lambda o\ bs)
blas.Copy(s, ws3)
err = scale(ws3, W, true, false)
blas.AxpyFloat(ws3, z, 1.0)
blas.ScalFloat(z, -1.0)
err = f3(x, y, z)
// Combine with solution of
//
// [ 0 A' G' ] [ x1 ] [ c ]
// [-A 0 0 ] [ y1 ] = -dgi * [ b ]
// [-G 0 W'*W ] [ W^{-1}*dzl ] [ h ]
//
// to satisfy
//
// -c'*x - b'*y - h'*W^{-1}*z + dg*tau = btau - bkappa/tau. '
// , kappa[0] := -kappa[0] / lmbd[-1] = -bkappa / lmbdag
kap_ := kappa.Float()
tau_ := tau.Float()
kap_ = -kap_ / lmbda.GetIndex(-1)
// tau[0] = tau[0] + kappa[0] / dgi = btau[0] - bkappa / tau
tau_ = tau_ + kap_/dgi
//tau[0] = dgi * ( tau[0] + xdot(c,x) + ydot(b,y) +
// misc.sdot(th, z, dims) ) / (1.0 + misc.sdot(z1, z1, dims))
//tau_ = tau_ + blas.DotFloat(c, x) + blas.DotFloat(b, y) + sdot(th, z, dims, 0)
tau_ += blas.DotFloat(c, x)
tau_ += blas.DotFloat(b, y)
tau_ += sdot(th, z, dims, 0)
tau_ = dgi * tau_ / (1.0 + sdot(z1, z1, dims, 0))
tau.SetValue(tau_)
blas.AxpyFloat(x1, x, tau_)
blas.AxpyFloat(y1, y, tau_)
blas.AxpyFloat(z1, z, tau_)
blas.AxpyFloat(z, s, -1.0)
kap_ = kap_ - tau_
kappa.SetValue(kap_)
return
}
// f6(x, y, z, tau, s, kappa) solves the same system as f6_no_ir,
// but applies iterative refinement. Following variables part of f6-closure
// and ~ 12 is the limit. We wrap them to a structure.
if iter == 0 {
if refinement > 0 || solopts.Debug {
WS.wx = c.Copy()
WS.wy = b.Copy()
WS.wz = matrix.FloatZeros(cdim, 1)
WS.ws = matrix.FloatZeros(cdim, 1)
WS.wtau = matrix.FloatValue(0.0)
WS.wkappa = matrix.FloatValue(0.0)
}
if refinement > 0 {
WS.wx2 = c.Copy()
WS.wy2 = b.Copy()
WS.wz2 = matrix.FloatZeros(cdim, 1)
WS.ws2 = matrix.FloatZeros(cdim, 1)
WS.wtau2 = matrix.FloatValue(0.0)
WS.wkappa2 = matrix.FloatValue(0.0)
}
}
f6 := func(x, y, z, tau, s, kappa *matrix.FloatMatrix) error {
var err error = nil
if refinement > 0 || solopts.Debug {
blas.Copy(x, WS.wx)
blas.Copy(y, WS.wy)
blas.Copy(z, WS.wz)
blas.Copy(s, WS.ws)
WS.wtau.SetValue(tau.Float())
WS.wkappa.SetValue(kappa.Float())
}
err = f6_no_ir(x, y, z, tau, s, kappa)
for i := 0; i < refinement; i++ {
blas.Copy(WS.wx, WS.wx2)
blas.Copy(WS.wy, WS.wy2)
blas.Copy(WS.wz, WS.wz2)
blas.Copy(WS.ws, WS.ws2)
WS.wtau2.SetValue(WS.wtau.Float())
WS.wkappa2.SetValue(WS.wkappa.Float())
err = res(x, y, z, tau, s, kappa, WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2, W, dg, lmbda)
err = f6_no_ir(WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2)
blas.AxpyFloat(WS.wx2, x, 1.0)
blas.AxpyFloat(WS.wy2, y, 1.0)
blas.AxpyFloat(WS.wz2, z, 1.0)
blas.AxpyFloat(WS.ws2, s, 1.0)
tau.SetValue(tau.Float() + WS.wtau2.Float())
kappa.SetValue(kappa.Float() + WS.wkappa2.Float())
}
if solopts.Debug {
res(x, y, z, tau, s, kappa, WS.wx, WS.wy, WS.wz, WS.wtau, WS.ws, WS.wkappa, W, dg, lmbda)
fmt.Printf("KKT residuals\n")
}
return err
}
var nrm float64 = blas.Nrm2(lmbda).Float()
mu := math.Pow(nrm, 2.0) / (1.0 + float64(cdim_diag))
sigma := 0.0
var step, tt, tk float64
for i := 0; i < 2; i++ {
// Solve
//
// [ 0 ] [ 0 A' G' c ] [ dx ]
// [ 0 ] [ -A 0 0 b ] [ dy ]
// [ W'*ds ] - [ -G 0 0 h ] [ W^{-1}*dz ]
// [ dg*dkappa ] [ -c' -b' -h' 0 ] [ dtau/dg ]
//
// [ rx ]
// [ ry ]
// = - (1-sigma) [ rz ]
// [ rtau ]
//
// lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e
// lmbdag * (dtau + dkappa) = - kappa * tau + sigma*mu
//
// ds = -lmbdasq if i is 0
// = -lmbdasq - dsa o dza + sigma*mu*e if i is 1
// dkappa = -lambdasq[-1] if i is 0
// = -lambdasq[-1] - dkappaa*dtaua + sigma*mu if i is 1.
