func sinv(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) (err error) {
/*DEBUGGED*/
err = nil
// For the nonlinear and 'l' blocks:
//
// yk o\ xk = yk .\ xk.
ind := mnl + dims.At("l")[0]
blas.Tbsv(y, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"ldA", 1})
// For the 'q' blocks:
//
// [ l0 -l1' ]
// yk o\ xk = 1/a^2 * [ ] * xk
// [ -l1 (a*I + l1*l1')/l0 ]
//
// where yk = (l0, l1) and a = l0^2 - l1'*l1.
for _, m := range dims.At("q") {
aa := blas.Nrm2Float(y, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offset", ind + 1})
ee := y.GetIndex(ind)
aa = (ee + aa) * (ee - aa)
cc := x.GetIndex(ind)
dd := blas.DotFloat(x, y, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offsetx", ind + 1},
&la_.IOpt{"offsety", ind + 1})
x.SetIndex(ind, cc*ee-dd)
blas.ScalFloat(x, aa/ee, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offset", ind + 1})
blas.AxpyFloat(y, x, dd/ee-cc, &la_.IOpt{"n", m - 1},
&la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1})
blas.ScalFloat(x, 1.0/aa, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
ind += m
}
// For the 's' blocks:
//
// yk o\ xk = xk ./ gamma
//
// where gammaij = .5 * (yk_i + yk_j).
ind2 := ind
for _, m := range dims.At("s") {
for j := 0; j < m; j++ {
u := matrix.FloatVector(y.FloatArray()[ind2+j : ind2+m])
u.Add(y.GetIndex(ind2 + j))
u.Scale(0.5)
blas.Tbsv(u, x, &la_.IOpt{"n", m - j}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1},
&la_.IOpt{"offsetx", ind + j*(m+1)})
}
ind += m * m
ind2 += m
}
return
}
func (g *matrixFs) Gf(x, y *matrix.FloatMatrix, alpha, beta float64, trans linalg.Option) error {
//
// y := alpha*(-diag(x)) + beta*y
//
if linalg.Equal(trans, linalg.OptNoTrans) {
blas.ScalFloat(y, beta)
blas.AxpyFloat(x, y, -alpha, &linalg.IOpt{"incy", g.n + 1})
} else {
blas.ScalFloat(y, beta)
blas.AxpyFloat(x, y, -alpha, &linalg.IOpt{"incx", g.n + 1})
}
return nil
}
/*
Sets upper triangular part of the 's' components of x equal to zero
and scales the strictly lower triangular part by 2.0.
*/
func trisc(x *matrix.FloatMatrix, dims *sets.DimensionSet, offset int) error {
//m := dims.Sum("l", "q") + dims.SumSquared("s")
ind := offset + dims.Sum("l", "q")
for _, mk := range dims.At("s") {
for j := 1; j < mk; j++ {
blas.ScalFloat(x, 0.0, la_.IntOpt("n", mk-j), la_.IntOpt("inc", mk),
la_.IntOpt("offset", ind+j*(mk+1)-1))
blas.ScalFloat(x, 2.0, la_.IntOpt("n", mk-j), la_.IntOpt("offset", ind+mk*(j-1)+j))
}
ind += mk * mk
}
return nil
}
/*
Copy x to y using packed storage.
The vector x is an element of S, with the 's' components stored in
unpacked storage. On return, x is copied to y with the 's' components
stored in packed storage and the off-diagonal entries scaled by
sqrt(2).
*/
func pack(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, opts ...la_.Option) (err error) {
/*DEBUGGED*/
err = nil
mnl := la_.GetIntOpt("mnl", 0, opts...)
offsetx := la_.GetIntOpt("offsetx", 0, opts...)
offsety := la_.GetIntOpt("offsety", 0, opts...)
nlq := mnl + dims.At("l")[0] + dims.Sum("q")
blas.Copy(x, y, &la_.IOpt{"n", nlq}, &la_.IOpt{"offsetx", offsetx},
&la_.IOpt{"offsety", offsety})
iu, ip := offsetx+nlq, offsety+nlq
for _, n := range dims.At("s") {
for k := 0; k < n; k++ {
blas.Copy(x, y, &la_.IOpt{"n", n - k}, &la_.IOpt{"offsetx", iu + k*(n+1)},
&la_.IOpt{"offsety", ip})
y.SetIndex(ip, (y.GetIndex(ip) / math.Sqrt(2.0)))
ip += n - k
}
iu += n * n
}
np := dims.SumPacked("s")
blas.ScalFloat(y, math.Sqrt(2.0), &la_.IOpt{"n", np}, &la_.IOpt{"offset", offsety + nlq})
return
}
func (gp *gpConvexProg) F1(x *matrix.FloatMatrix) (f, Df *matrix.FloatMatrix, err error) {
f = nil
Df = nil
err = nil
f = matrix.FloatZeros(gp.mnl+1, 1)
Df = matrix.FloatZeros(gp.mnl+1, gp.n)
y := gp.g.Copy()
blas.GemvFloat(gp.F, x, y, 1.0, 1.0)
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 = exp(y[start:stop] - ymax)
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})
}
return
}
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
}
/*
Scales the strictly lower triangular part of the 's' components of x
by 0.5.
*/
func triusc(x *matrix.FloatMatrix, dims *sets.DimensionSet, offset int) error {
//m := dims.Sum("l", "q") + dims.SumSquared("s")
ind := offset + dims.Sum("l", "q")
for _, mk := range dims.At("s") {
for j := 1; j < mk; j++ {
blas.ScalFloat(x, 0.5, &la_.IOpt{"n", mk - j}, &la_.IOpt{"offset", ind + mk*(j-1) + j})
}
ind += mk * mk
}
return nil
}
// The product x := y o y. The 's' components of y are diagonal and
// only the diagonals of x and y are stored.
func ssqr(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) (err error) {
/*DEBUGGED*/
blas.Copy(y, x)
ind := mnl + dims.At("l")[0]
err = blas.Tbmv(y, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
if err != nil {
return
}
for _, m := range dims.At("q") {
v := blas.Nrm2Float(y, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
x.SetIndex(ind, v*v)
blas.ScalFloat(x, 2.0*y.GetIndex(ind), &la_.IOpt{"n", m - 1}, &la_.IOpt{"offset", ind + 1})
ind += m
}
err = blas.Tbmv(y, x, &la_.IOpt{"n", dims.Sum("s")}, &la_.IOpt{"k", 0},
&la_.IOpt{"lda", 1}, &la_.IOpt{"offseta", ind}, &la_.IOpt{"offsetx", ind})
return
}
/*
The vector x is an element of S, with the 's' components stored
in unpacked storage and off-diagonal entries scaled by sqrt(2).
On return, x is copied to y with the 's' components stored in
unpacked storage.
*/
func unpack(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, opts ...la_.Option) (err error) {
/*DEBUGGED*/
err = nil
mnl := la_.GetIntOpt("mnl", 0, opts...)
offsetx := la_.GetIntOpt("offsetx", 0, opts...)
offsety := la_.GetIntOpt("offsety", 0, opts...)
nlq := mnl + dims.At("l")[0] + dims.Sum("q")
err = blas.Copy(x, y, &la_.IOpt{"n", nlq}, &la_.IOpt{"offsetx", offsetx},
&la_.IOpt{"offsety", offsety})
if err != nil {
return
}
ip, iu := offsetx+nlq, offsety+nlq
for _, n := range dims.At("s") {
for k := 0; k < n; k++ {
err = blas.Copy(x, y, &la_.IOpt{"n", n - k}, &la_.IOpt{"offsetx", ip},
&la_.IOpt{"offsety", iu + k*(n+1)})
if err != nil {
return
}
ip += n - k
blas.ScalFloat(y, 1.0/math.Sqrt(2.0),
&la_.IOpt{"n", n - k - 1}, &la_.IOpt{"offset", iu + k*(n+1) + 1})
}
iu += n * n
}
/*
nu := dims.SumSquared("s")
fmt.Printf("-- UnPack: nu=%d, offset=%d\n", nu, offsety+nlq)
err = blas.ScalFloat(y,
&la_.IOpt{"n", nu}, &la_.IOpt{"offset", offsety+nlq})
*/
return
}
// Internal CPL solver for CP and CLP problems. Everything is wrapped to proper interfaces
func cpl_solver(F ConvexVarProg, c MatrixVariable, G MatrixVarG, h *matrix.FloatMatrix,
A MatrixVarA, b MatrixVariable, dims *sets.DimensionSet, kktsolver KKTCpSolverVar,
solopts *SolverOptions, x0 MatrixVariable, mnl int) (sol *Solution, err error) {
const (
STEP = 0.99
BETA = 0.5
ALPHA = 0.01
EXPON = 3
MAX_RELAXED_ITERS = 8
)
var refinement int
sol = &Solution{Unknown,
nil,
0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0}
feasTolerance := FEASTOL
absTolerance := ABSTOL
relTolerance := RELTOL
maxIter := MAXITERS
if solopts.FeasTol > 0.0 {
feasTolerance = solopts.FeasTol
}
if solopts.AbsTol > 0.0 {
absTolerance = solopts.AbsTol
}
if solopts.RelTol > 0.0 {
relTolerance = solopts.RelTol
}
if solopts.Refinement > 0 {
refinement = solopts.Refinement
} else {
refinement = 1
}
if solopts.MaxIter > 0 {
maxIter = solopts.MaxIter
}
if x0 == nil {
mnl, x0, err = F.F0()
if err != nil {
return
}
}
if c == nil {
err = errors.New("Must define objective.")
return
}
if h == nil {
h = matrix.FloatZeros(0, 1)
}
if dims == nil {
err = errors.New("Problem dimensions not defined.")
return
}
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
}
if G == nil {
err = errors.New("'G' must be non-nil MatrixG interface.")
return
}
fG := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error {
return G.Gf(x, y, alpha, beta, trans)
}
// Check A and set defaults if it is nil
if A == nil {
err = errors.New("'A' must be non-nil MatrixA interface.")
