// Inner product of two vectors in S.
func sdot(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) float64 {
/*DEBUGGED*/
ind := mnl + dims.At("l")[0] + dims.Sum("q")
a := blas.DotFloat(x, y, &la_.IOpt{"n", ind})
for _, m := range dims.At("s") {
a += blas.DotFloat(x, y, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind},
&la_.IOpt{"incx", m + 1}, &la_.IOpt{"incy", m + 1}, &la_.IOpt{"n", m})
for j := 1; j < m; j++ {
a += 2.0 * blas.DotFloat(x, y, &la_.IOpt{"offsetx", ind + j}, &la_.IOpt{"offsety", ind + j},
&la_.IOpt{"incx", m + 1}, &la_.IOpt{"incy", m + 1}, &la_.IOpt{"n", m - j})
}
ind += m * m
}
return a
}
/*
Returns x' * J * y, where J = [1, 0; 0, -I].
*/
func jdot(x, y *matrix.FloatMatrix, n, offsetx, offsety int) float64 {
if n <= 0 {
n = x.NumElements()
}
a := blas.DotFloat(x, y, &la_.IOpt{"n", n - 1}, &la_.IOpt{"offsetx", offsetx + 1},
&la_.IOpt{"offsety", offsety + 1})
return x.GetIndex(offsetx)*y.GetIndex(offsety) - a
}
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 main() {
Sdata := [][]float64{
[]float64{4e-2, 6e-3, -4e-3, 0.0},
[]float64{6e-3, 1e-2, 0.0, 0.0},
[]float64{-4e-3, 0.0, 2.5e-3, 0.0},
[]float64{0.0, 0.0, 0.0, 0.0}}
pbar := matrix.FloatVector([]float64{.12, .10, .07, .03})
S := matrix.FloatMatrixFromTable(Sdata)
n := pbar.Rows()
G := matrix.FloatDiagonal(n, -1.0)
h := matrix.FloatZeros(n, 1)
A := matrix.FloatWithValue(1, n, 1.0)
b := matrix.FloatNew(1, 1, []float64{1.0})
var solopts cvx.SolverOptions
solopts.MaxIter = 30
solopts.ShowProgress = true
mu := 1.0
Smu := matrix.Scale(S, mu)
pbarNeg := matrix.Scale(pbar, -1.0)
fmt.Printf("Smu=\n%v\n", Smu.String())
fmt.Printf("-pbar=\n%v\n", pbarNeg.String())
sol, err := cvx.Qp(Smu, pbarNeg, G, h, A, b, &solopts, nil)
fmt.Printf("status: %v\n", err)
if sol != nil && sol.Status == cvx.Optimal {
x := sol.Result.At("x")[0]
ret := blas.DotFloat(x, pbar)
risk := math.Sqrt(blas.DotFloat(x, S.Times(x)))
fmt.Printf("ret=%.3f, risk=%.3f\n", ret, risk)
fmt.Printf("x=\n%v\n", x)
}
}
// Computes analytic center of A*x <= b with A m by n of rank n.
// We assume that b > 0 and the feasible set is bounded.
func Acent(A, b *matrix.FloatMatrix, niters int) (*matrix.FloatMatrix, []float64) {
if niters <= 0 {
niters = MAXITERS
}
ntdecrs := make([]float64, 0, niters)
if A.Rows() != b.Rows() {
return nil, nil
}
m, n := A.Size()
x := matrix.FloatZeros(n, 1)
H := matrix.FloatZeros(n, n)
// Helper m*n matrix
Dmn := matrix.FloatZeros(m, n)
for i := 0; i < niters; i++ {
// Gradient is g = A^T * (1.0/(b - A*x)). d = 1.0/(b - A*x)
// d is m*1 matrix, g is n*1 matrix
d := matrix.Minus(b, matrix.Times(A, x)).Inv()
g := matrix.Times(A.Transpose(), d)
// Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
// in the original python code expression d[:,n*[0]] creates
// a m*n matrix where each column is copy of column 0.
// We do it here manually.
for i := 0; i < n; i++ {
Dmn.SetColumn(i, d)
}
// Function mul creates element wise product of matrices.
