/*
Matrix-vector multiplication.
A is a matrix or spmatrix of size (m, n) where
N = dims['l'] + sum(dims['q']) + sum( k**2 for k in dims['s'] )
representing a mapping from R^n to S.
If trans is 'N':
y := alpha*A*x + beta * y (trans = 'N').
x is a vector of length n. y is a vector of length N.
If trans is 'T':
y := alpha*A'*x + beta * y (trans = 'T').
x is a vector of length N. y is a vector of length n.
The 's' components in S are stored in unpacked 'L' storage.
*/
func sgemv(A, x, y *matrix.FloatMatrix, alpha, beta float64, dims *sets.DimensionSet, opts ...la_.Option) error {
m := dims.Sum("l", "q") + dims.SumSquared("s")
n := la_.GetIntOpt("n", -1, opts...)
if n == -1 {
n = A.Cols()
}
trans := la_.GetIntOpt("trans", int(la_.PNoTrans), opts...)
offsetX := la_.GetIntOpt("offsetx", 0, opts...)
offsetY := la_.GetIntOpt("offsety", 0, opts...)
offsetA := la_.GetIntOpt("offseta", 0, opts...)
if trans == int(la_.PTrans) && alpha != 0.0 {
trisc(x, dims, offsetX)
//fmt.Printf("trisc x=\n%v\n", x.ConvertToString())
}
//fmt.Printf("alpha=%.4f beta=%.4f m=%d n=%d\n", alpha, beta, m, n)
//fmt.Printf("A=\n%v\nx=\n%v\ny=\n%v\n", A, x.ConvertToString(), y.ConvertToString())
err := blas.GemvFloat(A, x, y, alpha, beta, &la_.IOpt{"trans", trans},
&la_.IOpt{"n", n}, &la_.IOpt{"m", m}, &la_.IOpt{"offseta", offsetA},
&la_.IOpt{"offsetx", offsetX}, &la_.IOpt{"offsety", offsetY})
//fmt.Printf("gemv y=\n%v\n", y.ConvertToString())
if trans == int(la_.PTrans) && alpha != 0.0 {
triusc(x, dims, offsetX)
}
return err
}
// Solves a convex optimization problem with a linear objective
//
// minimize c'*x
// subject to f(x) <= 0
// G*x <= h
// A*x = b.
//
// f is vector valued, convex and twice differentiable. The linear
// inequalities are with respect to a cone C defined as the Cartesian
// product of N + M + 1 cones:
//
// C = C_0 x C_1 x .... x C_N x C_{N+1} x ... x C_{N+M}.
//
// The first cone C_0 is the nonnegative orthant of dimension ml. The
// next N cones are second order cones of dimension r[0], ..., r[N-1].
// The second order cone of dimension m is defined as
//
// { (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.
//
// The next M cones are positive semidefinite cones of order t[0], ..., t[M-1] >= 0.
//
// The structure of C is specified by DimensionSet dims which holds following sets
//
// dims.At("l") l, the dimension of the nonnegative orthant (array of length 1)
// dims.At("q") r[0], ... r[N-1], list with the dimesions of the second-order cones
// dims.At("s") t[0], ... t[M-1], array with the dimensions of the positive
// semidefinite cones
//
// The default value for dims is l: []int{h.Rows()}, q: []int{}, s: []int{}.
//
// On exit Solution contains the result and information about the accurancy of the
// solution. if SolutionStatus is Optimal then Solution.Result contains solutions
// for the problems.
//
// Result.At("x")[0] primal solution
// Result.At("snl")[0] non-linear constraint slacks
// Result.At("sl")[0] linear constraint slacks
// Result.At("y")[0] values for linear equality constraints y
// Result.At("znl")[0] values of dual variables for nonlinear inequalities
// Result.At("zl")[0] values of dual variables for linear inequalities
//
// If err is non-nil then sol is nil and err contains information about the argument or
// computation error.
