Exemple #1
0
func _TestPartition1Dh(t *testing.T) {
	var AL, AR, A0, a1, A2 matrix.FloatMatrix
	A := matrix.FloatZeros(1, 6)
	partition1x2(&AL, &AR, A, 0, pRIGHT)
	t.Logf("m(AL)=%d, m(AR)=%d\n", AL.Cols(), AR.Cols())
	for AL.Cols() < A.Cols() {
		AR.Add(1.0)
		t.Logf("m(AR)=%d; %v\n", AR.Cols(), AR)
		repartition1x2to1x3(&AL, &A0, &a1, &A2, A, 1, pRIGHT)
		t.Logf("m(A0)=%d, m(A2)=%d, a1=%.1f\n", A0.Cols(), A2.Cols(), a1.Float())
		continue1x3to1x2(&AL, &AR, &A0, &a1, A, pRIGHT)
	}
}
Exemple #2
0
func (F *cpProg) F1(xa MatrixVariable) (f MatrixVariable, Df MatrixVarDf, err error) {
	f = nil
	Df = nil
	err = nil
	x, x_ok := xa.(*epigraph)
	if !x_ok {
		err = errors.New("'x' argument not an epigraph")
	}
	var f0, Df0 *matrix.FloatMatrix
	f0, Df0, err = F.convexF.F1(x.m())
	if err != nil {
		fmt.Printf("'cpProg.F1' error: %v\n%s\n", err, x)
		return
	}
	f0.Add(-x.t(), 0)
	f = &matrixVar{f0}
	Df = &epigraphDf{Df0}
	return
}
Exemple #3
0
func (F *cpProg) F2(xa MatrixVariable, z *matrix.FloatMatrix) (f MatrixVariable, Df MatrixVarDf, H MatrixVarH, err error) {
	f = nil
	Df = nil
	H = nil
	err = nil
	x, x_ok := xa.(*epigraph)
	if !x_ok {
		err = errors.New("'x' argument not an epigraph")
		return
	}
	var f0, Df0, H0 *matrix.FloatMatrix
	f0, Df0, H0, err = F.convexF.F2(x.m(), z)
	if err != nil {
		return
	}
	f0.Add(-x.t(), 0)
	f = &matrixVar{f0}
	Df = &epigraphDf{Df0}
	H = &epigraphH{H0}
	return
}
Exemple #4
0
func _TestPartition2D(t *testing.T) {
	var ATL, ATR, ABL, ABR, As matrix.FloatMatrix
	var A00, a01, A02, a10, a11, a12, A20, a21, A22 matrix.FloatMatrix

	A := matrix.FloatZeros(6, 6)
	As.SubMatrixOf(A, 1, 1, 4, 4)
	As.SetIndexes(1.0)
	partition2x2(&ATL, &ATR, &ABL, &ABR, &As, 0)
	t.Logf("ATL:\n%v\n", &ATL)

	for ATL.Rows() < As.Rows() {
		repartition2x2to3x3(&ATL,
			&A00, &a01, &A02,
			&a10, &a11, &a12,
			&A20, &a21, &A22, &As, 1)
		t.Logf("m(a12)=%d [%d], m(a11)=%d\n", a12.Cols(), a12.NumElements(), a11.NumElements())
		a11.Add(1.0)
		a21.Add(-2.0)

		continue3x3to2x2(&ATL, &ATR, &ABL, &ABR, &A00, &a11, &A22, &As)
	}
	t.Logf("A:\n%v\n", A)
}
Exemple #5
0
// unblocked LU decomposition with pivots: FLAME LU variant 3
func unblockedLUpiv(A *matrix.FloatMatrix, p *pPivots) error {
	var err error
	var ATL, ATR, ABL, ABR matrix.FloatMatrix
	var A00, a01, A02, a10, a11, a12, A20, a21, A22 matrix.FloatMatrix
	var AL, AR, A0, a1, A2, aB1, AB0 matrix.FloatMatrix
	var pT, pB, p0, p1, p2 pPivots

	err = nil
	partition2x2(
		&ATL, &ATR,
		&ABL, &ABR, A, 0, 0, pTOPLEFT)
	partition1x2(
		&AL, &AR, A, 0, pLEFT)
	partitionPivot2x1(
		&pT,
		&pB, p, 0, pTOP)

	for ATL.Rows() < A.Rows() && ATL.Cols() < A.Cols() {
		repartition2x2to3x3(&ATL,
			&A00, &a01, &A02,
			&a10, &a11, &a12,
			&A20, &a21, &A22 /**/, A, 1, pBOTTOMRIGHT)
		repartition1x2to1x3(&AL,
			&A0, &a1, &A2 /**/, A, 1, pRIGHT)
		repartPivot2x1to3x1(&pT,
			&p0, &p1, &p2 /**/, p, 1, pBOTTOM)

