// 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 }
// Internal CPL solver for CP and CLP problems. Everything is wrapped to proper interfaces func cpl_solver(F ConvexVarProg, c MatrixVariable, G MatrixVarG, h *matrix.FloatMatrix, A MatrixVarA, b MatrixVariable, dims *sets.DimensionSet, kktsolver KKTCpSolverVar, solopts *SolverOptions, x0 MatrixVariable, mnl int) (sol *Solution, err error) { const ( STEP = 0.99 BETA = 0.5 ALPHA = 0.01 EXPON = 3 MAX_RELAXED_ITERS = 8 ) var refinement int sol = &Solution{Unknown, nil, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0} feasTolerance := FEASTOL absTolerance := ABSTOL relTolerance := RELTOL maxIter := MAXITERS if solopts.FeasTol > 0.0 { feasTolerance = solopts.FeasTol } if solopts.AbsTol > 0.0 { absTolerance = solopts.AbsTol } if solopts.RelTol > 0.0 { relTolerance = solopts.RelTol } if solopts.Refinement > 0 { refinement = solopts.Refinement } else { refinement = 1 } if solopts.MaxIter > 0 { maxIter = solopts.MaxIter } if x0 == nil { mnl, x0, err = F.F0() if err != nil { return } } if c == nil { err = errors.New("Must define objective.") return } if h == nil { h = matrix.FloatZeros(0, 1) } if dims == nil { err = errors.New("Problem dimensions not defined.") return } if err = checkConeLpDimensions(dims); err != nil { return } cdim := dims.Sum("l", "q") + dims.SumSquared("s") cdim_diag := dims.Sum("l", "q", "s") if h.Rows() != cdim { err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim)) return } if G == nil { err = errors.New("'G' must be non-nil MatrixG interface.") return } fG := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error { return G.Gf(x, y, alpha, beta, trans) } // Check A and set defaults if it is nil if A == nil { err = errors.New("'A' must be non-nil MatrixA interface.") return } fA := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error { return A.Af(x, y, alpha, beta, trans) } if b == nil { err = errors.New("'b' must be non-nil MatrixVariable interface.") return } if kktsolver == nil { err = errors.New("nil kktsolver not allowed.") return } x := x0.Copy() y := b.Copy() y.Scal(0.0) z := matrix.FloatZeros(mnl+cdim, 1) s := matrix.FloatZeros(mnl+cdim, 1) ind := mnl + dims.At("l")[0] z.SetIndexes(1.0, matrix.MakeIndexSet(0, ind, 1)...) s.SetIndexes(1.0, matrix.MakeIndexSet(0, ind, 1)...) for _, m := range dims.At("q") { z.SetIndexes(1.0, ind) s.SetIndexes(1.0, ind) ind += m } for _, m := range dims.At("s") { iset := matrix.MakeIndexSet(ind, ind+m*m, m+1) z.SetIndexes(1.0, iset...) s.SetIndexes(1.0, iset...) ind += m * m } rx := x0.Copy() ry := b.Copy() dx := x.Copy() dy := y.Copy() rznl := matrix.FloatZeros(mnl, 1) rzl := matrix.FloatZeros(cdim, 1) dz := matrix.FloatZeros(mnl+cdim, 1) ds := matrix.FloatZeros(mnl+cdim, 1) lmbda := matrix.FloatZeros(mnl+cdim_diag, 1) lmbdasq := matrix.FloatZeros(mnl+cdim_diag, 1) sigs := matrix.FloatZeros(dims.Sum("s"), 1) sigz := matrix.FloatZeros(dims.Sum("s"), 1) dz2 := matrix.FloatZeros(mnl+cdim, 1) ds2 := matrix.FloatZeros(mnl+cdim, 1) newx := x.Copy() newy := y.Copy() newrx := x0.Copy() newz := matrix.FloatZeros(mnl+cdim, 1) news := matrix.FloatZeros(mnl+cdim, 1) newrznl := matrix.FloatZeros(mnl, 1) rx0 := rx.Copy() ry0 := ry.Copy() rznl0 := matrix.FloatZeros(mnl, 1) rzl0 := matrix.FloatZeros(cdim, 1) x0, dx0 := x.Copy(), dx.Copy() y0, dy0 := y.Copy(), dy.Copy() z0 := matrix.FloatZeros(mnl+cdim, 1) dz0 := matrix.FloatZeros(mnl+cdim, 1) dz20 := matrix.FloatZeros(mnl+cdim, 1) s0 := matrix.FloatZeros(mnl+cdim, 1) ds0 := matrix.FloatZeros(mnl+cdim, 1) ds20 := matrix.FloatZeros(mnl+cdim, 1) checkpnt.AddMatrixVar("z", z) checkpnt.AddMatrixVar("s", s) checkpnt.AddMatrixVar("dz", dz) checkpnt.AddMatrixVar("ds", ds) checkpnt.AddMatrixVar("rznl", rznl) checkpnt.AddMatrixVar("rzl", rzl) checkpnt.AddMatrixVar("lmbda", lmbda) checkpnt.AddMatrixVar("lmbdasq", lmbdasq) checkpnt.AddMatrixVar("z0", z0) checkpnt.AddMatrixVar("dz0", dz0) checkpnt.AddVerifiable("c", c) checkpnt.AddVerifiable("x", x) checkpnt.AddVerifiable("rx", rx) checkpnt.AddVerifiable("dx", dx) checkpnt.AddVerifiable("newrx", newrx) checkpnt.AddVerifiable("newx", newx) checkpnt.AddVerifiable("x0", x0) checkpnt.AddVerifiable("dx0", dx0) checkpnt.AddVerifiable("rx0", rx0) checkpnt.AddVerifiable("y", y) checkpnt.AddVerifiable("dy", dy) W0 := sets.NewFloatSet("d", "di", "dnl", "dnli", "v", "r", "rti", "beta") W0.Set("dnl", matrix.FloatZeros(mnl, 1)) W0.Set("dnli", matrix.FloatZeros(mnl, 1)) W0.Set("d", matrix.FloatZeros(dims.At("l")[0], 1)) W0.Set("di", matrix.FloatZeros(dims.At("l")[0], 1)) W0.Set("beta", matrix.FloatZeros(len(dims.At("q")), 1)) for _, n := range dims.At("q") { W0.Append("v", matrix.FloatZeros(n, 1)) } for _, n := range dims.At("s") { W0.Append("r", matrix.FloatZeros(n, n)) W0.Append("rti", matrix.FloatZeros(n, n)) } lmbda0 := matrix.FloatZeros(mnl+dims.Sum("l", "q", "s"), 1) lmbdasq0 := matrix.FloatZeros(mnl+dims.Sum("l", "q", "s"), 1) var f MatrixVariable = nil var Df MatrixVarDf = nil var H MatrixVarH = nil var ws3, wz3, wz2l, wz2nl *matrix.FloatMatrix var ws, wz, wz2, ws2 *matrix.FloatMatrix var wx, wx2, wy, wy2 MatrixVariable var gap, gap0, theta1, theta2, theta3, ts, tz, phi, phi0, mu, sigma, eta float64 var resx, resy, reszl, resznl, pcost, dcost, dres, pres, relgap float64 var resx0, resznl0, dres0, pres0 float64 var dsdz, dsdz0, step, step0, dphi, dphi0, sigma0, eta0 float64 var newresx, newresznl, newgap, newphi float64 var W *sets.FloatMatrixSet var f3 KKTFuncVar