Example #1
0
// Solves a pair of primal and dual SDPs
//
//        minimize    c'*x
//        subject to  Gl*x + sl = hl
//                    mat(Gs[k]*x) + ss[k] = hs[k], k = 0, ..., N-1
//                    A*x = b
//                    sl >= 0,  ss[k] >= 0, k = 0, ..., N-1
//
//        maximize    -hl'*z - sum_k trace(hs[k]*zs[k]) - b'*y
//        subject to  Gl'*zl + sum_k Gs[k]'*vec(zs[k]) + A'*y + c = 0
//                    zl >= 0,  zs[k] >= 0, k = 0, ..., N-1.
//
// The inequalities sl >= 0 and zl >= 0 are elementwise vector
// inequalities.  The inequalities ss[k] >= 0, zs[k] >= 0 are matrix
// inequalities, i.e., the symmetric matrices ss[k] and zs[k] must be
// positive semidefinite.  mat(Gs[k]*x) is the symmetric matrix X with
// X[:] = Gs[k]*x.  For a symmetric matrix, zs[k], vec(zs[k]) is the
// vector zs[k][:].
//
func Sdp(c, Gl, hl, A, b *matrix.FloatMatrix, Ghs *sets.FloatMatrixSet, solopts *SolverOptions,
	primalstart, dualstart *sets.FloatMatrixSet) (sol *Solution, err error) {
	if c == nil {
		err = errors.New("'c' must a column matrix")
		return
	}
	n := c.Rows()
	if n < 1 {
		err = errors.New("Number of variables must be at least 1")
		return
	}
	if Gl == nil {
		Gl = matrix.FloatZeros(0, n)
	}
	if Gl.Cols() != n {
		err = errors.New(fmt.Sprintf("'G' must be matrix with %d columns", n))
		return
	}
	ml := Gl.Rows()
	if hl == nil {
		hl = matrix.FloatZeros(0, 1)
	}
	if !hl.SizeMatch(ml, 1) {
		err = errors.New(fmt.Sprintf("'hl' must be matrix of size (%d,1)", ml))
		return
	}
	Gsset := Ghs.At("Gs")
	ms := make([]int, 0)
	for i, Gs := range Gsset {
		if Gs.Cols() != n {
			err = errors.New(fmt.Sprintf("'Gs' must be list of matrices with %d columns", n))
			return
		}
		sz := int(math.Sqrt(float64(Gs.Rows())))
		if Gs.Rows() != sz*sz {
			err = errors.New(fmt.Sprintf("the squareroot of the number of rows of 'Gq[%d]' is not an integer", i))
			return
		}
		ms = append(ms, sz)
	}

	hsset := Ghs.At("hs")
	if len(Gsset) != len(hsset) {
		err = errors.New(fmt.Sprintf("'hs' must be a list of %d matrices", len(Gsset)))
		return
	}
	for i, hs := range hsset {
		if !hs.SizeMatch(ms[i], ms[i]) {
			s := fmt.Sprintf("hq[%d] has size (%d,%d). Expected size is (%d,%d)",
				i, hs.Rows(), hs.Cols(), ms[i], ms[i])
			err = errors.New(s)
			return
		}
	}
	if A == nil {
		A = matrix.FloatZeros(0, n)
	}
	if A.Cols() != n {
		err = errors.New(fmt.Sprintf("'A' must be matrix with %d columns", n))
		return
	}
	p := A.Rows()
	if b == nil {
		b = matrix.FloatZeros(0, 1)
	}
	if !b.SizeMatch(p, 1) {
		err = errors.New(fmt.Sprintf("'b' must be matrix of size (%d,1)", p))
		return
	}
	dims := sets.NewDimensionSet("l", "q", "s")
	dims.Set("l", []int{ml})
	dims.Set("s", ms)
	N := dims.Sum("l") + dims.SumSquared("s")

	// Map hs matrices to h vector
	h := matrix.FloatZeros(N, 1)
	h.SetIndexesFromArray(hl.FloatArray()[:ml], matrix.MakeIndexSet(0, ml, 1)...)
	ind := ml
	for k, hs := range hsset {
		h.SetIndexesFromArray(hs.FloatArray(), matrix.MakeIndexSet(ind, ind+ms[k]*ms[k], 1)...)
		ind += ms[k] * ms[k]
	}

	Gargs := make([]*matrix.FloatMatrix, 0)
	Gargs = append(Gargs, Gl)
	Gargs = append(Gargs, Gsset...)
	G, sizeg := matrix.FloatMatrixStacked(matrix.StackDown, Gargs...)

	var pstart, dstart *sets.FloatMatrixSet = nil, nil
	if primalstart != nil {
		pstart = sets.NewFloatSet("x", "s")
		pstart.Set("x", primalstart.At("x")[0])
		slset := primalstart.At("sl")
		margs := make([]*matrix.FloatMatrix, 0, len(slset)+1)
		margs = append(margs, primalstart.At("s")[0])
		margs = append(margs, slset...)
		sl, _ := matrix.FloatMatrixStacked(matrix.StackDown, margs...)
		pstart.Set("s", sl)
	}

	if dualstart != nil {
		dstart = sets.NewFloatSet("y", "z")
		dstart.Set("y", dualstart.At("y")[0])
		zlset := primalstart.At("zl")
		margs := make([]*matrix.FloatMatrix, 0, len(zlset)+1)
		margs = append(margs, dualstart.At("z")[0])
		margs = append(margs, zlset...)
		zl, _ := matrix.FloatMatrixStacked(matrix.StackDown, margs...)
		dstart.Set("z", zl)
	}