		// apply previously computed pivots
		applyPivots(&a1, &p0)

		// a01 = trilu(A00) \ a01 (TRSV)
		MVSolveTrm(&a01, &A00, 1.0, LOWER|UNIT)
		// a11 = a11 - a10 *a01
		a11.Add(Dot(&a10, &a01, -1.0))
		// a21 = a21 -A20*a01
		MVMult(&a21, &A20, &a01, -1.0, 1.0, NOTRANS)

		// pivot index on current column [a11, a21].T
		aB1.SubMatrixOf(&ABR, 0, 0, ABR.Rows(), 1)
		pivotIndex(&aB1, &p1)

		// pivots to current column
		applyPivots(&aB1, &p1)

		// a21 = a21 / a11
		InvScale(&a21, a11.Float())

		// apply pivots to previous columns
		AB0.SubMatrixOf(&ABL, 0, 0)
		applyPivots(&AB0, &p1)
		// scale last pivots to origin matrix row numbers
		p1.pivots[0] += ATL.Rows()

		continue3x3to2x2(
			&ATL, &ATR,
			&ABL, &ABR, &A00, &a11, &A22, A, pBOTTOMRIGHT)
		continue1x3to1x2(
			&AL, &AR, &A0, &a1, A, pRIGHT)
		contPivot3x1to2x1(
			&pT,
			&pB, &p0, &p1, p, pBOTTOM)
	}
	if ATL.Cols() < A.Cols() {
		applyPivots(&ATR, p)
		SolveTrm(&ATR, &ATL, 1.0, LEFT|UNIT|LOWER)
	}
	return err
}
Exemple #6
0
func coneqp_solver(P MatrixVarP, q MatrixVariable, G MatrixVarG, h *matrix.FloatMatrix,
	A MatrixVarA, b MatrixVariable, dims *sets.DimensionSet, kktsolver KKTConeSolverVar,
	solopts *SolverOptions, initvals *sets.FloatMatrixSet) (sol *Solution, err error) {

	err = nil
	EXPON := 3
	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 kktsolver func(*sets.FloatMatrixSet)(KKTFunc, error) = nil
	var refinement int
	var correction bool = true

	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 q == nil {
		err = errors.New("'q' must be non-nil MatrixVariable with one column")
		return
	}

	if h == nil {
		h = matrix.FloatZeros(0, 1)
	}
	if h.Cols() != 1 {
		err = errors.New("'h' must be non-nil matrix with one column")
		return
	}
	if dims == nil {
		dims = sets.NewDimensionSet("l", "q", "s")
		dims.Set("l", []int{h.Rows()})
	}

	err = checkConeQpDimensions(dims)
	if 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)
	}

	if P == nil {
		err = errors.New("'P' must be non-nil MatrixVarP interface.")
		return
	}
	fP := func(u, v MatrixVariable, alpha, beta float64) error {
		return P.Pf(u, v, alpha, beta)
	}

	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
	fA := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error {
		return A.Af(x, y, alpha, beta, trans)
	}

	// Check b and set defaults if it is nil
	if b == nil {
		err = errors.New("'b' must be non-nil MatrixVariable interface.")
		return
	}

	// kktsolver(W) returns a routine for solving 3x3 block KKT system
	//
	//     [ 0   A'  G'*W^{-1} ] [ ux ]   [ bx ]
	//     [ A   0   0         ] [ uy ] = [ by ].
	//     [ G   0   -W'       ] [ uz ]   [ bz ]

	if kktsolver == nil {
		err = errors.New("nil kktsolver not allowed.")
		return
	}

	ws3 := matrix.FloatZeros(cdim, 1)
	wz3 := matrix.FloatZeros(cdim, 1)
	checkpnt.AddMatrixVar("ws3", ws3)
	checkpnt.AddMatrixVar("wz3", wz3)