ind := dims.Sum("l", "q")
ind2 := ind
blas.Copy(lmbdasq, ds, &la.IOpt{"n", ind})
blas.ScalFloat(ds, 0.0, &la.IOpt{"offset", ind})
for _, m := range dims.At("s") {
blas.Copy(lmbdasq, ds, &la.IOpt{"n", m}, &la.IOpt{"offsetx", ind2},
&la.IOpt{"offsety", ind}, &la.IOpt{"incy", m + 1})
ind += m * m
ind2 += m
}
// dkappa[0] = lmbdasq[-1]
dkappa.SetValue(lmbdasq.GetIndex(-1))
if i == 1 {
blas.AxpyFloat(ws3, ds, 1.0)
ind = dims.Sum("l", "q")
is := make([]int, 0)
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
is = append(is, indq[:len(indq)-1]...)
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
}
for _, k := range is {
ds.SetIndex(k, ds.GetIndex(k)-sigma*mu)
}
dk_ := dkappa.Float()
wk_ := wkappa3.Float()
dkappa.SetValue(dk_ + wk_ - sigma*mu)
}
// (dx, dy, dz, dtau) = (1-sigma)*(rx, ry, rz, rt)
blas.Copy(rx, dx)
blas.ScalFloat(dx, 1.0-sigma)
blas.Copy(ry, dy)
blas.ScalFloat(dy, 1.0-sigma)
blas.Copy(rz, dz)
blas.ScalFloat(dz, 1.0-sigma)
// dtau[0] = (1.0 - sigma) * rt
dtau.SetValue((1.0 - sigma) * rt)
err = f6(dx, dy, dz, dtau, ds, dkappa)
// Save ds o dz and dkappa * dtau for Mehrotra correction
if i == 0 {
blas.Copy(ds, ws3)
sprod(ws3, dz, dims, 0)
wkappa3.SetValue(dtau.Float() * dkappa.Float())
}
// Maximum step to boundary.
//
// If i is 1, also compute eigenvalue decomposition of the 's'
// blocks in ds, dz. The eigenvectors Qs, Qz are stored in
// dsk, dzk. The eigenvalues are stored in sigs, sigz.
var ts, tz float64
scale2(lmbda, ds, dims, 0, false)
scale2(lmbda, dz, dims, 0, false)
if i == 0 {
ts, _ = maxStep(ds, dims, 0, nil)
tz, _ = maxStep(dz, dims, 0, nil)
} else {
ts, _ = maxStep(ds, dims, 0, sigs)
tz, _ = maxStep(dz, dims, 0, sigz)
}
dt_ := dtau.Float()
dk_ := dkappa.Float()
tt = -dt_ / lmbda.GetIndex(-1)
tk = -dk_ / lmbda.GetIndex(-1)
t := maxvec([]float64{0.0, ts, tz, tt, tk})
if t == 0.0 {
step = 1.0
} else {
if i == 0 {
step = math.Min(1.0, 1.0/t)
} else {
step = math.Min(1.0, STEP/t)
}
}
if i == 0 {
// sigma = (1 - step)^3
sigma = (1.0 - step) * (1.0 - step) * (1.0 - step)
//sigma = math.Pow((1.0 - step), EXPON)
}
}
//fmt.Printf("** tau = %.17f, kappa = %.17f\n", tau.Float(), kappa.Float())
//fmt.Printf("** step = %.17f, sigma = %.17f\n", step, sigma)
// Update x, y
blas.AxpyFloat(dx, x, step)
blas.AxpyFloat(dy, y, step)
// Replace 'l' and 'q' blocks of ds and dz with the updated
// variables in the current scaling.