//
func Cpl(F ConvexProg, c, G, h, A, b *matrix.FloatMatrix, dims *sets.DimensionSet, solopts *SolverOptions) (sol *Solution, err error) {
var mnl int
var x0 *matrix.FloatMatrix
mnl, x0, err = F.F0()
if err != nil {
return
}
if x0.Cols() != 1 {
err = errors.New("'x0' must be matrix with one column")
return
}
if c == nil {
err = errors.New("'c' must be non nil matrix")
return
}
if !c.SizeMatch(x0.Size()) {
err = errors.New(fmt.Sprintf("'c' must be matrix of size (%d,1)", x0.Rows()))
return
}
if h == nil {
h = matrix.FloatZeros(0, 1)
}
if h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = sets.NewDimensionSet("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
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
}
if G == nil {
G = matrix.FloatZeros(0, c.Rows())
}
if !G.SizeMatch(cdim, c.Rows()) {
estr := fmt.Sprintf("'G' must be of size (%d,%d)", cdim, c.Rows())
err = errors.New(estr)
return
}
// Check A and set defaults if it is nil
if A == nil {
// zeros rows reduces Gemv to vector products
A = matrix.FloatZeros(0, c.Rows())
}
if A.Cols() != c.Rows() {
estr := fmt.Sprintf("'A' must have %d columns", c.Rows())
err = errors.New(estr)
return
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() != A.Rows() {
estr := fmt.Sprintf("'b' must have length %d", A.Rows())
err = errors.New(estr)
return
}
var mc = matrixVar{c}
var mb = matrixVar{b}
var mA = matrixVarA{A}
var mG = matrixVarG{G, dims}
solvername := solopts.KKTSolverName
if len(solvername) == 0 {
if len(dims.At("q")) > 0 || len(dims.At("s")) > 0 {
solvername = "chol"
} else {
solvername = "chol2"
}
}
var factor kktFactor
var kktsolver KKTCpSolver = nil
if kktfunc, ok := solvers[solvername]; ok {
// kkt function returns us problem spesific factor function.
factor, err = kktfunc(G, dims, A, mnl)
// solver is
kktsolver = func(W *sets.FloatMatrixSet, x, z *matrix.FloatMatrix) (KKTFunc, error) {
_, Df, H, err := F.F2(x, z)
if err != nil {
return nil, err
}
return factor(W, H, Df)
}
} else {
err = errors.New(fmt.Sprintf("solver '%s' not known", solvername))
return
}
//return CplCustom(F, c, &mG, h, &mA, b, dims, kktsolver, solopts)
return cpl_problem(F, &mc, &mG, h, &mA, &mb, dims, kktsolver, solopts, x0, mnl)
}
// 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
}
// Solves a convex optimization problem with a linear objective
//
// minimize c'*x
// subject to f(x) <= 0
// G*x <= h
// A*x = b.
//
// using custom KTT equation solver and custom constraints G and A.
//
func CplCustomMatrix(F ConvexProg, c *matrix.FloatMatrix, G MatrixG, h *matrix.FloatMatrix,
A MatrixA, b *matrix.FloatMatrix, dims *sets.DimensionSet, kktsolver KKTCpSolver,
solopts *SolverOptions) (sol *Solution, err error) {
var mnl int
var x0 *matrix.FloatMatrix
mnl, x0, err = F.F0()
if err != nil {
return
}
if x0.Cols() != 1 {
err = errors.New("'x0' must be matrix with one column")
return
}
if c == nil {
err = errors.New("'c' must be non nil matrix")
return
}
if !c.SizeMatch(x0.Size()) {
err = errors.New(fmt.Sprintf("'c' must be matrix of size (%d,1)", x0.Rows()))
return
}
if h == nil {
h = matrix.FloatZeros(0, 1)
}
if h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = sets.NewDimensionSet("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
cdim := dims.Sum("l", "q") + dims.SumSquared("s")
if h.Rows() != cdim {
err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim))
return
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
mc := matrixVar{c}
mb := matrixVar{b}
var mG MatrixVarG
var mA MatrixVarA
if G == nil {
mG = &matrixVarG{matrix.FloatZeros(0, c.Rows()), dims}
} else {
mG = &matrixIfG{G}
}
if A == nil {
mA = &matrixVarA{matrix.FloatZeros(0, c.Rows())}
} else {
mA = &matrixIfA{A}
}
return cpl_problem(F, &mc, mG, h, mA, &mb, dims, kktsolver, solopts, x0, mnl)
}
// Solves a convex optimization problem with a linear objective
//
// minimize c'*x
// subject to f(x) <= 0
// G*x <= h
// A*x = b.
//
// using custom KTT equation solver.