	//
	res := func(ux, uy MatrixVariable, uz, us *matrix.FloatMatrix, vx, vy MatrixVariable, vz, vs *matrix.FloatMatrix, W *sets.FloatMatrixSet, lmbda *matrix.FloatMatrix) (err error) {
		// Evaluates residual in Newton equations:
		//
		//      [ vx ]    [ vx ]   [ 0     ]   [ P  A'  G' ]   [ ux        ]
		//      [ vy ] := [ vy ] - [ 0     ] - [ A  0   0  ] * [ uy        ]
		//      [ vz ]    [ vz ]   [ W'*us ]   [ G  0   0  ]   [ W^{-1}*uz ]
		//
		//      vs := vs - lmbda o (uz + us).

		// vx := vx - P*ux - A'*uy - G'*W^{-1}*uz
		minor := checkpnt.MinorTop()
		checkpnt.Check("00res", minor)
		fP(ux, vx, -1.0, 1.0)
		fA(uy, vx, -1.0, 1.0, la.OptTrans)
		blas.Copy(uz, wz3)
		scale(wz3, W, true, false)
		fG(&matrixVar{wz3}, vx, -1.0, 1.0, la.OptTrans)
		// vy := vy - A*ux
		fA(ux, vy, -1.0, 1.0, la.OptNoTrans)
		checkpnt.Check("50res", minor)

		// vz := vz - G*ux - W'*us
		fG(ux, &matrixVar{vz}, -1.0, 1.0, la.OptNoTrans)
		blas.Copy(us, ws3)
		scale(ws3, W, true, false)
		blas.AxpyFloat(ws3, vz, -1.0)

		// vs := vs - lmbda o (uz + us)
		blas.Copy(us, ws3)
		blas.AxpyFloat(uz, ws3, 1.0)
		sprod(ws3, lmbda, dims, 0, la.OptDiag)
		blas.AxpyFloat(ws3, vs, -1.0)
		checkpnt.Check("90res", minor)
		return
	}

	resx0 := math.Max(1.0, math.Sqrt(q.Dot(q)))
	resy0 := math.Max(1.0, math.Sqrt(b.Dot(b)))
	resz0 := math.Max(1.0, snrm2(h, dims, 0))
	//fmt.Printf("resx0: %.17f, resy0: %.17f, resz0: %.17f\n", resx0, resy0, resz0)

	var x, y, dx, dy, rx, ry MatrixVariable
	var z, s, ds, dz, rz *matrix.FloatMatrix
	var lmbda, lmbdasq, sigs, sigz *matrix.FloatMatrix
	var W *sets.FloatMatrixSet
	var f, f3 KKTFuncVar
	var resx, resy, resz, step, sigma, mu, eta float64
	var gap, pcost, dcost, relgap, pres, dres, f0 float64

	if cdim == 0 {
		// Solve
		//
		//     [ P  A' ] [ x ]   [ -q ]
		//     [       ] [   ] = [    ].
		//     [ A  0  ] [ y ]   [  b ]
		//
		Wtmp := sets.NewFloatSet("d", "di", "beta", "v", "r", "rti")
		Wtmp.Set("d", matrix.FloatZeros(0, 1))
		Wtmp.Set("di", matrix.FloatZeros(0, 1))
		f3, err = kktsolver(Wtmp)
		if err != nil {
			s := fmt.Sprintf("kkt error: %s", err)
			err = errors.New("2: Rank(A) < p or Rank(([P; A; G;]) < n : " + s)
			return
		}
		x = q.Copy()
		x.Scal(0.0)
		y = b.Copy()
		f3(x, y, matrix.FloatZeros(0, 1))