Asc := matrix.Mul(Dmn, A)
blas.SyrkFloat(Asc, H, 1.0, 0.0, linalg.OptTrans)
// Newton step is v = H^-1 * g.
v := g.Copy().Scale(-1.0)
lapack.PosvFloat(H, v)
// Directional derivative and Newton decrement.
lam := blas.DotFloat(g, v)
ntdecrs = append(ntdecrs, math.Sqrt(-lam))
if ntdecrs[len(ntdecrs)-1] < TOL {
fmt.Printf("last Newton decrement < TOL(%v)\n", TOL)
return x, ntdecrs
}
// Backtracking line search.
// y = d .* A*v
y := d.Mul(A.Times(v))
step := 1.0
for 1-step*y.Max() < 0 {
step *= BETA
}
search:
for {
// t = -step*y
t := y.Copy().Scale(-step)
// t = (1 + t) [e.g. t = 1 - step*y]
t.Add(1.0)
// ts = sum(log(1-step*y))
ts := t.Log().Sum()
if -ts < ALPHA*step*lam {
break search
}
step *= BETA
}
v.Scale(step)
x = x.Plus(v)
}
// no solution !!
fmt.Printf("Iteration %d exhausted\n", niters)
return x, ntdecrs
}
// 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
}
func qcl1(A, b *matrix.FloatMatrix) (*cvx.Solution, error) {
// Returns the solution u, z of
//
// (primal) minimize || u ||_1
// subject to || A * u - b ||_2 <= 1
//
// (dual) maximize b^T z - ||z||_2
// subject to || A'*z ||_inf <= 1.
//
// Exploits structure, assuming A is m by n with m >= n.
m, n := A.Size()
Fkkt := func(W *sets.FloatMatrixSet) (f cvx.KKTFunc, err error) {
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
}
err = nil
f = nil
beta := W.At("beta")[0].GetIndex(0)
v := W.At("v")[0]
// As = 2 * v *(v[1:].T * A)
//v_1 := matrix.FloatNew(1, v.NumElements()-1, v.FloatArray()[1:])
v_1 := v.SubMatrix(1, 0).Transpose()
As := matrix.Times(v, matrix.Times(v_1, A)).Scale(2.0)
//As_1 := As.GetSubMatrix(1, 0, m, n)
//As_1.Scale(-1.0)
//As.SetSubMatrix(1, 0, matrix.Minus(As_1, A))
As_1 := As.SubMatrix(1, 0, m, n)
As_1.Scale(-1.0)
As_1.Minus(A)
As.Scale(1.0 / beta)
S := matrix.Times(As.Transpose(), As)
checkpnt.AddMatrixVar("S", S)
d1 := W.At("d")[0].SubMatrix(0, 0, n, 1).Copy()
d2 := W.At("d")[0].SubMatrix(n, 0).Copy()
// D = 4.0 * (d1**2 + d2**2)**-1
d := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2)).Inv().Scale(4.0)
// S[::n+1] += d
S.Diag().Plus(d.Transpose())
err = lapack.Potrf(S)
checkpnt.Check("00-Fkkt", minor)
if err != nil {
return
}
f = func(x, y, z *matrix.FloatMatrix) (err error) {
minor := 0
if !checkpnt.MinorEmpty() {
minor = checkpnt.MinorTop()
} else {
loopf += 1
minor = loopf
}
checkpnt.Check("00-f", minor)
// -- z := - W**-T * z
// z[:n] = -div( z[:n], d1 )
z_val := z.SubMatrix(0, 0, n, 1)
z_res := matrix.Div(z_val, d1).Scale(-1.0)
z.SubMatrix(0, 0, n, 1).Set(z_res)
// z[n:2*n] = -div( z[n:2*n], d2 )
z_val = z.SubMatrix(n, 0, n, 1)
z_res = matrix.Div(z_val, d2).Scale(-1.0)
z.SubMatrix(n, 0, n, 1).Set(z_res)
// z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
v0_z2n := v.GetIndex(0) * z.GetIndex(2*n)
v1_z2n := blas.DotFloat(v, z, &linalg.IOpt{"offsetx", 1}, &linalg.IOpt{"offsety", 2*n + 1})
z_res = matrix.Scale(v, -2.0*(v0_z2n-v1_z2n))
z.SubMatrix(2*n, 0, z_res.NumElements(), 1).Plus(z_res)
// z[2*n+1:] *= -1.0
z.SubMatrix(2*n+1, 0).Scale(-1.0)
// z[2*n:] /= beta
z.SubMatrix(2*n, 0).Scale(1.0 / beta)
// -- x := x - G' * W**-1 * z