// Replace 's' blocks of ds and dz with the factors Ls, Lz in a
// factorization Ls*Ls', Lz*Lz' of the updated variables in the
// current scaling.
//
// ds := e + step*ds for 'l' and 'q' blocks.
// dz := e + step*dz for 'l' and 'q' blocks.
blas.ScalFloat(ds, step, &la.IOpt{"n", dims.Sum("l", "q")})
blas.ScalFloat(dz, step, &la.IOpt{"n", dims.Sum("l", "q")})
is := make([]int, 0)
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
is = append(is, indq[:len(indq)-1]...)
for _, k := range is {
ds.SetIndex(k, 1.0+ds.GetIndex(k))
dz.SetIndex(k, 1.0+dz.GetIndex(k))
}
// ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz.
//
// This replaces the 'l' and 'q' components of ds and dz with the
// updated variables in the current scaling.
// The 's' components of ds and dz are replaced with
//
// diag(lmbda_k)^{1/2} * Qs * diag(lmbda_k)^{1/2}
// diag(lmbda_k)^{1/2} * Qz * diag(lmbda_k)^{1/2}
scale2(lmbda, ds, dims, 0, true)
scale2(lmbda, dz, dims, 0, true)
// sigs := ( e + step*sigs ) ./ lambda for 's' blocks.
// sigz := ( e + step*sigz ) ./ lambda for 's' blocks.
blas.ScalFloat(sigs, step)
blas.ScalFloat(sigz, step)
sigs.Add(1.0)
sigz.Add(1.0)
sdimsum := dims.Sum("s")
qdimsum := dims.Sum("l", "q")
blas.TbsvFloat(lmbda, sigs, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0},
&la.IOpt{"lda", 1}, &la.IOpt{"offseta", qdimsum})
blas.TbsvFloat(lmbda, sigz, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0},
&la.IOpt{"lda", 1}, &la.IOpt{"offseta", qdimsum})
ind2 := qdimsum
ind3 := 0
sdims := dims.At("s")
for k := 0; k < len(sdims); k++ {
m := sdims[k]
for i := 0; i < m; i++ {
a := math.Sqrt(sigs.GetIndex(ind3 + i))
blas.ScalFloat(ds, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m})
a = math.Sqrt(sigz.GetIndex(ind3 + i))
blas.ScalFloat(dz, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m})
}
ind2 += m * m
ind3 += m
}
err = updateScaling(W, lmbda, ds, dz)
// For kappa, tau block:
//
// dg := sqrt( (kappa + step*dkappa) / (tau + step*dtau) )
// = dg * sqrt( (1 - step*tk) / (1 - step*tt) )
//
// lmbda[-1] := sqrt((tau + step*dtau) * (kappa + step*dkappa))
// = lmbda[-1] * sqrt(( 1 - step*tt) * (1 - step*tk))
dg *= math.Sqrt(1.0-step*tk) / math.Sqrt(1.0-step*tt)
dgi = 1.0 / dg
a := math.Sqrt(1.0-step*tk) * math.Sqrt(1.0-step*tt)
lmbda.SetIndex(-1, a*lmbda.GetIndex(-1))
// Unscale s, z, tau, kappa (unscaled variables are used only to
// compute feasibility residuals).
ind := dims.Sum("l", "q")
ind2 = ind
blas.Copy(lmbda, s, &la.IOpt{"n", ind})
for _, m := range dims.At("s") {
blas.ScalFloat(s, 0.0, &la.IOpt{"offset", ind2})
blas.Copy(lmbda, s, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2},
&la.IOpt{"n", m}, &la.IOpt{"incy", m + 1})
ind += m
ind2 += m * m
}
scale(s, W, true, false)
ind = dims.Sum("l", "q")
ind2 = ind
blas.Copy(lmbda, z, &la.IOpt{"n", ind})
for _, m := range dims.At("s") {
blas.ScalFloat(z, 0.0, &la.IOpt{"offset", ind2})
blas.Copy(lmbda, z, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2},
&la.IOpt{"n", m}, &la.IOpt{"incy", m + 1})
ind += m
ind2 += m * m
}
scale(z, W, false, true)
kappa.SetValue(lmbda.GetIndex(-1) / dgi)
tau.SetValue(lmbda.GetIndex(-1) * dgi)
g := blas.Nrm2Float(lmbda, &la.IOpt{"n", lmbda.Rows() - 1}) / tau.Float()
gap = g * g
//fmt.Printf(" ** kappa=%.10f, tau=%.10f, gap=%.10f\n", kappa.Float(), tau.Float(), gap)
}
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
}