//
func CplCustomKKT(F ConvexProg, c *matrix.FloatMatrix, G, h, A, b *matrix.FloatMatrix,
dims *sets.DimensionSet, kktsolver KKTCpSolver,
solopts *SolverOptions) (sol *Solution, err error) {
var mnl int
var x0 *matrix.FloatMatrix
mnl, x0, err = F.F0()
if err != nil {
return
}
if x0.Cols() != 1 {
err = errors.New("'x0' must be matrix with one column")
return
}
if c == nil {
err = errors.New("'c' must be non nil matrix")
return
}
if !c.SizeMatch(x0.Size()) {
err = errors.New(fmt.Sprintf("'c' must be matrix of size (%d,1)", x0.Rows()))
return
}
if h == nil {
h = matrix.FloatZeros(0, 1)
}
if h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = sets.NewDimensionSet("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
cdim := dims.Sum("l", "q") + dims.SumSquared("s")
if h.Rows() != cdim {
err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim))
return
}
if G == nil {
G = matrix.FloatZeros(0, c.Rows())
}
if !G.SizeMatch(cdim, c.Rows()) {
estr := fmt.Sprintf("'G' must be of size (%d,%d)", cdim, c.Rows())
err = errors.New(estr)
return
}
// Check A and set defaults if it is nil
if A == nil {
// zeros rows reduces Gemv to vector products
A = matrix.FloatZeros(0, c.Rows())
}
if A.Cols() != c.Rows() {
estr := fmt.Sprintf("'A' must have %d columns", c.Rows())
err = errors.New(estr)
return
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() != A.Rows() {
estr := fmt.Sprintf("'b' must have length %d", A.Rows())
err = errors.New(estr)
return
}
var mc = matrixVar{c}
var mb = matrixVar{b}
var mA = matrixVarA{A}
var mG = matrixVarG{G, dims}
return cpl_problem(F, &mc, &mG, h, &mA, &mb, dims, kktsolver, solopts, x0, mnl)
}
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
}
// Solves a pair of primal and dual cone programs using custom KKT solver and constraint
// interfaces MatrixG and MatrixA
//
func ConeLpCustomMatrix(c *matrix.FloatMatrix, G MatrixG, h *matrix.FloatMatrix,
A MatrixA, b *matrix.FloatMatrix, dims *sets.DimensionSet, kktsolver KKTConeSolver,
solopts *SolverOptions, primalstart, dualstart *sets.FloatMatrixSet) (sol *Solution, err error) {
err = nil
if c == nil || c.Cols() > 1 {
err = errors.New("'c' must be matrix with 1 column")
return
}
if h == nil || h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if 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)
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() > c.Rows() || b.Rows()+cdim_pckd < c.Rows() {
err = errors.New("Rank(A) < p or Rank([G; A]) < n")
return
}
if kktsolver == nil {
err = errors.New("nil kktsolver not allowed.")
return
}
var mA MatrixVarA
var mG MatrixVarG
if G == nil {
mG = &matrixVarG{matrix.FloatZeros(0, c.Rows()), dims}
} else {
mG = &matrixIfG{G}
}
if A == nil {
mA = &matrixVarA{matrix.FloatZeros(0, c.Rows())}
} else {
mA = &matrixIfA{A}
}
var mc = &matrixVar{c}
var mb = &matrixVar{b}
return conelp_problem(mc, mG, h, mA, mb, dims, kktsolver, solopts, primalstart, dualstart)
}
// Solves a pair of primal and dual cone programs using custom KKT solver.
//
func ConeLpCustomKKT(c, G, h, A, b *matrix.FloatMatrix, dims *sets.DimensionSet,
kktsolver KKTConeSolver, solopts *SolverOptions, primalstart,
dualstart *sets.FloatMatrixSet) (sol *Solution, err error) {
if c == nil || c.Cols() > 1 {
err = errors.New("'c' must be matrix with 1 column")
return
}
if h == nil {
h = matrix.FloatZeros(0, 1)
}
if h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = sets.NewDimensionSet("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
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 G == nil {
G = matrix.FloatZeros(0, c.Rows())
}
if !G.SizeMatch(cdim, c.Rows()) {
estr := fmt.Sprintf("'G' must be of size (%d,%d)", cdim, c.Rows())
err = errors.New(estr)
return
}
// Check A and set defaults if it is nil
if A == nil {
// zeros rows reduces Gemv to vector products
A = matrix.FloatZeros(0, c.Rows())
}
if A.Cols() != c.Rows() {
estr := fmt.Sprintf("'A' must have %d columns", c.Rows())
err = errors.New(estr)
return
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() != A.Rows() {
estr := fmt.Sprintf("'b' must have length %d", A.Rows())
err = errors.New(estr)
return
}
if b.Rows() > c.Rows() || b.Rows()+cdim_pckd < c.Rows() {
err = errors.New("Rank(A) < p or Rank([G; A]) < n")
return
}
mA := &matrixVarA{A}
mG := &matrixVarG{G, dims}
mc := &matrixVar{c}
mb := &matrixVar{b}
return conelp_problem(mc, mG, h, mA, mb, dims, kktsolver, solopts, primalstart, dualstart)
}
// Solves a pair of primal and dual cone programs
//
// minimize c'*x
// subject to G*x + s = h
// A*x = b
// s >= 0
//
// maximize -h'*z - b'*y
// subject to G'*z + A'*y + c = 0
// z >= 0.
//
// The inequalities are with respect to a cone C defined as the Cartesian
// product of N + M + 1 cones:
//
// C = C_0 x C_1 x .... x C_N x C_{N+1} x ... x C_{N+M}.