return
}
fA := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error {
return A.Af(x, y, alpha, beta, trans)
}
if b == nil {
err = errors.New("'b' must be non-nil MatrixVariable interface.")
return
}
if kktsolver == nil {
err = errors.New("nil kktsolver not allowed.")
return
}
x := x0.Copy()
y := b.Copy()
y.Scal(0.0)
z := matrix.FloatZeros(mnl+cdim, 1)
s := matrix.FloatZeros(mnl+cdim, 1)
ind := mnl + dims.At("l")[0]
z.SetIndexes(1.0, matrix.MakeIndexSet(0, ind, 1)...)
s.SetIndexes(1.0, matrix.MakeIndexSet(0, ind, 1)...)
for _, m := range dims.At("q") {
z.SetIndexes(1.0, ind)
s.SetIndexes(1.0, ind)
ind += m
}
for _, m := range dims.At("s") {
iset := matrix.MakeIndexSet(ind, ind+m*m, m+1)
z.SetIndexes(1.0, iset...)
s.SetIndexes(1.0, iset...)
ind += m * m
}
rx := x0.Copy()
ry := b.Copy()
dx := x.Copy()
dy := y.Copy()
rznl := matrix.FloatZeros(mnl, 1)
rzl := matrix.FloatZeros(cdim, 1)
dz := matrix.FloatZeros(mnl+cdim, 1)
ds := matrix.FloatZeros(mnl+cdim, 1)
lmbda := matrix.FloatZeros(mnl+cdim_diag, 1)
lmbdasq := matrix.FloatZeros(mnl+cdim_diag, 1)
sigs := matrix.FloatZeros(dims.Sum("s"), 1)
sigz := matrix.FloatZeros(dims.Sum("s"), 1)
dz2 := matrix.FloatZeros(mnl+cdim, 1)
ds2 := matrix.FloatZeros(mnl+cdim, 1)
newx := x.Copy()
newy := y.Copy()
newrx := x0.Copy()
newz := matrix.FloatZeros(mnl+cdim, 1)
news := matrix.FloatZeros(mnl+cdim, 1)
newrznl := matrix.FloatZeros(mnl, 1)
rx0 := rx.Copy()
ry0 := ry.Copy()
rznl0 := matrix.FloatZeros(mnl, 1)
rzl0 := matrix.FloatZeros(cdim, 1)
x0, dx0 := x.Copy(), dx.Copy()
y0, dy0 := y.Copy(), dy.Copy()
z0 := matrix.FloatZeros(mnl+cdim, 1)
dz0 := matrix.FloatZeros(mnl+cdim, 1)
dz20 := matrix.FloatZeros(mnl+cdim, 1)
s0 := matrix.FloatZeros(mnl+cdim, 1)
ds0 := matrix.FloatZeros(mnl+cdim, 1)
ds20 := matrix.FloatZeros(mnl+cdim, 1)
checkpnt.AddMatrixVar("z", z)
checkpnt.AddMatrixVar("s", s)
checkpnt.AddMatrixVar("dz", dz)
checkpnt.AddMatrixVar("ds", ds)
checkpnt.AddMatrixVar("rznl", rznl)
checkpnt.AddMatrixVar("rzl", rzl)
checkpnt.AddMatrixVar("lmbda", lmbda)
checkpnt.AddMatrixVar("lmbdasq", lmbdasq)
checkpnt.AddMatrixVar("z0", z0)
checkpnt.AddMatrixVar("dz0", dz0)
checkpnt.AddVerifiable("c", c)
checkpnt.AddVerifiable("x", x)
checkpnt.AddVerifiable("rx", rx)
checkpnt.AddVerifiable("dx", dx)
checkpnt.AddVerifiable("newrx", newrx)
checkpnt.AddVerifiable("newx", newx)
checkpnt.AddVerifiable("x0", x0)
checkpnt.AddVerifiable("dx0", dx0)
checkpnt.AddVerifiable("rx0", rx0)
checkpnt.AddVerifiable("y", y)
checkpnt.AddVerifiable("dy", dy)
W0 := sets.NewFloatSet("d", "di", "dnl", "dnli", "v", "r", "rti", "beta")
W0.Set("dnl", matrix.FloatZeros(mnl, 1))
W0.Set("dnli", matrix.FloatZeros(mnl, 1))
W0.Set("d", matrix.FloatZeros(dims.At("l")[0], 1))
W0.Set("di", matrix.FloatZeros(dims.At("l")[0], 1))
W0.Set("beta", matrix.FloatZeros(len(dims.At("q")), 1))
for _, n := range dims.At("q") {
W0.Append("v", matrix.FloatZeros(n, 1))
}
for _, n := range dims.At("s") {
W0.Append("r", matrix.FloatZeros(n, n))
W0.Append("rti", matrix.FloatZeros(n, n))
}
lmbda0 := matrix.FloatZeros(mnl+dims.Sum("l", "q", "s"), 1)
lmbdasq0 := matrix.FloatZeros(mnl+dims.Sum("l", "q", "s"), 1)
var f MatrixVariable = nil
var Df MatrixVarDf = nil
var H MatrixVarH = nil
var ws3, wz3, wz2l, wz2nl *matrix.FloatMatrix
var ws, wz, wz2, ws2 *matrix.FloatMatrix
var wx, wx2, wy, wy2 MatrixVariable
var gap, gap0, theta1, theta2, theta3, ts, tz, phi, phi0, mu, sigma, eta float64
var resx, resy, reszl, resznl, pcost, dcost, dres, pres, relgap float64
var resx0, resznl0, dres0, pres0 float64
var dsdz, dsdz0, step, step0, dphi, dphi0, sigma0, eta0 float64
var newresx, newresznl, newgap, newphi float64
var W *sets.FloatMatrixSet
var f3 KKTFuncVar
checkpnt.AddFloatVar("gap", &gap)
checkpnt.AddFloatVar("pcost", &pcost)
checkpnt.AddFloatVar("dcost", &dcost)
checkpnt.AddFloatVar("pres", &pres)
checkpnt.AddFloatVar("dres", &dres)
checkpnt.AddFloatVar("relgap", &relgap)
checkpnt.AddFloatVar("step", &step)
checkpnt.AddFloatVar("dsdz", &dsdz)
checkpnt.AddFloatVar("resx", &resx)
checkpnt.AddFloatVar("resy", &resy)
checkpnt.AddFloatVar("reszl", &reszl)
checkpnt.AddFloatVar("resznl", &resznl)
// Declare fDf and fH here, they bind to Df and H as they are already declared.
// ??really??
var fDf func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error = nil
var fH func(u, v MatrixVariable, alpha, beta float64) error = nil
relaxed_iters := 0
for iters := 0; iters <= maxIter+1; iters++ {
checkpnt.MajorNext()
checkpnt.Check("loopstart", 10)
checkpnt.MinorPush(10)
if refinement != 0 || solopts.Debug {
f, Df, H, err = F.F2(x, matrix.FloatVector(z.FloatArray()[:mnl]))
fDf = func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error {
return Df.Df(u, v, alpha, beta, trans)
}
fH = func(u, v MatrixVariable, alpha, beta float64) error {
return H.Hf(u, v, alpha, beta)
}
} else {
f, Df, err = F.F1(x)
fDf = func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error {
return Df.Df(u, v, alpha, beta, trans)
}
}
checkpnt.MinorPop()
gap = sdot(s, z, dims, mnl)
// these are helpers, copies of parts of z,s
z_mnl := matrix.FloatVector(z.FloatArray()[:mnl])
z_mnl2 := matrix.FloatVector(z.FloatArray()[mnl:])
s_mnl := matrix.FloatVector(s.FloatArray()[:mnl])
s_mnl2 := matrix.FloatVector(s.FloatArray()[mnl:])
// rx = c + A'*y + Df'*z[:mnl] + G'*z[mnl:]
// -- y, rx MatrixArg
mCopy(c, rx)
fA(y, rx, 1.0, 1.0, la.OptTrans)
fDf(&matrixVar{z_mnl}, rx, 1.0, 1.0, la.OptTrans)
fG(&matrixVar{z_mnl2}, rx, 1.0, 1.0, la.OptTrans)
resx = math.Sqrt(rx.Dot(rx))
// rznl = s[:mnl] + f
blas.Copy(s_mnl, rznl)
blas.AxpyFloat(f.Matrix(), rznl, 1.0)
resznl = blas.Nrm2Float(rznl)
// rzl = s[mnl:] + G*x - h
blas.Copy(s_mnl2, rzl)
blas.AxpyFloat(h, rzl, -1.0)
fG(x, &matrixVar{rzl}, 1.0, 1.0, la.OptNoTrans)
reszl = snrm2(rzl, dims, 0)
// Statistics for stopping criteria
// pcost = c'*x
// dcost = c'*x + y'*(A*x-b) + znl'*f(x) + zl'*(G*x-h)
// = c'*x + y'*(A*x-b) + znl'*(f(x)+snl) + zl'*(G*x-h+sl)
// - z'*s
// = c'*x + y'*ry + znl'*rznl + zl'*rzl - gap
//pcost = blas.DotFloat(c, x)
pcost = c.Dot(x)
dcost = pcost + blas.DotFloat(y.Matrix(), ry.Matrix()) + blas.DotFloat(z_mnl, rznl)
dcost += sdot(z_mnl2, rzl, dims, 0) - gap
if pcost < 0.0 {
relgap = gap / -pcost
} else if dcost > 0.0 {
relgap = gap / dcost
} else {
relgap = math.NaN()
}
pres = math.Sqrt(resy*resy + resznl*resznl + reszl*reszl)
dres = resx
if iters == 0 {
resx0 = math.Max(1.0, resx)
resznl0 = math.Max(1.0, resznl)
pres0 = math.Max(1.0, pres)
dres0 = math.Max(1.0, dres)
gap0 = gap
theta1 = 1.0 / gap0
theta2 = 1.0 / resx0
theta3 = 1.0 / resznl0
}
phi = theta1*gap + theta2*resx + theta3*resznl
pres = pres / pres0
dres = dres / dres0
if solopts.ShowProgress {
if iters == 0 {
// some headers
fmt.Printf("% 10s% 12s% 10s% 8s% 7s\n",
"pcost", "dcost", "gap", "pres", "dres")
}
fmt.Printf("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e\n",
iters, pcost, dcost, gap, pres, dres)
}
checkpnt.Check("checkgap", 50)
// Stopping criteria
if (pres <= feasTolerance && dres <= feasTolerance &&
(gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance))) ||
iters == maxIter {
if iters == maxIter {
s := "Terminated (maximum number of iterations reached)"
if solopts.ShowProgress {
fmt.Printf(s + "\n")
}
err = errors.New(s)
sol.Status = Unknown
} else {
err = nil
sol.Status = Optimal
}
sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl")
sol.Result.Set("x", x.Matrix())
sol.Result.Set("y", y.Matrix())
sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl]))
sol.Result.Set("zl", matrix.FloatVector(z.FloatArray()[mnl:]))
sol.Result.Set("sl", matrix.FloatVector(s.FloatArray()[mnl:]))
sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl]))
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
return
}
// Compute initial scaling W:
//
// W * z = W^{-T} * s = lambda.