// z_n = z[:n], z_2n = z[n:2*n], z_m = z[-(m+1):],
z_n := z.SubMatrix(0, 0, n, 1)
z_2n := z.SubMatrix(n, 0, n, 1)
z_m := z.SubMatrix(z.NumElements()-(m+1), 0)
// x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
z_res = matrix.Minus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
a_res := matrix.Times(As.Transpose(), z_m)
z_res.Plus(a_res).Scale(-1.0)
x.SubMatrix(0, 0, n, 1).Plus(z_res)
// x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
z_res = matrix.Plus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
x.SubMatrix(n, 0, z_res.NumElements(), 1).Plus(z_res)
checkpnt.Check("15-f", minor)
// Solve for x[:n]:
//
// S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]
// w1 = (d1**2 - d2**2), w2 = (d1**2 + d2**2)
w1 := matrix.Minus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))
w2 := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))
// x[:n] += -mul( div(w1, w2), x[n:])
x_n := x.SubMatrix(n, 0)
x_val := matrix.Mul(matrix.Div(w1, w2), x_n).Scale(-1.0)
x.SubMatrix(0, 0, n, 1).Plus(x_val)
checkpnt.Check("25-f", minor)
// Solve for x[n:]:
//
// (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
err = lapack.Potrs(S, x)
if err != nil {
fmt.Printf("Potrs error: %s\n", err)
}
checkpnt.Check("30-f", minor)
// Solve for x[n:]:
//
// (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]
// w1 = (d1**-2 - d2**-2), w2 = (d1**-2 + d2**-2)
w1 = matrix.Minus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
w2 = matrix.Plus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
x_n = x.SubMatrix(0, 0, n, 1)
// x[n:] += mul( d1**-2 - d2**-2, x[:n])
x_val = matrix.Mul(w1, x_n)
x.SubMatrix(n, 0, x_val.NumElements(), 1).Plus(x_val)
checkpnt.Check("35-f", minor)
// x[n:] = div( x[n:], d1**-2 + d2**-2)
x_n = x.SubMatrix(n, 0)
x_val = matrix.Div(x_n, w2)
x.SubMatrix(n, 0, x_val.NumElements(), 1).Set(x_val)
checkpnt.Check("40-f", minor)
// x_n = x[:n], x-2n = x[n:2*n]
x_n = x.SubMatrix(0, 0, n, 1)
x_2n := x.SubMatrix(n, 0, n, 1)
// z := z + W^-T * G*x
// z[:n] += div( x[:n] - x[n:2*n], d1)
x_val = matrix.Div(matrix.Minus(x_n, x_2n), d1)
z.SubMatrix(0, 0, n, 1).Plus(x_val)
checkpnt.Check("44-f", minor)
// z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
x_val = matrix.Div(matrix.Plus(x_n, x_2n).Scale(-1.0), d2)
z.SubMatrix(n, 0, n, 1).Plus(x_val)
checkpnt.Check("48-f", minor)
// z[2*n:] += As*x[:n]
x_val = matrix.Times(As, x_n)
z.SubMatrix(2*n, 0, x_val.NumElements(), 1).Plus(x_val)
checkpnt.Check("50-f", minor)
return nil
}
return
}
// matrix(n*[0.0] + n*[1.0])
c := matrix.FloatZeros(2*n, 1)
c.SubMatrix(n, 0).SetIndexes(1.0)
h := matrix.FloatZeros(2*n+m+1, 1)
h.SetIndexes(1.0, 2*n)
// h[2*n+1:] = -b
h.SubMatrix(2*n+1, 0).Plus(b).Scale(-1.0)
G := &matrixFs{A}
dims := sets.DSetNew("l", "q", "s")
dims.Set("l", []int{2 * n})
dims.Set("q", []int{m + 1})
var solopts cvx.SolverOptions
solopts.ShowProgress = true
if maxIter > 0 {
solopts.MaxIter = maxIter
}
if len(solver) > 0 {
solopts.KKTSolverName = solver
}
return cvx.ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, nil, nil)
}
/*
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
}