checkpnt.AddFloatVar("gap", &gap) checkpnt.AddFloatVar("pcost", &pcost) checkpnt.AddFloatVar("dcost", &dcost) checkpnt.AddFloatVar("pres", &pres) checkpnt.AddFloatVar("dres", &dres) checkpnt.AddFloatVar("relgap", &relgap) checkpnt.AddFloatVar("step", &step) checkpnt.AddFloatVar("dsdz", &dsdz) checkpnt.AddFloatVar("resx", &resx) checkpnt.AddFloatVar("resy", &resy) checkpnt.AddFloatVar("reszl", &reszl) checkpnt.AddFloatVar("resznl", &resznl) // Declare fDf and fH here, they bind to Df and H as they are already declared. // ??really?? var fDf func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error = nil var fH func(u, v MatrixVariable, alpha, beta float64) error = nil relaxed_iters := 0 for iters := 0; iters <= maxIter+1; iters++ { checkpnt.MajorNext() checkpnt.Check("loopstart", 10) checkpnt.MinorPush(10) if refinement != 0 || solopts.Debug { f, Df, H, err = F.F2(x, matrix.FloatVector(z.FloatArray()[:mnl])) fDf = func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error { return Df.Df(u, v, alpha, beta, trans) } fH = func(u, v MatrixVariable, alpha, beta float64) error { return H.Hf(u, v, alpha, beta) } } else { f, Df, err = F.F1(x) fDf = func(u, v MatrixVariable, alpha, beta float64, trans la.Option) error { return Df.Df(u, v, alpha, beta, trans) } } checkpnt.MinorPop() gap = sdot(s, z, dims, mnl) // these are helpers, copies of parts of z,s z_mnl := matrix.FloatVector(z.FloatArray()[:mnl]) z_mnl2 := matrix.FloatVector(z.FloatArray()[mnl:]) s_mnl := matrix.FloatVector(s.FloatArray()[:mnl]) s_mnl2 := matrix.FloatVector(s.FloatArray()[mnl:]) // rx = c + A'*y + Df'*z[:mnl] + G'*z[mnl:] // -- y, rx MatrixArg mCopy(c, rx) fA(y, rx, 1.0, 1.0, la.OptTrans) fDf(&matrixVar{z_mnl}, rx, 1.0, 1.0, la.OptTrans) fG(&matrixVar{z_mnl2}, rx, 1.0, 1.0, la.OptTrans) resx = math.Sqrt(rx.Dot(rx)) // rznl = s[:mnl] + f blas.Copy(s_mnl, rznl) blas.AxpyFloat(f.Matrix(), rznl, 1.0) resznl = blas.Nrm2Float(rznl) // rzl = s[mnl:] + G*x - h blas.Copy(s_mnl2, rzl) blas.AxpyFloat(h, rzl, -1.0) fG(x, &matrixVar{rzl}, 1.0, 1.0, la.OptNoTrans) reszl = snrm2(rzl, dims, 0) // Statistics for stopping criteria // pcost = c'*x // dcost = c'*x + y'*(A*x-b) + znl'*f(x) + zl'*(G*x-h) // = c'*x + y'*(A*x-b) + znl'*(f(x)+snl) + zl'*(G*x-h+sl) // - z'*s // = c'*x + y'*ry + znl'*rznl + zl'*rzl - gap //pcost = blas.DotFloat(c, x) pcost = c.Dot(x) dcost = pcost + blas.DotFloat(y.Matrix(), ry.Matrix()) + blas.DotFloat(z_mnl, rznl) dcost += sdot(z_mnl2, rzl, dims, 0) - gap if pcost < 0.0 { relgap = gap / -pcost } else if dcost > 0.0 { relgap = gap / dcost } else { relgap = math.NaN() } pres = math.Sqrt(resy*resy + resznl*resznl + reszl*reszl) dres = resx if iters == 0 { resx0 = math.Max(1.0, resx) resznl0 = math.Max(1.0, resznl) pres0 = math.Max(1.0, pres) dres0 = math.Max(1.0, dres) gap0 = gap theta1 = 1.0 / gap0 theta2 = 1.0 / resx0 theta3 = 1.0 / resznl0 } phi = theta1*gap + theta2*resx + theta3*resznl pres = pres / pres0 dres = dres / dres0 if