//
// lmbdasq = lambda o lambda
if iters == 0 {
W, _ = computeScaling(s, z, lmbda, dims, mnl)
checkpnt.AddScaleVar(W)
}
ssqr(lmbdasq, lmbda, dims, mnl)
checkpnt.Check("lmbdasq", 90)
// f3(x, y, z) solves
//
// [ H A' GG'*W^{-1} ] [ ux ] [ bx ]
// [ A 0 0 ] [ uy ] = [ by ].
// [ GG 0 -W' ] [ uz ] [ bz ]
//
// On entry, x, y, z contain bx, by, bz.
// On exit, they contain ux, uy, uz.
checkpnt.MinorPush(95)
f3, err = kktsolver(W, x, z_mnl)
checkpnt.MinorPop()
checkpnt.Check("f3", 100)
if err != nil {
// ?? z_mnl is really copy of z[:mnl] ... should we copy here back to z??
singular_kkt_matrix := false
if iters == 0 {
err = errors.New("Rank(A) < p or Rank([H(x); A; Df(x); G] < n")
return
} else if relaxed_iters > 0 && relaxed_iters < MAX_RELAXED_ITERS {
// The arithmetic error may be caused by a relaxed line
// search in the previous iteration. Therefore we restore
// the last saved state and require a standard line search.
phi, gap = phi0, gap0
mu = gap / float64(mnl+dims.Sum("l", "s")+len(dims.At("q")))
blas.Copy(W0.At("dnl")[0], W.At("dnl")[0])
blas.Copy(W0.At("dnli")[0], W.At("dnli")[0])
blas.Copy(W0.At("d")[0], W.At("d")[0])
blas.Copy(W0.At("di")[0], W.At("di")[0])
blas.Copy(W0.At("beta")[0], W.At("beta")[0])
for k, _ := range dims.At("q") {
blas.Copy(W0.At("v")[k], W.At("v")[k])
}
for k, _ := range dims.At("s") {
blas.Copy(W0.At("r")[k], W.At("r")[k])
blas.Copy(W0.At("rti")[k], W.At("rti")[k])
}
//blas.Copy(x0, x)
//x0.CopyTo(x)
mCopy(x0, x)
//blas.Copy(y0, y)
mCopy(y0, y)
blas.Copy(s0, s)
blas.Copy(z0, z)
blas.Copy(lmbda0, lmbda)
blas.Copy(lmbdasq0, lmbdasq) // ???
//blas.Copy(rx0, rx)
//rx0.CopyTo(rx)
mCopy(rx0, rx)
//blas.Copy(ry0, ry)
mCopy(ry0, ry)
//resx = math.Sqrt(blas.DotFloat(rx, rx))
resx = math.Sqrt(rx.Dot(rx))
blas.Copy(rznl0, rznl)
blas.Copy(rzl0, rzl)
resznl = blas.Nrm2Float(rznl)
relaxed_iters = -1
// How about z_mnl here???
checkpnt.MinorPush(120)
f3, err = kktsolver(W, x, z_mnl)
checkpnt.MinorPop()
if err != nil {
singular_kkt_matrix = true
}
} else {
singular_kkt_matrix = true
}
if singular_kkt_matrix {
msg := "Terminated (singular KKT matrix)."
if solopts.ShowProgress {
fmt.Printf(msg + "\n")
}
zl := matrix.FloatVector(z.FloatArray()[mnl:])
sl := matrix.FloatVector(s.FloatArray()[mnl:])
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(sl, m, ind)
symm(zl, m, ind)
ind += m * m
}
ts, _ = maxStep(s, dims, mnl, nil)
tz, _ = maxStep(z, dims, mnl, nil)
err = errors.New(msg)
sol.Status = Unknown
sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl")
sol.Result.Set("x", x.Matrix())
sol.Result.Set("y", y.Matrix())
sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl]))
sol.Result.Set("zl", zl)
sol.Result.Set("sl", sl)
sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl]))
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
return
}
}
// f4_no_ir(x, y, z, s) solves
//
// [ 0 ] [ H A' GG' ] [ ux ] [ bx ]
// [ 0 ] + [ A 0 0 ] [ uy ] = [ by ]
// [ W'*us ] [ GG 0 0 ] [ W^{-1}*uz ] [ bz ]
//
// lmbda o (uz + us) = bs.
//
// On entry, x, y, z, x, contain bx, by, bz, bs.
// On exit, they contain ux, uy, uz, us.
if iters == 0 {
ws3 = matrix.FloatZeros(mnl+cdim, 1)
wz3 = matrix.FloatZeros(mnl+cdim, 1)
checkpnt.AddMatrixVar("ws3", ws3)
checkpnt.AddMatrixVar("wz3", wz3)
}
f4_no_ir := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) (err error) {
// Solve
//
// [ H A' GG' ] [ ux ] [ bx ]
// [ A 0 0 ] [ uy ] = [ by ]
// [ GG 0 -W'*W ] [ W^{-1}*uz ] [ bz - W'*(lmbda o\ bs) ]
//
// us = lmbda o\ bs - uz.
err = nil
// s := lmbda o\ s
// = lmbda o\ bs
sinv(s, lmbda, dims, mnl)
// z := z - W'*s
// = bz - W' * (lambda o\ bs)
blas.Copy(s, ws3)
scale(ws3, W, true, false)
blas.AxpyFloat(ws3, z, -1.0)
// Solve for ux, uy, uz
err = f3(x, y, z)
// s := s - z
// = lambda o\ bs - z.
blas.AxpyFloat(z, s, -1.0)
return
}
if iters == 0 {
wz2nl = matrix.FloatZeros(mnl, 1)
wz2l = matrix.FloatZeros(cdim, 1)
checkpnt.AddMatrixVar("wz2nl", wz2nl)
checkpnt.AddMatrixVar("wz2l", wz2l)
}
res := func(ux, uy MatrixVariable, uz, us *matrix.FloatMatrix, vx, vy MatrixVariable, vz, vs *matrix.FloatMatrix) (err error) {
// Evaluates residuals in Newton equations:
//
// [ vx ] [ 0 ] [ H A' GG' ] [ ux ]
// [ vy ] -= [ 0 ] + [ A 0 0 ] [ uy ]
// [ vz ] [ W'*us ] [ GG 0 0 ] [ W^{-1}*uz ]
//
// vs -= lmbda o (uz + us).
err = nil
minor := checkpnt.MinorTop()
// vx := vx - H*ux - A'*uy - GG'*W^{-1}*uz
fH(ux, vx, -1.0, 1.0)
fA(uy, vx, -1.0, 1.0, la.OptTrans)
blas.Copy(uz, wz3)
scale(wz3, W, false, true)
wz3_nl := matrix.FloatVector(wz3.FloatArray()[:mnl])
wz3_l := matrix.FloatVector(wz3.FloatArray()[mnl:])
fDf(&matrixVar{wz3_nl}, vx, -1.0, 1.0, la.OptTrans)
fG(&matrixVar{wz3_l}, vx, -1.0, 1.0, la.OptTrans)
checkpnt.Check("10res", minor+10)
// vy := vy - A*ux
fA(ux, vy, -1.0, 1.0, la.OptNoTrans)
// vz := vz - W'*us - GG*ux
err = fDf(ux, &matrixVar{wz2nl}, 1.0, 0.0, la.OptNoTrans)
checkpnt.Check("15res", minor+10)
blas.AxpyFloat(wz2nl, vz, -1.0)
fG(ux, &matrixVar{wz2l}, 1.0, 0.0, la.OptNoTrans)
checkpnt.Check("20res", minor+10)
blas.AxpyFloat(wz2l, vz, -1.0, &la.IOpt{"offsety", mnl})
blas.Copy(us, ws3)
scale(ws3, W, true, false)
blas.AxpyFloat(ws3, vz, -1.0)
checkpnt.Check("30res", minor+10)
// vs -= lmbda o (uz + us)
blas.Copy(us, ws3)
blas.AxpyFloat(uz, ws3, 1.0)
sprod(ws3, lmbda, dims, mnl, &la.SOpt{"diag", "D"})
blas.AxpyFloat(ws3, vs, -1.0)
checkpnt.Check("90res", minor+10)
return
}
// f4(x, y, z, s) solves the same system as f4_no_ir, but applies
// iterative refinement.