	//fmt.Printf("h=\n%v\n", h.ToString("%.3f"))
	//fmt.Printf("G=\n%v\n", G.ToString("%.3f"))

	sol, err = ConeLp(c, G, h, A, b, dims, solopts, pstart, dstart)
	// unpack sol.Result
	if err == nil {
		s := sol.Result.At("s")[0]
		sl := matrix.FloatVector(s.FloatArray()[:ml])
		sol.Result.Append("sl", sl)
		ind := ml
		for _, m := range ms {
			sk := matrix.FloatNew(m, m, s.FloatArray()[ind:ind+m*m])
			sol.Result.Append("ss", sk)
			ind += m * m
		}

		z := sol.Result.At("z")[0]
		zl := matrix.FloatVector(s.FloatArray()[:ml])
		sol.Result.Append("zl", zl)
		ind = ml
		for i, k := range sizeg[1:] {
			zk := matrix.FloatNew(ms[i], ms[i], z.FloatArray()[ind:ind+k])
			sol.Result.Append("zs", zk)
			ind += k
		}
	}
	sol.Result.Remove("s")
	sol.Result.Remove("z")

	return

}
Example #2
0
File: cpl.go Project: hrautila/cvx
// 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
}
Example #3
0
func mcsdp(w *matrix.FloatMatrix) (*Solution, error) {
	//
	// Returns solution x, z to
	//
	//    (primal)  minimize    sum(x)
	//              subject to  w + diag(x) >= 0
	//
	//    (dual)    maximize    -tr(w*z)
	//              subject to  diag(z) = 1
	//                          z >= 0.
	//
	n := w.Rows()
	G := &matrixFs{n}

	cngrnc := func(r, x *matrix.FloatMatrix, alpha float64) (err error) {
		// Congruence transformation
		//
		//    x := alpha * r'*x*r.
		//
		// r and x are square matrices.
		//
		err = nil

		// tx = matrix(x, (n,n)) is copying and reshaping
		// scale diagonal of x by 1/2, (x is (n,n))
		tx := x.Copy()
		matrix.Reshape(tx, n, n)
		tx.Diag().Scale(0.5)

		// a := tril(x)*r
		// (python: a = +r is really making a copy of r)
		a := r.Copy()

		err = blas.TrmmFloat(tx, a, 1.0, linalg.OptLeft)

		// x := alpha*(a*r' + r*a')
		err = blas.Syr2kFloat(r, a, tx, alpha, 0.0, linalg.OptTrans)

		// x[:] = tx[:]
		tx.CopyTo(x)
		return
	}

	Fkkt := func(W *sets.FloatMatrixSet) (KKTFunc, error) {

		//    Solve
		//                  -diag(z)                           = bx
		//        -diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
		//
		//    On entry, x and z contain bx and bs.
		//    On exit, they contain the solution, with z scaled
		//    (inv(rti)'*z*inv(rti) is returned instead of z).
		//
		//    We first solve
		//
		//        ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
		//
		//    and take z  = -rti' * (diag(x) + bs) * rti.

		var err error = nil
		rti := W.At("rti")[0]

		// t = rti*rti' as a nonsymmetric matrix.
		t := matrix.FloatZeros(n, n)
		err = blas.GemmFloat(rti, rti, t, 1.0, 0.0, linalg.OptTransB)
		if err != nil {
			return nil, err
		}

		// Cholesky factorization of tsq = t.*t.
		tsq := matrix.Mul(t, t)
		err = lapack.Potrf(tsq)
		if err != nil {
			return nil, err
		}

		f := func(x, y, z *matrix.FloatMatrix) (err error) {
			// tbst := t * zs * t = t * bs * t
			tbst := z.Copy()
			matrix.Reshape(tbst, n, n)
			cngrnc(t, tbst, 1.0)

			// x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
			diag := tbst.Diag().Transpose()
			x.Minus(diag)

			// x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
			err = lapack.Potrs(tsq, x)

			// z := z + diag(x) = bs + diag(x)
			// z, x are really column vectors here
			z.AddIndexes(matrix.MakeIndexSet(0, n*n, n+1), x.FloatArray())

			// z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
			cngrnc(rti, z, -1.0)
			return nil
		}
		return f, nil
	}

	c := matrix.FloatWithValue(n, 1, 1.0)

	// initial feasible x: x = 1.0 - min(lmbda(w))
	lmbda := matrix.FloatZeros(n, 1)
	wp := w.Copy()
	lapack.Syevx(wp, lmbda, nil, 0.0, nil, []int{1, 1}, linalg.OptRangeInt)
	x0 := matrix.FloatZeros(n, 1).Add(-lmbda.GetAt(0, 0) + 1.0)
	s0 := w.Copy()
	s0.Diag().Plus(x0.Transpose())
	matrix.Reshape(s0, n*n, 1)

	// initial feasible z is identity
	z0 := matrix.FloatIdentity(n)
	matrix.Reshape(z0, n*n, 1)

	dims := sets.DSetNew("l", "q", "s")
	dims.Set("s", []int{n})

	primalstart := sets.FloatSetNew("x", "s")
	dualstart := sets.FloatSetNew("z")
	primalstart.Set("x", x0)
	primalstart.Set("s", s0)
	dualstart.Set("z", z0)

	var solopts SolverOptions
	solopts.MaxIter = 30
	solopts.ShowProgress = false
	h := w.Copy()
	matrix.Reshape(h, h.NumElements(), 1)
	return ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, primalstart, dualstart)
}
Example #4
0
File: misc.go Project: hrautila/cvx
/*
   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
}
Example #5
0
File: misc.go Project: hrautila/cvx
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

}
Example #6
0
File: misc.go Project: hrautila/cvx
// 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
}
Example #7
0
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
}