		// dres = || P*x + q + A'*y || / resx0
		rx = q.Copy()
		fP(x, rx, 1.0, 1.0)
		pcost = 0.5 * (x.Dot(rx) + x.Dot(q))
		fA(y, rx, 1.0, 1.0, la.OptTrans)
		dres = math.Sqrt(rx.Dot(rx) / resx0)

		ry = b.Copy()
		fA(x, ry, 1.0, -1.0, la.OptNoTrans)
		pres = math.Sqrt(ry.Dot(ry) / resy0)

		relgap = 0.0
		if pcost == 0.0 {
			relgap = math.NaN()
		}

		sol.Result = sets.NewFloatSet("x", "y", "s", "z")
		sol.Result.Set("x", x.Matrix())
		sol.Result.Set("y", y.Matrix())
		sol.Result.Set("s", matrix.FloatZeros(0, 1))
		sol.Result.Set("z", matrix.FloatZeros(0, 1))
		sol.Status = Optimal
		sol.Gap = 0.0
		sol.RelativeGap = relgap
		sol.PrimalObjective = pcost
		sol.DualObjective = pcost
		sol.PrimalInfeasibility = pres
		sol.DualInfeasibility = dres
		sol.PrimalSlack = 0.0
		sol.DualSlack = 0.0
		return
	}
	x = q.Copy()
	y = b.Copy()
	s = matrix.FloatZeros(cdim, 1)
	z = matrix.FloatZeros(cdim, 1)

	checkpnt.AddVerifiable("x", x)
	checkpnt.AddVerifiable("y", y)
	checkpnt.AddMatrixVar("s", s)
	checkpnt.AddMatrixVar("z", z)

	var ts, tz, nrms, nrmz float64

	if initvals == nil {
		// Factor
		//
		//     [ 0   A'  G' ]
		//     [ A   0   0  ].
		//     [ G   0  -I  ]
		//
		W = sets.NewFloatSet("d", "di", "v", "beta", "r", "rti")
		W.Set("d", matrix.FloatOnes(dims.At("l")[0], 1))
		W.Set("di", matrix.FloatOnes(dims.At("l")[0], 1))
		W.Set("beta", matrix.FloatOnes(len(dims.At("q")), 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))
		}
		checkpnt.AddScaleVar(W)
		f, err = kktsolver(W)
		if err != nil {
			s := fmt.Sprintf("kkt error: %s", err)
			err = errors.New("3: Rank(A) < p or Rank([P; G; A]) < n : " + s)
			return
		}
		// Solve
		//
		//     [ P   A'  G' ]   [ x ]   [ -q ]
		//     [ A   0   0  ] * [ y ] = [  b ].
		//     [ G   0  -I  ]   [ z ]   [  h ]
		mCopy(q, x)
		x.Scal(-1.0)
		mCopy(b, y)
		blas.Copy(h, z)
		checkpnt.Check("00init", 1)
		err = f(x, y, z)
		if err != nil {
			s := fmt.Sprintf("kkt error: %s", err)
			err = errors.New("4: Rank(A) < p or Rank([P; G; A]) < n : " + s)
			return
		}
		blas.Copy(z, s)
		blas.ScalFloat(s, -1.0)
		checkpnt.Check("05init", 1)

		nrms = snrm2(s, dims, 0)
		ts, _ = maxStep(s, dims, 0, nil)
		//fmt.Printf("nrms = %.7f, ts = %.7f\n", nrms, ts)
		if ts >= -1e-8*math.Max(nrms, 1.0) {
			// a = 1.0 + ts
			a := 1.0 + ts
			is := make([]int, 0)
			// indexes s[:dims['l']]
			is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
			// indexes s[indq[:-1]]
			is = append(is, indq[:len(indq)-1]...)
			ind := dims.Sum("l", "q")
			// indexes s[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
			}
			for _, k := range is {
				s.SetIndex(k, a+s.GetIndex(k))
			}
		}

		nrmz = snrm2(z, dims, 0)
		tz, _ = maxStep(z, dims, 0, nil)
		//fmt.Printf("nrmz = %.7f, tz = %.7f\n", nrmz, tz)
		if tz >= -1e-8*math.Max(nrmz, 1.0) {
			a := 1.0 + tz
			is := make([]int, 0)
			is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
			is = append(is, indq[:len(indq)-1]...)
			ind := dims.Sum("l", "q")
			for _, m := range dims.At("s") {
				is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
				ind += m * m
			}
			for _, k := range is {
				z.SetIndex(k, a+z.GetIndex(k))
			}
		}