if iters == 0 {
if refinement > 0 || solopts.Debug {
wx = c.Copy()
wy = b.Copy()
wz = z.Copy()
ws = s.Copy()
checkpnt.AddVerifiable("wx", wx)
checkpnt.AddMatrixVar("ws", ws)
checkpnt.AddMatrixVar("wz", wz)
}
if refinement > 0 {
wx2 = c.Copy()
wy2 = b.Copy()
wz2 = matrix.FloatZeros(mnl+cdim, 1)
ws2 = matrix.FloatZeros(mnl+cdim, 1)
checkpnt.AddVerifiable("wx2", wx2)
checkpnt.AddMatrixVar("ws2", ws2)
checkpnt.AddMatrixVar("wz2", wz2)
}
}
f4 := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) (err error) {
if refinement > 0 || solopts.Debug {
mCopy(x, wx)
mCopy(y, wy)
blas.Copy(z, wz)
blas.Copy(s, ws)
}
minor := checkpnt.MinorTop()
checkpnt.Check("0_f4", minor+100)
checkpnt.MinorPush(minor + 100)
err = f4_no_ir(x, y, z, s)
checkpnt.MinorPop()
checkpnt.Check("1_f4", minor+200)
for i := 0; i < refinement; i++ {
mCopy(wx, wx2)
mCopy(wy, wy2)
blas.Copy(wz, wz2)
blas.Copy(ws, ws2)
checkpnt.Check("2_f4", minor+(1+i)*200)
checkpnt.MinorPush(minor + (1+i)*200)
res(x, y, z, s, wx2, wy2, wz2, ws2)
checkpnt.MinorPop()
checkpnt.Check("3_f4", minor+(1+i)*200+100)
err = f4_no_ir(wx2, wy2, wz2, ws2)
checkpnt.MinorPop()
checkpnt.Check("4_f4", minor+(1+i)*200+199)
wx2.Axpy(x, 1.0)
wy2.Axpy(y, 1.0)
blas.AxpyFloat(wz2, z, 1.0)
blas.AxpyFloat(ws2, s, 1.0)
}
if solopts.Debug {
res(x, y, z, s, wx, wy, wz, ws)
fmt.Printf("KKT residuals:\n")
}
return
}
sigma, eta = 0.0, 0.0
for i := 0; i < 2; i++ {
minor := (i + 2) * 1000
checkpnt.MinorPush(minor)
checkpnt.Check("loop01", minor)
// Solve
//
// [ 0 ] [ H A' GG' ] [ dx ]
// [ 0 ] + [ A 0 0 ] [ dy ] = -(1 - eta)*r
// [ W'*ds ] [ GG 0 0 ] [ W^{-1}*dz ]
//
// lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e.
//
mu = gap / float64(mnl+dims.Sum("l", "s")+len(dims.At("q")))
blas.ScalFloat(ds, 0.0)
blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", mnl + dims.Sum("l", "q")})
ind = mnl + dims.At("l")[0]
iset := matrix.MakeIndexSet(0, ind, 1)
ds.Add(sigma*mu, iset...)
for _, m := range dims.At("q") {
ds.Add(sigma*mu, ind)
ind += m
}
ind2 := ind
for _, m := range dims.At("s") {
blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", m}, &la.IOpt{"offsetx", ind2},
&la.IOpt{"offsety", ind}, &la.IOpt{"incy", m + 1})
ds.Add(sigma*mu, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
ind += m * m
ind2 += m
}
dx.Scal(0.0)
rx.Axpy(dx, -1.0+eta)
dy.Scal(0.0)
ry.Axpy(dy, -1.0+eta)
dz.Scale(0.0)
blas.AxpyFloat(rznl, dz, -1.0+eta)
blas.AxpyFloat(rzl, dz, -1.0+eta, &la.IOpt{"offsety", mnl})
//fmt.Printf("dx=\n%v\n", dx)
//fmt.Printf("dz=\n%v\n", dz.ToString("%.7f"))
//fmt.Printf("ds=\n%v\n", ds.ToString("%.7f"))
checkpnt.Check("pref4", minor)
checkpnt.MinorPush(minor)
err = f4(dx, dy, dz, ds)
if err != nil {
if iters == 0 {
s := fmt.Sprintf("Rank(A) < p or Rank([H(x); A; Df(x); G] < n (%s)", err)
err = errors.New(s)
return
}
msg := "Terminated (singular KKT matrix)."
if solopts.ShowProgress {
fmt.Printf(msg + "\n")
}
zl := matrix.FloatVector(z.FloatArray()[mnl:])
sl := matrix.FloatVector(s.FloatArray()[mnl:])
ind := dims.Sum("l", "q")
for _, m := range dims.At("s") {
symm(sl, m, ind)
symm(zl, m, ind)
ind += m * m
}
ts, _ = maxStep(s, dims, mnl, nil)
tz, _ = maxStep(z, dims, mnl, nil)
err = errors.New(msg)
sol.Status = Unknown
sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl")
sol.Result.Set("x", x.Matrix())
sol.Result.Set("y", y.Matrix())
sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl]))
sol.Result.Set("zl", zl)
sol.Result.Set("sl", sl)
sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl]))
sol.Gap = gap
sol.RelativeGap = relgap
sol.PrimalObjective = pcost
sol.DualObjective = dcost
sol.PrimalInfeasibility = pres
sol.DualInfeasibility = dres
sol.PrimalSlack = -ts
sol.DualSlack = -tz
return
}
checkpnt.MinorPop()
checkpnt.Check("postf4", minor+400)
// Inner product ds'*dz and unscaled steps are needed in the
// line search.
dsdz = sdot(ds, dz, dims, mnl)
blas.Copy(dz, dz2)
scale(dz2, W, false, true)
blas.Copy(ds, ds2)
scale(ds2, W, true, false)
checkpnt.Check("dsdz", minor+400)
// Maximum steps to boundary.
//
// Also compute the eigenvalue decomposition of 's' blocks in
// ds, dz. The eigenvectors Qs, Qz are stored in ds, dz.
// The eigenvalues are stored in sigs, sigz.
scale2(lmbda, ds, dims, mnl, false)
ts, _ = maxStep(ds, dims, mnl, sigs)
scale2(lmbda, dz, dims, mnl, false)
tz, _ = maxStep(dz, dims, mnl, sigz)
t := maxvec([]float64{0.0, ts, tz})
if t == 0 {
step = 1.0
} else {
step = math.Min(1.0, STEP/t)
}
checkpnt.Check("maxstep", minor+400)
var newDf MatrixVarDf = nil
var newf MatrixVariable = nil
// Backtrack until newx is in domain of f.
backtrack := true
for backtrack {
mCopy(x, newx)
dx.Axpy(newx, step)
newf, newDf, err = F.F1(newx)
if newf != nil {
backtrack = false
} else {
step *= BETA
}
}
// Merit function
//
// phi = theta1 * gap + theta2 * norm(rx) +
// theta3 * norm(rznl)
//
// and its directional derivative dphi.
phi = theta1*gap + theta2*resx + theta3*resznl
if i == 0 {
dphi = -phi
} else {
dphi = -theta1*(1-sigma)*gap - theta2*(1-eta)*resx - theta3*(1-eta)*resznl
}
var newfDf func(x, y MatrixVariable, a, b float64, trans la.Option) error
// Line search
backtrack = true
for backtrack {
mCopy(x, newx)
dx.Axpy(newx, step)
mCopy(y, newy)
dy.Axpy(newy, step)
blas.Copy(z, newz)
blas.AxpyFloat(dz2, newz, step)
blas.Copy(s, news)
blas.AxpyFloat(ds2, news, step)
newf, newDf, err = F.F1(newx)
newfDf = func(u, v MatrixVariable, a, b float64, trans la.Option) error {
return newDf.Df(u, v, a, b, trans)
}
// newrx = c + A'*newy + newDf'*newz[:mnl] + G'*newz[mnl:]
newz_mnl := matrix.FloatVector(newz.FloatArray()[:mnl])
newz_ml := matrix.FloatVector(newz.FloatArray()[mnl:])
//blas.Copy(c, newrx)
//c.CopyTo(newrx)
mCopy(c, newrx)
fA(newy, newrx, 1.0, 1.0, la.OptTrans)
newfDf(&matrixVar{newz_mnl}, newrx, 1.0, 1.0, la.OptTrans)
fG(&matrixVar{newz_ml}, newrx, 1.0, 1.0, la.OptTrans)
newresx = math.Sqrt(newrx.Dot(newrx))
// newrznl = news[:mnl] + newf
news_mnl := matrix.FloatVector(news.FloatArray()[:mnl])
//news_ml := matrix.FloatVector(news.FloatArray()[mnl:])
blas.Copy(news_mnl, newrznl)
blas.AxpyFloat(newf.Matrix(), newrznl, 1.0)
newresznl = blas.Nrm2Float(newrznl)
newgap = (1.0-(1.0-sigma)*step)*gap + step*step*dsdz
newphi = theta1*newgap + theta2*newresx + theta3*newresznl
if i == 0 {
if newgap <= (1.0-ALPHA*step)*gap &&
(relaxed_iters > 0 && relaxed_iters < MAX_RELAXED_ITERS ||
newphi <= phi+ALPHA*step*dphi) {
backtrack = false
sigma = math.Min(newgap/gap, math.Pow((newgap/gap), EXPON))
//fmt.Printf("break 1: sigma=%.7f\n", sigma)
eta = 0.0
} else {
step *= BETA
}
} else {
if relaxed_iters == -1 || (relaxed_iters == 0 && MAX_RELAXED_ITERS == 0) {
// Do a standard line search.
if newphi <= phi+ALPHA*step*dphi {
relaxed_iters = 0
backtrack = false
//fmt.Printf("break 2 : newphi=%.7f\n", newphi)
} else {
step *= BETA
}
} else if relaxed_iters == 0 && relaxed_iters < MAX_RELAXED_ITERS {
if newphi <= phi+ALPHA*step*dphi {
// Relaxed l.s. gives sufficient decrease.
relaxed_iters = 0
} else {
// Save state.