//
// The first cone C_0 is the nonnegative orthant of dimension ml.
// The next N cones are second order cones of dimension r[0], ..., r[N-1].
// The second order cone of dimension m is defined as
//
// { (u0, u1) in R x R^{m-1} | u0 >= ||u1||_2 }.
//
// The next M cones are positive semidefinite cones of order t[0], ..., t[M-1] >= 0.
//
// The structure of C is specified by DimensionSet dims which holds following sets
//
// dims.At("l") l, the dimension of the nonnegative orthant (array of length 1)
// dims.At("q") r[0], ... r[N-1], list with the dimesions of the second-order cones
// dims.At("s") t[0], ... t[M-1], array with the dimensions of the positive
// semidefinite cones
//
// The default value for dims is l: []int{G.Rows()}, q: []int{}, s: []int{}.
//
// Arguments primalstart, dualstart are optional starting points for primal and
// dual problems. If non-nil then primalstart is a FloatMatrixSet having two entries.
//
// primalstart.At("x")[0] starting point for x
// primalstart.At("s")[0] starting point for s
// dualstart.At("y")[0] starting point for y
// dualstart.At("z")[0] starting point for z
//
// On exit Solution contains the result and information about the accurancy of the
// solution. if SolutionStatus is Optimal then Solution.Result contains solutions
// for the problems.
//
// Result.At("x")[0] solution for x
// Result.At("y")[0] solution for y
// Result.At("s")[0] solution for s
// Result.At("z")[0] solution for z
//
func ConeLp(c, G, h, A, b *matrix.FloatMatrix, dims *sets.DimensionSet, solopts *SolverOptions,
primalstart, dualstart *sets.FloatMatrixSet) (sol *Solution, err error) {
if c == nil || c.Cols() > 1 {
err = errors.New("'c' must be matrix with 1 column")
return
}
if c.Rows() < 1 {
err = errors.New("No variables, 'c' must have at least one row")
return
}
if h == nil || h.Cols() > 1 {
err = errors.New("'h' must be matrix with 1 column")
return
}
if dims == nil {
dims = sets.NewDimensionSet("l", "q", "s")
dims.Set("l", []int{h.Rows()})
}
cdim := dims.Sum("l", "q") + dims.SumSquared("s")
cdim_pckd := dims.Sum("l", "q") + dims.SumPacked("s")
if h.Rows() != cdim {
err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim))
return
}
if G == nil {
G = matrix.FloatZeros(0, c.Rows())
}
if !G.SizeMatch(cdim, c.Rows()) {
estr := fmt.Sprintf("'G' must be of size (%d,%d)", cdim, c.Rows())
err = errors.New(estr)
return
}
// Check A and set defaults if it is nil
if A == nil {
// zeros rows reduces Gemv to vector products
A = matrix.FloatZeros(0, c.Rows())
}
if A.Cols() != c.Rows() {
estr := fmt.Sprintf("'A' must have %d columns", c.Rows())
err = errors.New(estr)
return
}
// Check b and set defaults if it is nil
if b == nil {
b = matrix.FloatZeros(0, 1)
}
if b.Cols() != 1 {
estr := fmt.Sprintf("'b' must be a matrix with 1 column")
err = errors.New(estr)
return
}
if b.Rows() != A.Rows() {
estr := fmt.Sprintf("'b' must have length %d", A.Rows())
err = errors.New(estr)
return
}
if b.Rows() > c.Rows() || b.Rows()+cdim_pckd < c.Rows() {
err = errors.New("Rank(A) < p or Rank([G; A]) < n")
return
}
solvername := solopts.KKTSolverName
if len(solvername) == 0 {
if len(dims.At("q")) > 0 || len(dims.At("s")) > 0 {
solvername = "qr"
} else {
solvername = "chol2"
}
}
var factor kktFactor
var kktsolver KKTConeSolver = nil
if kktfunc, ok := lpsolvers[solvername]; ok {
// kkt function returns us problem spesific factor function.
factor, err = kktfunc(G, dims, A, 0)
if err != nil {
return nil, err
}
kktsolver = func(W *sets.FloatMatrixSet) (KKTFunc, error) {
return factor(W, nil, nil)
}
} else {
err = errors.New(fmt.Sprintf("solver '%s' not known", solvername))
return
}
//return ConeLpCustom(c, &mG, h, &mA, b, dims, kktsolver, solopts, primalstart, dualstart)
c_e := &matrixVar{c}
G_e := &matrixVarG{G, dims}
A_e := &matrixVarA{A}
b_e := &matrixVar{b}
return conelp_problem(c_e, G_e, h, A_e, b_e, dims, kktsolver, solopts, primalstart, dualstart)
}