	} else {
		ix := initvals.At("x")[0]
		if ix != nil {
			mCopy(&matrixVar{ix}, x)
		} else {
			x.Scal(0.0)
		}

		is := initvals.At("s")[0]
		if is != nil {
			blas.Copy(is, s)
		} else {
			iset := make([]int, 0)
			iset = append(iset, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
			iset = append(iset, indq[:len(indq)-1]...)
			ind := dims.Sum("l", "q")
			for _, m := range dims.At("s") {
				iset = append(iset, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
				ind += m * m
			}
			for _, k := range iset {
				s.SetIndex(k, 1.0)
			}
		}

		iy := initvals.At("y")[0]
		if iy != nil {
			mCopy(&matrixVar{iy}, y)
		} else {
			y.Scal(0.0)
		}

		iz := initvals.At("z")[0]
		if iz != nil {
			blas.Copy(iz, z)
		} else {
			iset := make([]int, 0)
			iset = append(iset, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...)
			iset = append(iset, indq[:len(indq)-1]...)
			ind := dims.Sum("l", "q")
			for _, m := range dims.At("s") {
				iset = append(iset, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
				ind += m * m
			}
			for _, k := range iset {
				z.SetIndex(k, 1.0)
			}
		}
	}

	rx = q.Copy()
	ry = b.Copy()
	rz = matrix.FloatZeros(cdim, 1)
	dx = x.Copy()
	dy = y.Copy()
	dz = matrix.FloatZeros(cdim, 1)
	ds = matrix.FloatZeros(cdim, 1)
	lmbda = matrix.FloatZeros(cdim_diag, 1)
	lmbdasq = matrix.FloatZeros(cdim_diag, 1)
	sigs = matrix.FloatZeros(dims.Sum("s"), 1)
	sigz = matrix.FloatZeros(dims.Sum("s"), 1)

	checkpnt.AddVerifiable("rx", rx)
	checkpnt.AddVerifiable("ry", ry)
	checkpnt.AddVerifiable("dx", dx)
	checkpnt.AddVerifiable("dy", dy)
	//checkpnt.AddMatrixVar("rs", rs)
	checkpnt.AddMatrixVar("rz", rz)
	checkpnt.AddMatrixVar("ds", ds)
	checkpnt.AddMatrixVar("dz", dz)
	checkpnt.AddMatrixVar("lmbda", lmbda)
	checkpnt.AddMatrixVar("lmbdasq", lmbdasq)

	//var resx, resy, resz, step, sigma, mu, eta float64
	//var gap, pcost, dcost, relgap, pres, dres, f0 float64
	checkpnt.AddFloatVar("resx", &resx)
	checkpnt.AddFloatVar("resy", &resy)
	checkpnt.AddFloatVar("resz", &resz)
	checkpnt.AddFloatVar("step", &step)
	checkpnt.AddFloatVar("gap", &gap)
	checkpnt.AddFloatVar("dcost", &dcost)
	checkpnt.AddFloatVar("pcost", &pcost)
	checkpnt.AddFloatVar("dres", &dres)
	checkpnt.AddFloatVar("pres", &pres)
	checkpnt.AddFloatVar("relgap", &relgap)
	checkpnt.AddFloatVar("sigma", &sigma)

	var WS fVarClosure

	gap = sdot(s, z, dims, 0)
	for iter := 0; iter < maxIter+1; iter++ {
		checkpnt.MajorNext()
		checkpnt.Check("loopstart", 10)

		// f0 = (1/2)*x'*P*x + q'*x + r and  rx = P*x + q + A'*y + G'*z.
		mCopy(q, rx)
		fP(x, rx, 1.0, 1.0)
		f0 = 0.5 * (x.Dot(rx) + x.Dot(q))
		fA(y, rx, 1.0, 1.0, la.OptTrans)
		fG(&matrixVar{z}, rx, 1.0, 1.0, la.OptTrans)
		resx = math.Sqrt(rx.Dot(rx))

		// ry = A*x - b
		mCopy(b, ry)
		fA(x, ry, 1.0, -1.0, la.OptNoTrans)
		resy = math.Sqrt(ry.Dot(ry))

		// rz = s + G*x - h
		blas.Copy(s, rz)
		blas.AxpyFloat(h, rz, -1.0)
		fG(x, &matrixVar{rz}, 1.0, 1.0, la.OptNoTrans)
		resz = snrm2(rz, dims, 0)
		//fmt.Printf("resx: %.17f, resy: %.17f, resz: %.17f\n", resx, resy, resz)

		// Statistics for stopping criteria.