phi0, dphi0, gap0 = phi, dphi, gap
step0 = step
blas.Copy(W.At("dnl")[0], W0.At("dnl")[0])
blas.Copy(W.At("dnli")[0], W0.At("dnli")[0])
blas.Copy(W.At("d")[0], W0.At("d")[0])
blas.Copy(W.At("di")[0], W0.At("di")[0])
blas.Copy(W.At("beta")[0], W0.At("beta")[0])
for k, _ := range dims.At("q") {
blas.Copy(W.At("v")[k], W0.At("v")[k])
}
for k, _ := range dims.At("s") {
blas.Copy(W.At("r")[k], W0.At("r")[k])
blas.Copy(W.At("rti")[k], W0.At("rti")[k])
}
mCopy(x, x0)
mCopy(y, y0)
mCopy(dx, dx0)
mCopy(dy, dy0)
blas.Copy(s, s0)
blas.Copy(z, z0)
blas.Copy(ds, ds0)
blas.Copy(dz, dz0)
blas.Copy(ds2, ds20)
blas.Copy(dz2, dz20)
blas.Copy(lmbda, lmbda0)
blas.Copy(lmbdasq, lmbdasq0) // ???
mCopy(rx, rx0)
mCopy(ry, ry0)
blas.Copy(rznl, rznl0)
blas.Copy(rzl, rzl0)
dsdz0 = dsdz
sigma0, eta0 = sigma, eta
relaxed_iters = 1
}
backtrack = false
//fmt.Printf("break 3 : newphi=%.7f\n", newphi)
} else if relaxed_iters >= 0 && relaxed_iters < MAX_RELAXED_ITERS &&
MAX_RELAXED_ITERS > 0 {
if newphi <= phi0+ALPHA*step0*dphi0 {
// Relaxed l.s. gives sufficient decrease.
relaxed_iters = 0
} else {
// Relaxed line search
relaxed_iters += 1
}
backtrack = false
//fmt.Printf("break 4 : newphi=%.7f\n", newphi)
} else if relaxed_iters == MAX_RELAXED_ITERS && MAX_RELAXED_ITERS > 0 {
if newphi <= phi0+ALPHA*step0*dphi0 {
// Series of relaxed line searches ends
// with sufficient decrease w.r.t. phi0.
backtrack = false
relaxed_iters = 0
//fmt.Printf("break 5 : newphi=%.7f\n", newphi)
} else if newphi >= phi0 {
// Resume last saved line search
phi, dphi, gap = phi0, dphi0, gap0
step = step0
blas.Copy(W0.At("dnl")[0], W.At("dnl")[0])
blas.Copy(W0.At("dnli")[0], W.At("dnli")[0])
blas.Copy(W0.At("d")[0], W.At("d")[0])
blas.Copy(W0.At("di")[0], W.At("di")[0])
blas.Copy(W0.At("beta")[0], W.At("beta")[0])
for k, _ := range dims.At("q") {
blas.Copy(W0.At("v")[k], W.At("v")[k])
}
for k, _ := range dims.At("s") {
blas.Copy(W0.At("r")[k], W.At("r")[k])
blas.Copy(W0.At("rti")[k], W.At("rti")[k])
}
mCopy(x, x0)
mCopy(y, y0)
mCopy(dx, dx0)
mCopy(dy, dy0)
blas.Copy(s, s0)
blas.Copy(z, z0)
blas.Copy(ds2, ds20)
blas.Copy(dz2, dz20)
blas.Copy(lmbda, lmbda0)
blas.Copy(lmbdasq, lmbdasq0) // ???
mCopy(rx, rx0)
mCopy(ry, ry0)
blas.Copy(rznl, rznl0)
blas.Copy(rzl, rzl0)
dsdz = dsdz0
sigma, eta = sigma0, eta0
relaxed_iters = -1
} else if newphi <= phi+ALPHA*step*dphi {
// Series of relaxed line searches ends
// with sufficient decrease w.r.t. phi0.
backtrack = false
relaxed_iters = -1
//fmt.Printf("break 6 : newphi=%.7f\n", newphi)
}
}
}
} // end of line search
checkpnt.Check("eol", minor+900)
} // end for [0,1]
// Update x, y
dx.Axpy(x, step)
dy.Axpy(y, step)
checkpnt.Check("updatexy", 5000)
// Replace nonlinear, '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 nonlinear, 'l' and 'q' blocks.
// dz := e + step*dz for nonlinear, 'l' and 'q' blocks.
blas.ScalFloat(ds, step, &la.IOpt{"n", mnl + dims.Sum("l", "q")})
blas.ScalFloat(dz, step, &la.IOpt{"n", mnl + dims.Sum("l", "q")})
ind := mnl + dims.At("l")[0]
is := matrix.MakeIndexSet(0, ind, 1)
ds.Add(1.0, is...)
dz.Add(1.0, is...)
for _, m := range dims.At("q") {
ds.SetIndex(ind, 1.0+ds.GetIndex(ind))
dz.SetIndex(ind, 1.0+dz.GetIndex(ind))
ind += m
}
checkpnt.Check("updatedsdz", 5100)
// 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, mnl, true)
scale2(lmbda, dz, dims, mnl, true)
checkpnt.Check("scale2", 5200)
// 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", mnl + qdimsum})
blas.TbsvFloat(lmbda, sigz, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0},
&la.IOpt{"lda", 1}, &la.IOpt{"offseta", mnl + qdimsum})
checkpnt.Check("sigs", 5300)
ind2 := mnl + 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
}
checkpnt.Check("scaling", 5400)
err = updateScaling(W, lmbda, ds, dz)
checkpnt.Check("postscaling", 5500)
// Unscale s, z, tau, kappa (unscaled variables are used only to
// compute feasibility residuals).
ind = mnl + 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)
checkpnt.Check("unscale_s", 5600)
ind = mnl + 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)
checkpnt.Check("unscale_z", 5700)
gap = blas.DotFloat(lmbda, lmbda)
}
return
}
/*
Applies Nesterov-Todd scaling or its inverse.
Computes
x := W*x (trans is false 'N', inverse = false 'N')
x := W^T*x (trans is true 'T', inverse = false 'N')
x := W^{-1}*x (trans is false 'N', inverse = true 'T')
x := W^{-T}*x (trans is true 'T', inverse = true 'T').
x is a dense float matrix.
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].
The 'dnl' and 'dnli' entries are optional, and only present when the
function is called from the nonlinear solver.
*/
func scale(x *matrix.FloatMatrix, W *sets.FloatMatrixSet, trans, inverse bool) (err error) {
/*DEBUGGED*/
var wl []*matrix.FloatMatrix
var w *matrix.FloatMatrix
ind := 0
err = nil
// var minor int = 0
//if ! checkpnt.MinorEmpty() {
// minor = checkpnt.MinorTop()
//}
//fmt.Printf("\n%d.%04d scaling x=\n%v\n", checkpnt.Major(), minor, x.ToString("%.17f"))
// Scaling for nonlinear component xk is xk := dnl .* xk; inverse
// scaling is xk ./ dnl = dnli .* xk, where dnl = W['dnl'],
// dnli = W['dnli'].
if wl = W.At("dnl"); wl != nil {
if inverse {
w = W.At("dnli")[0]
} else {
w = W.At("dnl")[0]
}
for k := 0; k < x.Cols(); k++ {
err = blas.TbmvFloat(w, x, &la_.IOpt{"n", w.Rows()}, &la_.IOpt{"k", 0},
&la_.IOpt{"lda", 1}, &la_.IOpt{"offsetx", k * x.Rows()})
if err != nil {
//fmt.Printf("1. TbmvFloat: %v\n", err)
return
}
}
ind += w.Rows()
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("000scale", minor)
//}
// Scaling for linear 'l' component xk is xk := d .* xk; inverse
// scaling is xk ./ d = di .* xk, where d = W['d'], di = W['di'].
if inverse {
w = W.At("di")[0]
} else {
w = W.At("d")[0]
}
for k := 0; k < x.Cols(); k++ {
err = blas.TbmvFloat(w, x, &la_.IOpt{"n", w.Rows()}, &la_.IOpt{"k", 0},
&la_.IOpt{"lda", 1}, &la_.IOpt{"offsetx", k*x.Rows() + ind})
if err != nil {
//fmt.Printf("2. TbmvFloat: %v\n", err)
return
}
}
ind += w.Rows()
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("010scale", minor)
//}
// Scaling for 'q' component is
//
// xk := beta * (2*v*v' - J) * xk
// = beta * (2*v*(xk'*v)' - J*xk)
//
// where beta = W['beta'][k], v = W['v'][k], J = [1, 0; 0, -I].
//
//Inverse scaling is
//
// xk := 1/beta * (2*J*v*v'*J - J) * xk
// = 1/beta * (-J) * (2*v*((-J*xk)'*v)' + xk).
//wf := matrix.FloatZeros(x.Cols(), 1)
w = matrix.FloatZeros(x.Cols(), 1)
for k, v := range W.At("v") {
m := v.Rows()
if inverse {
blas.ScalFloat(x, -1.0, &la_.IOpt{"offset", ind}, &la_.IOpt{"inc", x.Rows()})
}
err = blas.GemvFloat(x, v, w, 1.0, 0.0, la_.OptTrans, &la_.IOpt{"m", m},
&la_.IOpt{"n", x.Cols()}, &la_.IOpt{"offsetA", ind},
&la_.IOpt{"lda", x.Rows()})
if err != nil {
//fmt.Printf("3. GemvFloat: %v\n", err)
return
}
err = blas.ScalFloat(x, -1.0, &la_.IOpt{"offset", ind}, &la_.IOpt{"inc", x.Rows()})
if err != nil {
return
}
err = blas.GerFloat(v, w, x, 2.0, &la_.IOpt{"m", m},
&la_.IOpt{"n", x.Cols()}, &la_.IOpt{"lda", x.Rows()},
&la_.IOpt{"offsetA", ind})
if err != nil {
//fmt.Printf("4. GerFloat: %v\n", err)
return
}
var a float64
if inverse {
blas.ScalFloat(x, -1.0,
&la_.IOpt{"offset", ind}, &la_.IOpt{"inc", x.Rows()})
// a[i,j] := 1.0/W[i,j]
a = 1.0 / W.At("beta")[0].GetIndex(k)
} else {
a = W.At("beta")[0].GetIndex(k)
}
for i := 0; i < x.Cols(); i++ {
blas.ScalFloat(x, a, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind + i*x.Rows()})
}
ind += m
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("020scale", minor)
//}
// Scaling for 's' component xk is
//
// xk := vec( r' * mat(xk) * r ) if trans = 'N'
// xk := vec( r * mat(xk) * r' ) if trans = 'T'.