		// pcost = (1/2)*x'*P*x + q'*x
		// dcost = (1/2)*x'*P*x + q'*x + y'*(A*x-b) + z'*(G*x-h) '
		//       = (1/2)*x'*P*x + q'*x + y'*(A*x-b) + z'*(G*x-h+s) - z'*s
		//       = (1/2)*x'*P*x + q'*x + y'*ry + z'*rz - gap
		pcost = f0
		dcost = f0 + y.Dot(ry) + sdot(z, rz, dims, 0) - gap
		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

		if solopts.ShowProgress {
			if iter == 0 {
				// show headers of something
				fmt.Printf("% 10s% 12s% 10s% 8s% 7s\n",
					"pcost", "dcost", "gap", "pres", "dres")
			}
			// show something
			fmt.Printf("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e\n",
				iter, pcost, dcost, gap, pres, dres)
		}
		checkpnt.Check("stoptest", 100)

		if pres <= feasTolerance && dres <= feasTolerance &&
			(gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance)) ||
			iter == maxIter {

			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 {
				// terminated on max iterations.
				sol.Status = Unknown
				err = errors.New("Terminated (maximum iterations reached)")
				fmt.Printf("Terminated (maximum iterations reached)\n")
				return
			}
			// optimal solution found
			//fmt.Print("Optimal solution.\n")
			err = nil
			sol.Result = sets.NewFloatSet("x", "y", "s", "z")
			sol.Result.Set("x", x.Matrix())
			sol.Result.Set("y", y.Matrix())
			sol.Result.Set("s", s)
			sol.Result.Set("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
		}

		// Compute initial scaling W and scaled iterates:
		//
		//     W * z = W^{-T} * s = lambda.
		//
		// lmbdasq = lambda o lambda.
		if iter == 0 {
			W, err = computeScaling(s, z, lmbda, dims, 0)
			checkpnt.AddScaleVar(W)
		}
		ssqr(lmbdasq, lmbda, dims, 0)

		f3, err = kktsolver(W)
		if err != nil {
			if iter == 0 {
				s := fmt.Sprintf("kkt error: %s", err)
				err = errors.New("5: Rank(A) < p or Rank([P; A; G]) < n : " + s)
				return
			} else {
				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)
				// terminated (singular KKT matrix)
				fmt.Printf("Terminated (singular KKT matrix).\n")
				err = errors.New("Terminated (singular KKT matrix).")
				sol.Result = sets.NewFloatSet("x", "y", "s", "z")
				sol.Result.Set("x", x.Matrix())
				sol.Result.Set("y", y.Matrix())
				sol.Result.Set("s", s)
				sol.Result.Set("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
			}
		}
		// f4_no_ir(x, y, z, s) solves
		//
		//     [ 0     ]   [ P  A'  G' ]   [ ux        ]   [ bx ]
		//     [ 0     ] + [ A  0   0  ] * [ uy        ] = [ by ]
		//     [ W'*us ]   [ G  0   0  ]   [ W^{-1}*uz ]   [ bz ]
		//
		//     lmbda o (uz + us) = bs.
		//
		// On entry, x, y, z, s contain bx, by, bz, bs.
		// On exit, they contain ux, uy, uz, us.

		f4_no_ir := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) error {
			// Solve
			//
			//     [ P A' G'   ] [ ux        ]    [ bx                    ]
			//     [ A 0  0    ] [ uy        ] =  [ by                    ]
			//     [ G 0 -W'*W ] [ W^{-1}*uz ]    [ bz - W'*(lmbda o\ bs) ]
			//
			//     us = lmbda o\ bs - uz.
			//
			// On entry, x, y, z, s  contains bx, by, bz, bs.
			// On exit they contain x, y, z, s.

			minor := checkpnt.MinorTop()
			checkpnt.Check("f4_no_ir_start", minor)
			// s := lmbda o\ s
			//    = lmbda o\ bs
			sinv(s, lmbda, dims, 0)