//
// r is kth element of W['r'].
//
// Inverse scaling is
//
// xk := vec( rti * mat(xk) * rti' ) if trans = 'N'
// xk := vec( rti' * mat(xk) * rti ) if trans = 'T'.
//
// rti is kth element of W['rti'].
maxn := 0
for _, r := range W.At("r") {
if r.Rows() > maxn {
maxn = r.Rows()
}
}
a := matrix.FloatZeros(maxn, maxn)
for k, v := range W.At("r") {
t := trans
var r *matrix.FloatMatrix
if !inverse {
r = v
t = !trans
} else {
r = W.At("rti")[k]
}
n := r.Rows()
for i := 0; i < x.Cols(); i++ {
// scale diagonal of xk by 0.5
blas.ScalFloat(x, 0.5, &la_.IOpt{"offset", ind + i*x.Rows()},
&la_.IOpt{"inc", n + 1}, &la_.IOpt{"n", n})
// a = r*tril(x) (t is 'N') or a = tril(x)*r (t is 'T')
blas.Copy(r, a)
if !t {
err = blas.TrmmFloat(x, a, 1.0, la_.OptRight, &la_.IOpt{"m", n},
&la_.IOpt{"n", n}, &la_.IOpt{"lda", n}, &la_.IOpt{"ldb", n},
&la_.IOpt{"offsetA", ind + i*x.Rows()})
if err != nil {
//fmt.Printf("5. TrmmFloat: %v\n", err)
return
}
// x := (r*a' + a*r') if t is 'N'
err = blas.Syr2kFloat(r, a, x, 1.0, 0.0, la_.OptNoTrans, &la_.IOpt{"n", n},
&la_.IOpt{"k", n}, &la_.IOpt{"ldb", n}, &la_.IOpt{"ldc", n},
&la_.IOpt{"offsetC", ind + i*x.Rows()})
if err != nil {
//fmt.Printf("6. Syr2kFloat: %v\n", err)
return
}
} else {
err = blas.TrmmFloat(x, a, 1.0, la_.OptLeft, &la_.IOpt{"m", n},
&la_.IOpt{"n", n}, &la_.IOpt{"lda", n}, &la_.IOpt{"ldb", n},
&la_.IOpt{"offsetA", ind + i*x.Rows()})
if err != nil {
//fmt.Printf("7. TrmmFloat: %v\n", err)
return
}
// x := (r'*a + a'*r) if t is 'T'
err = blas.Syr2kFloat(r, a, x, 1.0, 0.0, la_.OptTrans, &la_.IOpt{"n", n},
&la_.IOpt{"k", n}, &la_.IOpt{"ldb", n}, &la_.IOpt{"ldc", n},
&la_.IOpt{"offsetC", ind + i*x.Rows()})
if err != nil {
//fmt.Printf("8. Syr2kFloat: %v\n", err)
return
}
}
}
ind += n * n
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("030scale", minor)
//}
return
}
/*
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 *sets.DimensionSet, mnl int) (W *sets.FloatMatrixSet, err error) {
/*DEBUGGED*/
err = nil
W = sets.NewFloatSet("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)
dnl := matrix.Sqrt(matrix.Div(stmp, ztmp))
//dnli := dnl.Copy()
//dnli.Apply(dnli, func(a float64)float64 { return 1.0/a })
dnli := matrix.Inv(dnl)
W.Set("dnl", dnl)
W.Set("dnli", dnli)
//lmd = stmp.Mul(ztmp)
//lmd.Apply(lmd, math.Sqrt)
lmd = matrix.Sqrt(matrix.Mul(stmp, ztmp))
lmbda.SetIndexesFromArray(lmd.FloatArray(), matrix.MakeIndexSet(0, mnl, 1)...)
} else {
// set for empty matrices
//W.Set("dnl", matrix.FloatZeros(0, 1))
//W.Set("dnli", matrix.FloatZeros(0, 1))
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(s.FloatArray()[mnl : mnl+m])
//zd := z.FloatArray()
ztmp = matrix.FloatVector(z.FloatArray()[mnl : mnl+m])
//fmt.Printf(".Sqrt()=\n%v\n", matrix.Div(stmp, ztmp).Sqrt().ToString("%.17f"))
//d := stmp.Div(ztmp)
//d.Apply(d, math.Sqrt)
d := matrix.Div(stmp, ztmp).Sqrt()
//di := d.Copy()
//di.Apply(di, func(a float64)float64 { return 1.0/a })
di := matrix.Inv(d)
//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 = matrix.Mul(stmp, ztmp).Sqrt()
// lmd has indexes mnl:mnl+m and length of m
lmbda.SetIndexesFromArray(lmd.FloatArray(), matrix.MakeIndexSet(mnl, mnl+m, 1)...)
//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 *sets.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(math.Sqrt)
s.SetIndexesFromArray(stmp.FloatArray(), matrix.MakeIndexSet(0, m, 1)...)
ztmp = matrix.FloatVector(z.FloatArray()[:m])
ztmp.Apply(math.Sqrt)
z.SetIndexesFromArray(ztmp.FloatArray(), matrix.MakeIndexSet(0, m, 1)...)
// 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})
//--dnliset[0] = matrix.Inv(dnlset[0])
matrix.Set(dnliset[0], dnlset[0])
dnliset[0].Inv()
}
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})
//--diset[0] = matrix.Inv(dset[0])
matrix.Set(diset[0], dset[0])
diset[0].Inv()
// 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})
// '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
}
/*
Evaluates
x := H(lambda^{1/2}) * x (inverse is 'N')
x := H(lambda^{-1/2}) * x (inverse is 'I').
H is the Hessian of the logarithmic barrier.
*/
func scale2(lmbda, x *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int, inverse bool) (err error) {
err = nil
//var minor int = 0
//if ! checkpnt.MinorEmpty() {
// minor = checkpnt.MinorTop()
//}
//fmt.Printf("\n%d.%04d scale2 x=\n%v\nlmbda=\n%v\n", checkpnt.Major(), minor,
// x.ToString("%.17f"), lmbda.ToString("%.17f"))
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("000scale2", minor)
//}
// For the nonlinear and 'l' blocks,
//
// xk := xk ./ l (inverse is 'N')
// xk := xk .* l (inverse is 'I')
//
// where l is lmbda[:mnl+dims['l']].
ind := mnl + dims.Sum("l")
if !inverse {
blas.TbsvFloat(lmbda, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
} else {
blas.TbmvFloat(lmbda, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("010scale2", minor)
//}
// For 'q' blocks, if inverse is 'N',
//
// xk := 1/a * [ l'*J*xk;
// xk[1:] - (xk[0] + l'*J*xk) / (l[0] + 1) * l[1:] ].
//
// If inverse is 'I',
//
// xk := a * [ l'*xk;
// xk[1:] + (xk[0] + l'*xk) / (l[0] + 1) * l[1:] ].
//
// a = sqrt(lambda_k' * J * lambda_k), l = lambda_k / a.
for _, m := range dims.At("q") {
var lx, a, c, x0 float64
a = jnrm2(lmbda, m, ind) //&la_.IOpt{"n", m}, &la_.IOpt{"offset", ind})
if !inverse {
lx = jdot(lmbda, x, m, ind, ind) //&la_.IOpt{"n", m}, &la_.IOpt{"offsetx", ind},
//&la_.IOpt{"offsety", ind})
lx /= a
} else {
lx = blas.DotFloat(lmbda, x, &la_.IOpt{"n", m}, &la_.IOpt{"offsetx", ind},
&la_.IOpt{"offsety", ind})
lx /= a
}
x0 = x.GetIndex(ind)
x.SetIndex(ind, lx)
c = (lx + x0) / (lmbda.GetIndex(ind)/a + 1.0) / a
if !inverse {
c *= -1.0
}
blas.AxpyFloat(lmbda, x, c, &la_.IOpt{"n", m - 1}, &la_.IOpt{"offsetx", ind + 1},
&la_.IOpt{"offsety", ind + 1})
if !inverse {
a = 1.0 / a
}
blas.ScalFloat(x, a, &la_.IOpt{"offset", ind}, &la_.IOpt{"n", m})
ind += m
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("020scale2", minor)
//}
// For the 's' blocks, if inverse is 'N',
//
// xk := vec( diag(l)^{-1/2} * mat(xk) * diag(k)^{-1/2}).
//
// If inverse is true,
//
// xk := vec( diag(l)^{1/2} * mat(xk) * diag(k)^{1/2}).
//
// where l is kth block of lambda.
//
// We scale upper and lower triangular part of mat(xk) because the
// inverse operation will be applied to nonsymmetric matrices.
ind2 := ind
sdims := dims.At("s")
for k := 0; k < len(sdims); k++ {
m := sdims[k]
scaleF := func(v, x float64) float64 {
return math.Sqrt(v) * math.Sqrt(x)
}
for j := 0; j < m; j++ {
c := matrix.FloatVector(lmbda.FloatArray()[ind2 : ind2+m])
c.ApplyConst(lmbda.GetIndex(ind2+j), scaleF)
if !inverse {
blas.Tbsv(c, x, &la_.IOpt{"n", m}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1},
&la_.IOpt{"offsetx", ind + j*m})
} else {
blas.Tbmv(c, x, &la_.IOpt{"n", m}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1},
&la_.IOpt{"offsetx", ind + j*m})
}
}
ind += m * m
ind2 += m
}
//if ! checkpnt.MinorEmpty() {
// checkpnt.Check("030scale2", minor)
//}
return
}
// The product x := (y o x). If diag is 'D', the 's' part of y is
// diagonal and only the diagonal is stored.
func sprod(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int, opts ...la_.Option) (err error) {
err = nil
diag := la_.GetStringOpt("diag", "N", opts...)
// For the nonlinear and 'l' blocks:
//
// yk o xk = yk .* xk.
ind := mnl + dims.At("l")[0]
err = blas.Tbmv(y, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1})
if err != nil {
return
}
//fmt.Printf("Sprod l:x=\n%v\n", x)
// For 'q' blocks:
//
// [ l0 l1' ]
// yk o xk = [ ] * xk
// [ l1 l0*I ]
//
// where yk = (l0, l1).
for _, m := range dims.At("q") {
dd := blas.DotFloat(x, y, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind},
&la_.IOpt{"n", m})
//fmt.Printf("dd=%v\n", dd)
alpha := y.GetIndex(ind)
//fmt.Printf("scal=%v\n", alpha)
blas.ScalFloat(x, alpha, &la_.IOpt{"offset", ind + 1}, &la_.IOpt{"n", m - 1})
alpha = x.GetIndex(ind)
//fmt.Printf("axpy=%v\n", alpha)
blas.AxpyFloat(y, x, alpha, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1},
&la_.IOpt{"n", m - 1})
x.SetIndex(ind, dd)
ind += m
}
//fmt.Printf("Sprod q :x=\n%v\n", x)
// For the 's' blocks:
//
// yk o sk = .5 * ( Yk * mat(xk) + mat(xk) * Yk )
//
// where Yk = mat(yk) if diag is 'N' and Yk = diag(yk) if diag is 'D'.
if diag[0] == 'N' {
// DEBUGGED
maxm := maxdim(dims.At("s"))
A := matrix.FloatZeros(maxm, maxm)
for _, m := range dims.At("s") {
blas.Copy(x, A, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m * m})
for i := 0; i < m-1; i++ { // i < m-1 --> i < m
symm(A, m, 0)
symm(y, m, ind)
}
err = blas.Syr2kFloat(A, y, x, 0.5, 0.0, &la_.IOpt{"n", m}, &la_.IOpt{"k", m},
&la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m},
&la_.IOpt{"offsetb", ind}, &la_.IOpt{"offsetc", ind})
if err != nil {
return
}
ind += m * m
}
//fmt.Printf("Sprod diag=N s:x=\n%v\n", x)
} else {
ind2 := ind
for _, m := range dims.At("s") {
for i := 0; i < m; i++ {
// original: u = 0.5 * ( y[ind2+i:ind2+m] + y[ind2+i] )
// creates matrix of elements: [ind2+i ... ind2+m] then
// element wisely adds y[ind2+i] and scales by 0.5
iset := matrix.MakeIndexSet(ind2+i, ind2+m, 1)
u := matrix.FloatVector(y.GetIndexes(iset...))
u.Add(y.GetIndex(ind2 + i))
u.Scale(0.5)
err = blas.Tbmv(u, x, &la_.IOpt{"n", m - i}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1},
&la_.IOpt{"offsetx", ind + i*(m+1)})
if err != nil {
return
}
}
ind += m * m
ind2 += m
}
//fmt.Printf("Sprod diag=T s:x=\n%v\n", x)
}
return
}
func conelp_solver(c MatrixVariable, G MatrixVarG, h *matrix.FloatMatrix,
A MatrixVarA, b MatrixVariable, dims *sets.DimensionSet, kktsolver KKTConeSolverVar,
solopts *SolverOptions, primalstart, dualstart *sets.FloatMatrixSet) (sol *Solution, err error) {
err = nil
const EXPON = 3
const STEP = 0.99
sol = &Solution{Unknown,
nil,
0.0, 0.0, 0.0, 0.0, 0.0,
0.0, 0.0, 0.0, 0.0, 0.0, 0}
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
maxIter := MAXITERS
if solopts.FeasTol > 0.0 {
feasTolerance = solopts.FeasTol
}
if solopts.AbsTol > 0.0 {
absTolerance = solopts.AbsTol
}
if solopts.RelTol > 0.0 {
relTolerance = solopts.RelTol
}
if solopts.MaxIter > 0 {
maxIter = solopts.MaxIter
}
if err = checkConeLpDimensions(dims); err != nil {
return
}
cdim := dims.Sum("l", "q") + dims.SumSquared("s")
//cdim_pckd := dims.Sum("l", "q") + dims.SumPacked("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)
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)
inds = append(inds, indq[len(indq)-1])
for _, k := range dims.At("s") {
inds = append(inds, inds[len(inds)-1]+k*k)
}
Gf := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error {
return G.Gf(x, y, alpha, beta, trans)
}
Af := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error {
return A.Af(x, y, alpha, beta, trans)
}
// 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 ]
if kktsolver == nil {
err = errors.New("nil kktsolver not allowed.")
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)
checkpnt.AddMatrixVar("ws3", ws3)
checkpnt.AddMatrixVar("wz3", wz3)
//
res := func(ux, uy MatrixVariable, uz, utau, us, ukappa *matrix.FloatMatrix,
vx, vy MatrixVariable, vz, vtau, vs, vkappa *matrix.FloatMatrix,
W *sets.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(&matrixVar{wz3}, vx, -1.0, 1.0, la.OptTrans)
//blas.AxpyFloat(c, vx, -utau.Float()/dg)
c.Axpy(vx, -utau.Float()/dg)
// vy := vy + A*ux - b*utau/dg
Af(ux, vy, 1.0, 1.0, la.OptNoTrans)
//blas.AxpyFloat(b, vy, -utau.Float()/dg)
b.Axpy(vy, -utau.Float()/dg)
// vz := vz + G*ux - h*utau/dg + W'*us
Gf(ux, &matrixVar{vz}, 1.0, 1.0, la.OptNoTrans)
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() + c.Dot(ux) +
b.Dot(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(-1)
var vkplus float64 = lscale * (utau.Float() + ukappa.Float())
vkappa.SetValue(vkappa.Float() + vkplus)
return
}
resx0 := math.Max(1.0, math.Sqrt(c.Dot(c)))
resy0 := math.Max(1.0, math.Sqrt(b.Dot(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)
x.Scal(0.0)
y := b.Copy()
//blas.ScalFloat(y, 0.0)
y.Scal(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)
checkpnt.AddVerifiable("x", x)
checkpnt.AddMatrixVar("s", s)
checkpnt.AddMatrixVar("z", z)
checkpnt.AddVerifiable("dx", dx)
checkpnt.AddMatrixVar("ds", ds)
checkpnt.AddMatrixVar("dz", dz)
checkpnt.Check("00init", 1)
var W *sets.FloatMatrixSet
var f KKTFuncVar
if primalstart == nil || dualstart == nil {
// Factor
//
// [ 0 A' G' ]
// [ A 0 0 ].
// [ G 0 -I ]
//
W = sets.NewFloatSet("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)
if err != nil {
fmt.Printf("kktsolver error: %s\n", err)
return
}
checkpnt.AddScaleVar(W)
}
checkpnt.Check("05init", 5)
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(b, dy)
checkpnt.MinorPush(5)
x.Scal(0.0)
mCopy(b, 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.ToString("%.5f"))
checkpnt.MinorPop()
} else {
mCopy(&matrixVar{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
}
//fmt.Printf("initial ts=%.5f\n", ts)
checkpnt.Check("10init", 10)
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)
checkpnt.MinorPush(10)
mCopy(c, dx)
dx.Scal(-1.0)
y.Scal(0.0)
blas.ScalFloat(z, 0.0)
err = f(dx, y, z)
if err != nil {
fmt.Printf("f(dx,y,z): %s\n", err)
return
}
//fmt.Printf("initial z=\n%v\n", z.ToString("%.5f"))
checkpnt.MinorPop()
} else {
if len(dualstart.At("y")) > 0 {
mCopy(&matrixVar{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
}
//fmt.Printf("initial tz=%.5f\n", tz)
nrms := snrm2(s, dims, 0)
nrmz := snrm2(z, dims, 0)
gap := 0.0
pcost := 0.0
dcost := 0.0
relgap := 0.0
checkpnt.Check("20init", 0)
if primalstart == nil && dualstart == nil {
gap = sdot(s, z, dims, 0)
pcost = c.Dot(x)
dcost = -b.Dot(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(&matrixVar{z}, rx, 1.0, 1.0, la.OptTrans)
resx := math.Sqrt(rx.Dot(rx))
// ry = b - A*x
ry := b.Copy()
Af(x, ry, -1.0, -1.0, la.OptNoTrans)
resy := math.Sqrt(ry.Dot(ry))
// rz = s + G*x - h
rz := matrix.FloatZeros(cdim, 1)
Gf(x, &matrixVar{rz}, 1.0, 0.0, la.OptNoTrans)
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 := c.Dot(x)
by := b.Dot(y)
hz := sdot(h, z, dims, 0)
//sol.X = x; sol.Y = y; sol.S = s; sol.Z = z
sol.Result = sets.NewFloatSet("x", "y", "s", "x")
sol.Result.Append("x", x.Matrix())
sol.Result.Append("y", y.Matrix())
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']]
if dims.At("l")[0] > 0 {
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
}
// indexes s[indq[:-1]]
if len(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)...)
ind += m * m
}
for _, k := range is {
s.SetIndex(k, a+s.GetIndex(k))
}
}
if tz >= -1e-8*math.Max(nrmz, 1.0) {
a := 1.0 + tz
is := make([]int, 0)
// indexes z[:dims['l']]
if dims.At("l")[0] > 0 {
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
}
// indexes z[indq[:-1]]
if len(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)...)
ind += m * m
}
for _, k := range is {
z.SetIndex(k, a+z.GetIndex(k))
}
}
} else if primalstart == nil && dualstart != nil {
if ts >= -1e-8*math.Max(nrms, 1.0) {
a := 1.0 + ts
is := make([]int, 0)
if dims.At("l")[0] > 0 {
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
}
if len(indq) > 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)...)