			// z := z - W'*s
			//    = bz - W'*(lambda o\ bs)
			blas.Copy(s, ws3)
			scale(ws3, W, true, false)
			blas.AxpyFloat(ws3, z, -1.0)

			checkpnt.Check("f4_no_ir_f3", minor+50)
			err := f3(x, y, z)
			if err != nil {
				return err
			}
			checkpnt.Check("f4_no_ir_f3", minor+60)

			// s := s - z
			//    = lambda o\ bs - uz.
			blas.AxpyFloat(z, s, -1.0)
			checkpnt.Check("f4_no_ir_f3", minor+90)
			return nil
		}

		if iter == 0 {
			if refinement > 0 || solopts.Debug {
				WS.wx = q.Copy()
				WS.wy = y.Copy()
				WS.ws = matrix.FloatZeros(cdim, 1)
				WS.wz = matrix.FloatZeros(cdim, 1)
				checkpnt.AddVerifiable("wx", WS.wx)
				checkpnt.AddVerifiable("wy", WS.wy)
				checkpnt.AddMatrixVar("ws", WS.ws)
				checkpnt.AddMatrixVar("wz", WS.wz)
			}
			if refinement > 0 {
				WS.wx2 = q.Copy()
				WS.wy2 = y.Copy()
				WS.ws2 = matrix.FloatZeros(cdim, 1)
				WS.wz2 = matrix.FloatZeros(cdim, 1)
				checkpnt.AddVerifiable("wx2", WS.wx2)
				checkpnt.AddVerifiable("wy2", WS.wy2)
				checkpnt.AddMatrixVar("ws2", WS.ws2)
				checkpnt.AddMatrixVar("wz2", WS.wz2)
			}
		}

		f4 := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) (err error) {
			minor := checkpnt.MinorTop()
			checkpnt.Check("f4start", minor)
			err = nil
			if refinement > 0 || solopts.Debug {
				mCopy(x, WS.wx)
				mCopy(y, WS.wy)
				blas.Copy(z, WS.wz)
				blas.Copy(s, WS.ws)
			}

			checkpnt.MinorPush(minor + 100)
			err = f4_no_ir(x, y, z, s)
			checkpnt.MinorPop()

			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)

				checkpnt.MinorPush(minor + (i+1)*300)
				res(x, y, z, s, WS.wx2, WS.wy2, WS.wz2, WS.ws2, W, lmbda)
				checkpnt.MinorPop()

				checkpnt.MinorPush(minor + (i+1)*500)
				f4_no_ir(WS.wx2, WS.wy2, WS.wz2, WS.ws2)
				checkpnt.MinorPop()

				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)
			}
			checkpnt.Check("f4end", minor+1500)
			return
		}

		//var mu, sigma, eta float64
		mu = gap / float64(dims.Sum("l", "s")+len(dims.At("q")))
		sigma, eta = 0.0, 0.0

		for i := 0; i < 2; i++ {
			// Solve
			//
			//     [ 0     ]   [ P  A' G' ]   [ dx        ]
			//     [ 0     ] + [ A  0  0  ] * [ dy        ] = -(1 - eta) * r
			//     [ W'*ds ]   [ G  0  0  ]   [ W^{-1}*dz ]
			//
			//     lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e (i=0)
			//     lmbda o (dz + ds) = -lmbda o lmbda - dsa o dza
			//                         + sigma*mu*e (i=1) where dsa, dza
			//                         are the solution for i=0.

			minor_base := (i + 1) * 2000
			// ds = -lmbdasq + sigma * mu * e  (if i is 0)
			//    = -lmbdasq - dsa o dza + sigma * mu * e  (if i is 1),
			//    where ds, dz are solution for i is 0.
			blas.ScalFloat(ds, 0.0)
			if correction && i == 1 {
				blas.AxpyFloat(ws3, ds, -1.0)
			}
			blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", dims.Sum("l", "q")})
			ind := dims.At("l")[0]
			ds.Add(sigma*mu, matrix.MakeIndexSet(0, ind, 1)...)
			for _, m := range dims.At("q") {
				ds.SetIndex(ind, sigma*mu+ds.GetIndex(ind))
				ind += m
			}
			ind2 := ind
			for _, m := range dims.At("s") {
				blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", m}, &la.IOpt{"incy", m + 1},
					&la.IOpt{"offsetx", ind2}, &la.IOpt{"offsety", ind})
				ds.Add(sigma*mu, matrix.MakeIndexSet(ind, ind+m*m, m+1)...)
				ind += m * m
				ind2 += m
			}

			checkpnt.Check("00loop01", minor_base)

			// (dx, dy, dz) := -(1 - eta) * (rx, ry, rz)
			//blas.ScalFloat(dx, 0.0)
			//blas.AxpyFloat(rx, dx, -1.0+eta)
			dx.Scal(0.0)
			rx.Axpy(dx, -1.0+eta)
			dy.Scal(0.0)
			ry.Axpy(dy, -1.0+eta)
			blas.ScalFloat(dz, 0.0)
			blas.AxpyFloat(rz, dz, -1.0+eta)

			//fmt.Printf("== Calling f4 %d\n", i)
			//fmt.Printf("dx=\n%v\n", dx.ToString("%.17f"))
			//fmt.Printf("ds=\n%v\n", ds.ToString("%.17f"))
			//fmt.Printf("dz=\n%v\n", dz.ToString("%.17f"))
			//fmt.Printf("== Entering f4 %d\n", i)
			checkpnt.MinorPush(minor_base)
			err = f4(dx, dy, dz, ds)
			checkpnt.MinorPop()
			if err != nil {
				if iter == 0 {
					s := fmt.Sprintf("kkt error: %s", err)
					err = errors.New("6: Rank(A) < p or Rank([P; A; G]) < n : " + s)
					return
				} else {
					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)
					return
				}
			}

			dsdz := sdot(ds, dz, dims, 0)
			if correction && i == 0 {
				blas.Copy(ds, ws3)
				sprod(ws3, dz, dims, 0)
			}

			// 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.
			scale2(lmbda, ds, dims, 0, false)
			scale2(lmbda, dz, dims, 0, false)
			checkpnt.Check("maxstep", minor_base+1500)
			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)
			}
			t := maxvec([]float64{0.0, ts, tz})
			//fmt.Printf("== t=%.17f from %v\n", t, []float64{ts, tz})
			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 {
				m := math.Max(0.0, 1.0-step+dsdz/gap*(step*step))
				sigma = math.Pow(math.Min(1.0, m), float64(EXPON))
				eta = 0.0
			}
			//fmt.Printf("== step=%.17f sigma=%.17f dsdz=%.17f\n", step, sigma, dsdz)

		}

		checkpnt.Check("updatexy", 8000)
		dx.Axpy(x, step)
		dy.Axpy(y, step)
		//fmt.Printf("x=\n%v\n", x.ConvertToString())
		//fmt.Printf("y=\n%v\n", y.ConvertToString())
		//fmt.Printf("ds=\n%v\n", ds.ConvertToString())
		//fmt.Printf("dz=\n%v\n", dz.ConvertToString())

		// We will now replace the 'l' and 'q' blocks of ds and dz with
		// the updated iterates in the current scaling.
		// We also replace the '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", dims.Sum("l", "q")})
		blas.ScalFloat(dz, step, &la.IOpt{"n", dims.Sum("l", "q")})
		ind := 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", 8010)

		// ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz.
		//
		// This replaces the 'l' and 'q' components of ds and dz with the
		// updated variables in the current scaling.
		// The 's' components of ds and dz are replaced with
		//
		// diag(lmbda_k)^{1/2} * Qs * diag(lmbda_k)^{1/2}
		// diag(lmbda_k)^{1/2} * Qz * diag(lmbda_k)^{1/2}
		scale2(lmbda, ds, dims, 0, true)
		scale2(lmbda, dz, dims, 0, true)
		checkpnt.Check("scale2", 8030)

		// 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("updatescaling", 8050)
		err = updateScaling(W, lmbda, ds, dz)
		checkpnt.Check("afterscaling", 8060)

		// 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)

		gap = blas.DotFloat(lmbda, lmbda)
		checkpnt.Check("eol", 8900)
		//fmt.Printf("== gap = %.17f\n", gap)
	}
	return
}