ind += m * m
}
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)
if dims.At("l")[0] > 0 {
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
}
if len(indq) > 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)...)
ind += m * m
}
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 MatrixVariable
var z1 *matrix.FloatMatrix
var dg, dgi float64
var th *matrix.FloatMatrix
var WS fVarClosure
var f3 KKTFuncVar
var cx, by, hz, rt float64
var hresx, resx, hresy, resy, hresz, resz float64
var dres, pres, dinfres, pinfres float64
// check point variables
checkpnt.AddMatrixVar("lmbda", lmbda)
checkpnt.AddMatrixVar("lmbdasq", lmbdasq)
checkpnt.AddVerifiable("rx", rx)
checkpnt.AddVerifiable("ry", ry)
checkpnt.AddMatrixVar("rz", rz)
checkpnt.AddFloatVar("resx", &resx)
checkpnt.AddFloatVar("resy", &resy)
checkpnt.AddFloatVar("resz", &resz)
checkpnt.AddFloatVar("hresx", &hresx)
checkpnt.AddFloatVar("hresy", &hresy)
checkpnt.AddFloatVar("hresz", &hresz)
checkpnt.AddFloatVar("cx", &cx)
checkpnt.AddFloatVar("by", &by)
checkpnt.AddFloatVar("hz", &hz)
checkpnt.AddFloatVar("gap", &gap)
checkpnt.AddFloatVar("pres", &pres)
checkpnt.AddFloatVar("dres", &dres)
for iter := 0; iter < maxIter+1; iter++ {
checkpnt.MajorNext()
checkpnt.Check("loop-start", 100)
// hrx = -A'*y - G'*z
Af(y, hrx, -1.0, 0.0, la.OptTrans)
Gf(&matrixVar{z}, hrx, -1.0, 1.0, la.OptTrans)
hresx = math.Sqrt(hrx.Dot(hrx))
// rx = hrx - c*tau
// = -A'*y - G'*z - c*tau
mCopy(hrx, rx)
c.Axpy(rx, -tau.Float())
resx = math.Sqrt(rx.Dot(rx)) / tau.Float()
// hry = A*x
Af(x, hry, 1.0, 0.0, la.OptNoTrans)
hresy = math.Sqrt(hry.Dot(hry))
// ry = hry - b*tau
// = A*x - b*tau
mCopy(hry, ry)
b.Axpy(ry, -tau.Float())
resy = math.Sqrt(ry.Dot(ry)) / tau.Float()
// hrz = s + G*x
Gf(x, &matrixVar{hrz}, 1.0, 0.0, la.OptNoTrans)
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 = c.Dot(x)
by = b.Dot(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))
}
checkpnt.Check("isready", 200)
if (pres <= feasTolerance && dres <= feasTolerance &&
(gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance))) ||
iter == maxIter {
// done
x.Scal(1.0 / tau.Float())
y.Scal(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 == 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 = sets.NewFloatSet("x", "y", "s", "x")
sol.Result.Append("x", x.Matrix())
sol.Result.Append("y", y.Matrix())
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 = sets.NewFloatSet("x", "y", "s", "x")
sol.Result.Append("x", x.Matrix())
sol.Result.Append("y", y.Matrix())
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")
y.Scal(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 = sets.NewFloatSet("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")
x.Scal(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 = sets.NewFloatSet("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 {
//fmt.Printf("compute scaling: lmbda=\n%v\n", lmbda.ToString("%.17f"))
//fmt.Printf("s=\n%v\n", s.ToString("%.17f"))
//fmt.Printf("z=\n%v\n", z.ToString("%.17f"))
W, err = computeScaling(s, z, lmbda, dims, 0)
checkpnt.AddScaleVar(W)
// 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())))
//fmt.Printf("lmbda=\n%v\n", lmbda.ToString("%.17f"))
//W.Print()
checkpnt.Check("compute_scaling", 300)
}
// 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)
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)
checkpnt.AddVerifiable("x1", x1)
checkpnt.AddMatrixVar("z1", z1)
}
mCopy(c, x1)
x1.Scal(-1.0)
mCopy(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)
x1.Scal(dgi)
y1.Scal(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()
x.Scal(t_)
y.Scal(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 = sets.NewFloatSet("x", "y", "s", "x")
sol.Result.Append("x", x.Matrix())
sol.Result.Append("y", y.Matrix())
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)
checkpnt.AddMatrixVar("th", th)
}
blas.Copy(h, th)
scale(th, W, true, true)
f6_no_ir := func(x, y MatrixVariable, 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) ]
minor := checkpnt.MinorTop()
err = nil
// y := -y = -by
y.Scal(-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)
checkpnt.Check("prescale", minor+5)
checkpnt.MinorPush(minor + 5)
err = scale(ws3, W, true, false)
checkpnt.MinorPop()
if err != nil {
fmt.Printf("scale error: %s\n", err)
}
blas.AxpyFloat(ws3, z, 1.0)
blas.ScalFloat(z, -1.0)
checkpnt.Check("f3-call", minor+20)
checkpnt.MinorPush(minor + 20)
err = f3(x, y, z)
checkpnt.MinorPop()
checkpnt.Check("f3-return", minor+40)
// 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_ += c.Dot(x)
tau_ += b.Dot(y)
tau_ += sdot(th, z, dims, 0)
tau_ = dgi * tau_ / (1.0 + sdot(z1, z1, dims, 0))
tau.SetValue(tau_)
x1.Axpy(x, tau_)
y1.Axpy(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)
checkpnt.AddVerifiable("wx", WS.wx)
checkpnt.AddMatrixVar("wz", WS.wz)
checkpnt.AddMatrixVar("ws", WS.ws)
}
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)
checkpnt.AddVerifiable("wx2", WS.wx2)
checkpnt.AddMatrixVar("wz2", WS.wz2)
checkpnt.AddMatrixVar("ws2", WS.ws2)
}
}
f6 := func(x, y MatrixVariable, z, tau, s, kappa *matrix.FloatMatrix) error {
var err error = nil
minor := checkpnt.MinorTop()
checkpnt.Check("startf6", minor+100)
if refinement > 0 || solopts.Debug {
mCopy(x, WS.wx)
mCopy(y, WS.wy)
blas.Copy(z, WS.wz)
blas.Copy(s, WS.ws)
WS.wtau.SetValue(tau.Float())
WS.wkappa.SetValue(kappa.Float())
}
checkpnt.Check("pref6_no_ir", minor+200)
err = f6_no_ir(x, y, z, tau, s, kappa)
checkpnt.Check("postf6_no_ir", minor+399)
for i := 0; i < refinement; i++ {
mCopy(WS.wx, WS.wx2)
mCopy(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())
checkpnt.Check("res-call", minor+400)
checkpnt.MinorPush(minor + 400)
err = res(x, y, z, tau, s, kappa, WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2, W, dg, lmbda)
checkpnt.MinorPop()
checkpnt.Check("refine_pref6_no_ir", minor+500)
checkpnt.MinorPush(minor + 500)
err = f6_no_ir(WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2)
checkpnt.MinorPop()
checkpnt.Check("refine_postf6_no_ir", minor+600)
WS.wx2.Axpy(x, 1.0)
WS.wy2.Axpy(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 {
checkpnt.MinorPush(minor + 700)
res(x, y, z, tau, s, kappa, WS.wx, WS.wy, WS.wz, WS.wtau, WS.ws, WS.wkappa, W, dg, lmbda)
checkpnt.MinorPop()
fmt.Printf("KKT residuals\n")
fmt.Printf(" 'x' : %.6e\n", math.Sqrt(WS.wx.Dot(WS.wx)))
fmt.Printf(" 'y' : %.6e\n", math.Sqrt(WS.wy.Dot(WS.wy)))
fmt.Printf(" 'z' : %.6e\n", snrm2(WS.wz, dims, 0))
fmt.Printf(" 'tau' : %.6e\n", math.Abs(WS.wtau.Float()))
fmt.Printf(" 's' : %.6e\n", snrm2(WS.ws, dims, 0))
fmt.Printf(" 'kappa': %.6e\n", math.Abs(WS.wkappa.Float()))
}
return err
}
var nrm float64 = blas.Nrm2Float(lmbda)
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)
// indexes: [:dims['l']]
if dims.At("l")[0] > 0 {
is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
}
// ...[indq[:-1]]
if len(indq) > 1 {
is = append(is, indq[:len(indq)-1]...)
}
// ...[ind : ind+m*m : m+1] (diagonal)
for _, m := range dims.At("s") {
is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
ind += m * m
}
//ds.Add(-sigma*mu, is...)
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)
mCopy(rx, dx)
dx.Scal(1.0 - sigma)
mCopy(ry, dy)
dy.Scal(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)
checkpnt.Check("pref6", (1+i)*1000)
checkpnt.MinorPush((1 + i) * 1000)
err = f6(dx, dy, dz, dtau, ds, dkappa)
checkpnt.MinorPop()
checkpnt.Check("postf6", (1+i)*1000+800)
// 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
checkpnt.MinorPush((1+i)*1000 + 900)
scale2(lmbda, ds, dims, 0, false)
scale2(lmbda, dz, dims, 0, false)
checkpnt.MinorPop()
checkpnt.Check("post-scale2", (1+i)*1000+990)
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)
checkpnt.Check("update-xy", 7000)
// Update x, y
dx.Axpy(x, step)
dy.Axpy(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))
}
checkpnt.Check("update-dsdz", 7500)
// 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}
checkpnt.MinorPush(7500)
scale2(lmbda, ds, dims, 0, true)
scale2(lmbda, dz, dims, 0, true)
checkpnt.MinorPop()
// 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
}
checkpnt.Check("pre-update-scaling", 7700)
err = updateScaling(W, lmbda, ds, dz)
checkpnt.Check("post-update-scaling", 7800)
// 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
checkpnt.Check("end-of-loop", 8000)
//fmt.Printf(" ** kappa=%.10f, tau=%.10f, gap=%.10f\n", kappa.Float(), tau.Float(), gap)
}
return
}