solopts.ShowProgress { if iters == 0 { // some headers fmt.Printf("% 10s% 12s% 10s% 8s% 7s\n", "pcost", "dcost", "gap", "pres", "dres") } fmt.Printf("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e\n", iters, pcost, dcost, gap, pres, dres) } checkpnt.Check("checkgap", 50) // Stopping criteria if (pres <= feasTolerance && dres <= feasTolerance && (gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance))) || iters == maxIter { if iters == maxIter { s := "Terminated (maximum number of iterations reached)" if solopts.ShowProgress { fmt.Printf(s + "\n") } err = errors.New(s) sol.Status = Unknown } else { err = nil sol.Status = Optimal } sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl") sol.Result.Set("x", x.Matrix()) sol.Result.Set("y", y.Matrix()) sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl])) sol.Result.Set("zl", matrix.FloatVector(z.FloatArray()[mnl:])) sol.Result.Set("sl", matrix.FloatVector(s.FloatArray()[mnl:])) sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl])) sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz return } // Compute initial scaling W: // // W * z = W^{-T} * s = lambda. // // lmbdasq = lambda o lambda if iters == 0 { W, _ = computeScaling(s, z, lmbda, dims, mnl) checkpnt.AddScaleVar(W) } ssqr(lmbdasq, lmbda, dims, mnl) checkpnt.Check("lmbdasq", 90) // f3(x, y, z) solves // // [ H A' GG'*W^{-1} ] [ ux ] [ bx ] // [ A 0 0 ] [ uy ] = [ by ]. // [ GG 0 -W' ] [ uz ] [ bz ] // // On entry, x, y, z contain bx, by, bz. // On exit, they contain ux, uy, uz. checkpnt.MinorPush(95) f3, err = kktsolver(W, x, z_mnl) checkpnt.MinorPop() checkpnt.Check("f3", 100) if err != nil { // ?? z_mnl is really copy of z[:mnl] ... should we copy here back to z?? singular_kkt_matrix := false if iters == 0 { err = errors.New("Rank(A) < p or Rank([H(x); A; Df(x); G] < n") return } else if relaxed_iters > 0 && relaxed_iters < MAX_RELAXED_ITERS { // The arithmetic error may be caused by a relaxed line // search in the previous iteration. Therefore we restore // the last saved state and require a standard line search. phi, gap = phi0, gap0 mu = gap / float64(mnl+dims.Sum("l", "s")+len(dims.At("q"))) blas.Copy(W0.At("dnl")[0], W.At("dnl")[0]) blas.Copy(W0.At("dnli")[0], W.At("dnli")[0]) blas.Copy(W0.At("d")[0], W.At("d")[0]) blas.Copy(W0.At("di")[0], W.At("di")[0]) blas.Copy(W0.At("beta")[0], W.At("beta")[0]) for k, _ := range dims.At("q") { blas.Copy(W0.At("v")[k], W.At("v")[k]) } for k, _ := range dims.At("s") { blas.Copy(W0.At("r")[k], W.At("r")[k]) blas.Copy(W0.At("rti")[k], W.At("rti")[k]) } //blas.Copy(x0, x) //x0.CopyTo(x) mCopy(x0, x) //blas.Copy(y0, y) mCopy(y0, y) blas.Copy(s0, s) blas.Copy(z0, z) blas.Copy(lmbda0, lmbda) blas.Copy(lmbdasq0, lmbdasq) // ??? //blas.Copy(rx0, rx) //rx0.CopyTo(rx) mCopy(rx0, rx) //blas.Copy(ry0, ry) mCopy(ry0, ry) //resx = math.Sqrt(blas.DotFloat(rx, rx)) resx = math.Sqrt(rx.Dot(rx)) blas.Copy(rznl0, rznl) blas.Copy(rzl0, rzl) resznl = blas.Nrm2Float(rznl) relaxed_iters = -1 // How about z_mnl here??? checkpnt.MinorPush(120) f3, err = kktsolver(W, x, z_mnl) checkpnt.MinorPop() if err != nil { singular_kkt_matrix = true } } else { singular_kkt_matrix = true } if singular_kkt_matrix { msg := "Terminated (singular KKT matrix)." if solopts.ShowProgress { fmt.Printf(msg + "\n") } zl := matrix.FloatVector(z.FloatArray()[mnl:]) sl := matrix.FloatVector(s.FloatArray()[mnl:]) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(sl, m, ind) symm(zl, m, ind) ind += m * m } ts, _ = maxStep(s, dims, mnl, nil) tz, _ = maxStep(z, dims, mnl, nil) err = errors.New(msg) sol.Status = Unknown sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl") sol.Result.Set("x", x.Matrix()) sol.Result.Set("y", y.Matrix()) sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl])) sol.Result.Set("zl", zl) sol.Result.Set("sl", sl) sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl])) sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz return } } // f4_no_ir(x, y, z, s) solves // // [ 0 ] [ H A' GG' ] [ ux ] [ bx ] // [ 0 ] + [ A 0 0 ] [ uy ] = [ by ] // [ W'*us ] [ GG 0 0 ] [ W^{-1}*uz ] [ bz ] // // lmbda o (uz + us) = bs. // // On entry, x, y, z, x, contain bx, by, bz, bs. // On exit, they contain ux, uy, uz, us. if iters == 0 { ws3 = matrix.FloatZeros(mnl+cdim, 1) wz3 = matrix.FloatZeros(mnl+cdim, 1) checkpnt.AddMatrixVar("ws3", ws3) checkpnt.AddMatrixVar("wz3", wz3) } f4_no_ir := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) (err error) { // Solve // // [ H A' GG' ] [ ux ] [ bx ] // [ A 0 0 ] [ uy ] = [ by ] // [ GG 0 -W'*W ] [ W^{-1}*uz ] [ bz - W'*(lmbda o\ bs) ] // // us = lmbda o\ bs - uz. err = nil // s := lmbda o\ s // = lmbda o\ bs sinv(s, lmbda, dims, mnl) // z := z - W'*s // = bz - W' * (lambda o\ bs) blas.Copy(s, ws3) scale(ws3, W, true, false) blas.AxpyFloat(ws3, z, -1.0) // Solve for ux, uy, uz err = f3(x, y, z) // s := s - z // = lambda o\ bs - z. blas.AxpyFloat(z, s, -1.0) return } if iters == 0 { wz2nl = matrix.FloatZeros(mnl, 1) wz2l = matrix.FloatZeros(cdim, 1) checkpnt.AddMatrixVar("wz2nl", wz2nl) checkpnt.AddMatrixVar("wz2l", wz2l) } res := func(ux, uy MatrixVariable, uz, us *matrix.FloatMatrix, vx, vy MatrixVariable, vz, vs *matrix.FloatMatrix) (err error) { // Evaluates residuals in Newton equations: // // [ vx ] [ 0 ] [ H A' GG' ] [ ux ] // [ vy ] -= [ 0 ] + [ A 0 0 ] [ uy ] // [ vz ] [ W'*us ] [ GG 0 0 ] [ W^{-1}*uz ] // // vs -= lmbda o (uz + us). err = nil minor := checkpnt.MinorTop() // vx := vx - H*ux - A'*uy - GG'*W^{-1}*uz fH(ux, vx, -1.0, 1.0) fA(uy, vx, -1.0, 1.0, la.OptTrans) blas.Copy(uz, wz3) scale(wz3, W, false, true) wz3_nl := matrix.FloatVector(wz3.FloatArray()[:mnl]) wz3_l := matrix.FloatVector(wz3.FloatArray()[mnl:]) fDf(&matrixVar{wz3_nl}, vx, -1.0, 1.0, la.OptTrans) fG(&matrixVar{wz3_l}, vx, -1.0, 1.0, la.OptTrans) checkpnt.Check("10res", minor+10) // vy := vy - A*ux fA(ux, vy, -1.0, 1.0, la.OptNoTrans) // vz := vz - W'*us - GG*ux err = fDf(ux, &matrixVar{wz2nl}, 1.0, 0.0, la.OptNoTrans) checkpnt.Check("15res", minor+10) blas.AxpyFloat(wz2nl, vz, -1.0) fG(ux, &matrixVar{wz2l}, 1.0, 0.0, la.OptNoTrans) checkpnt.Check("20res", minor+10) blas.AxpyFloat(wz2l, vz, -1.0, &la.IOpt{"offsety", mnl}) blas.Copy(us, ws3) scale(ws3, W, true, false) blas.AxpyFloat(ws3, vz, -1.0) checkpnt.Check("30res", minor+10) // vs -= lmbda o (uz + us) blas.Copy(us, ws3) blas.AxpyFloat(uz, ws3, 1.0) sprod(ws3, lmbda, dims, mnl, &la.SOpt{"diag", "D"}) blas.AxpyFloat(ws3, vs, -1.0) checkpnt.Check("90res", minor+10) return } // f4(x, y, z, s) solves the same system as f4_no_ir, but applies // iterative refinement. if iters == 0 { if refinement > 0 || solopts.Debug { wx = c.Copy() wy = b.Copy() wz = z.Copy() ws = s.Copy() checkpnt.AddVerifiable("wx", wx) checkpnt.AddMatrixVar("ws", ws) checkpnt.AddMatrixVar("wz", wz) } if refinement > 0 { wx2 = c.Copy() wy2 = b.Copy() wz2 = matrix.FloatZeros(mnl+cdim, 1) ws2 = matrix.FloatZeros(mnl+cdim, 1) checkpnt.AddVerifiable("wx2", wx2) checkpnt.AddMatrixVar("ws2", ws2) checkpnt.AddMatrixVar("wz2", wz2) } } f4 := func(x, y MatrixVariable, z, s *matrix.FloatMatrix) (err error) { if refinement > 0 || solopts.Debug { mCopy(x, wx) mCopy(y, wy) blas.Copy(z, wz) blas.Copy(s, ws) } minor := checkpnt.MinorTop() checkpnt.Check("0_f4", minor+100) checkpnt.MinorPush(minor + 100) err = f4_no_ir(x, y, z, s) checkpnt.MinorPop() checkpnt.Check("1_f4", minor+200) for i := 0; i < refinement; i++ { mCopy(wx, wx2) mCopy(wy, wy2) blas.Copy(wz, wz2) blas.Copy(ws, ws2) checkpnt.Check("2_f4", minor+(1+i)*200) checkpnt.MinorPush(minor + (1+i)*200) res(x, y, z, s, wx2, wy2, wz2, ws2) checkpnt.MinorPop() checkpnt.Check("3_f4", minor+(1+i)*200+100) err = f4_no_ir(wx2, wy2, wz2, ws2) checkpnt.MinorPop() checkpnt.Check("4_f4", minor+(1+i)*200+199) wx2.Axpy(x, 1.0) wy2.Axpy(y, 1.0) blas.AxpyFloat(wz2, z, 1.0) blas.AxpyFloat(ws2, s, 1.0) } if solopts.Debug { res(x, y, z, s, wx, wy, wz, ws) fmt.Printf("KKT residuals:\n") } return } sigma, eta = 0.0, 0.0 for i := 0; i < 2; i++ { minor := (i + 2) * 1000 checkpnt.MinorPush(minor) checkpnt.Check("loop01", minor) // Solve // // [ 0 ] [ H A' GG' ] [ dx ] // [ 0 ] + [ A 0 0 ] [ dy ] = -(1 - eta)*r // [ W'*ds ] [ GG 0 0 ] [ W^{-1}*dz ] // // lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e. // mu = gap / float64(mnl+dims.Sum("l", "s")+len(dims.At("q"))) blas.ScalFloat(ds, 0.0) blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", mnl + dims.Sum("l", "q")}) ind = mnl + dims.At("l")[0] iset := matrix.MakeIndexSet(0, ind, 1) ds.Add(sigma*mu, iset...) for _, m := range dims.At("q") { ds.Add(sigma*mu, ind) ind += m } ind2 := ind for _, m := range dims.At("s") { blas.AxpyFloat(lmbdasq, ds, -1.0, &la.IOpt{"n", m}, &la.IOpt{"offsetx", ind2}, &la.IOpt{"offsety", ind}, &la.IOpt{"incy", m + 1}) ds.Add(sigma*mu, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m ind2 += m } dx.Scal(0.0) rx.Axpy(dx, -1.0+eta) dy.Scal(0.0) ry.Axpy(dy, -1.0+eta) dz.Scale(0.0) blas.AxpyFloat(rznl, dz, -1.0+eta) blas.AxpyFloat(rzl, dz, -1.0+eta, &la.IOpt{"offsety", mnl}) //fmt.Printf("dx=\n%v\n", dx) //fmt.Printf("dz=\n%v\n", dz.ToString("%.7f")) //fmt.Printf("ds=\n%v\n", ds.ToString("%.7f")) checkpnt.Check("pref4", minor) checkpnt.MinorPush(minor) err = f4(dx, dy, dz, ds) if err != nil { if iters == 0 { s := fmt.Sprintf("Rank(A) < p or Rank([H(x); A; Df(x); G] < n (%s)", err) err = errors.New(s) return } msg := "Terminated (singular KKT matrix)." if solopts.ShowProgress { fmt.Printf(msg + "\n") } zl := matrix.FloatVector(z.FloatArray()[mnl:]) sl := matrix.FloatVector(s.FloatArray()[mnl:]) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(sl, m, ind) symm(zl, m, ind) ind += m * m } ts, _ = maxStep(s, dims, mnl, nil) tz, _ = maxStep(z, dims, mnl, nil) err = errors.New(msg) sol.Status = Unknown sol.Result = sets.NewFloatSet("x", "y", "znl", "zl", "snl", "sl") sol.Result.Set("x", x.Matrix()) sol.Result.Set("y", y.Matrix()) sol.Result.Set("znl", matrix.FloatVector(z.FloatArray()[:mnl])) sol.Result.Set("zl", zl) sol.Result.Set("sl", sl) sol.Result.Set("snl", matrix.FloatVector(s.FloatArray()[:mnl])) sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz return } checkpnt.MinorPop() checkpnt.Check("postf4", minor+400) // Inner product ds'*dz and unscaled steps are needed in the // line search. dsdz = sdot(ds, dz, dims, mnl) blas.Copy(dz, dz2) scale(dz2, W, false, true) blas.Copy(ds, ds2) scale(ds2, W, true, false) checkpnt.Check("dsdz", minor+400) // Maximum steps to boundary. // // Also compute the eigenvalue decomposition of 's' blocks in // ds, dz. The eigenvectors Qs, Qz are stored in ds, dz. // The eigenvalues are stored in sigs, sigz. scale2(lmbda, ds, dims, mnl, false) ts, _ = maxStep(ds, dims, mnl, sigs) scale2(lmbda, dz, dims, mnl, false) tz, _ = maxStep(dz, dims, mnl, sigz) t := maxvec([]float64{0.0, ts, tz}) if t == 0 { step = 1.0 } else { step = math.Min(1.0, STEP/t) } checkpnt.Check("maxstep", minor+400) var newDf MatrixVarDf = nil var newf MatrixVariable = nil // Backtrack until newx is in domain of f. backtrack := true for backtrack { mCopy(x, newx) dx.Axpy(newx, step) newf, newDf, err = F.F1(newx) if newf != nil { backtrack = false } else { step *= BETA } } // Merit function // // phi = theta1 * gap + theta2 * norm(rx) + // theta3 * norm(rznl) // // and its directional derivative dphi. phi = theta1*gap + theta2*resx + theta3*resznl if i == 0 { dphi = -phi } else { dphi = -theta1*(1-sigma)*gap - theta2*(1-eta)*resx - theta3*(1-eta)*resznl } var newfDf func(x, y MatrixVariable, a, b float64, trans la.Option) error // Line search backtrack = true for backtrack { mCopy(x, newx) dx.Axpy(newx, step) mCopy(y, newy) dy.Axpy(newy, step) blas.Copy(z, newz) blas.AxpyFloat(dz2, newz, step) blas.Copy(s, news) blas.AxpyFloat(ds2, news, step) newf, newDf, err = F.F1(newx) newfDf = func(u, v MatrixVariable, a, b float64, trans la.Option) error { return newDf.Df(u, v, a, b, trans) } // newrx = c + A'*newy + newDf'*newz[:mnl] + G'*newz[mnl:] newz_mnl := matrix.FloatVector(newz.FloatArray()[:mnl]) newz_ml := matrix.FloatVector(newz.FloatArray()[mnl:]) //blas.Copy(c, newrx) //c.CopyTo(newrx) mCopy(c, newrx) fA(newy, newrx, 1.0, 1.0, la.OptTrans) newfDf(&matrixVar{newz_mnl}, newrx, 1.0, 1.0, la.OptTrans) fG(&matrixVar{newz_ml}, newrx, 1.0, 1.0, la.OptTrans) newresx = math.Sqrt(newrx.Dot(newrx)) // newrznl = news[:mnl] + newf news_mnl := matrix.FloatVector(news.FloatArray()[:mnl]) //news_ml := matrix.FloatVector(news.FloatArray()[mnl:]) blas.Copy(news_mnl, newrznl) blas.AxpyFloat(newf.Matrix(), newrznl, 1.0) newresznl = blas.Nrm2Float(newrznl) newgap = (1.0-(1.0-sigma)*step)*gap + step*step*dsdz newphi = theta1*newgap + theta2*newresx + theta3*newresznl if i == 0 { if newgap <= (1.0-ALPHA*step)*gap && (relaxed_iters > 0 && relaxed_iters < MAX_RELAXED_ITERS || newphi <= phi+ALPHA*step*dphi) { backtrack = false sigma = math.Min(newgap/gap, math.Pow((newgap/gap), EXPON)) //fmt.Printf("break 1: sigma=%.7f\n", sigma) eta = 0.0 } else { step *= BETA } } else { if relaxed_iters == -1 || (relaxed_iters == 0 && MAX_RELAXED_ITERS == 0) { // Do a standard line search. if newphi <= phi+ALPHA*step*dphi { relaxed_iters = 0 backtrack = false //fmt.Printf("break 2 : newphi=%.7f\n", newphi) } else { step *= BETA } } else if relaxed_iters == 0 && relaxed_iters < MAX_RELAXED_ITERS { if newphi <= phi+ALPHA*step*dphi { // Relaxed l.s. gives sufficient decrease. relaxed_iters = 0 } else { // Save state. phi0, dphi0, gap0 = phi, dphi, gap step0 = step blas.Copy(W.At("dnl")[0], W0.At("dnl")[0]) blas.Copy(W.At("dnli")[0], W0.At("dnli")[0]) blas.Copy(W.At("d")[0], W0.At("d")[0]) blas.Copy(W.At("di")[0], W0.At("di")[0]) blas.Copy(W.At("beta")[0], W0.At("beta")[0]) for k, _ := range dims.At("q") { blas.Copy(W.At("v")[k], W0.At("v")[k]) } for k, _ := range dims.At("s") { blas.Copy(W.At("r")[k], W0.At("r")[k]) blas.Copy(W.At("rti")[k], W0.At("rti")[k]) } mCopy(x, x0) mCopy(y, y0) mCopy(dx, dx0) mCopy(dy, dy0) blas.Copy(s, s0) blas.Copy(z, z0) blas.Copy(ds, ds0) blas.Copy(dz, dz0) blas.Copy(ds2, ds20) blas.Copy(dz2, dz20) blas.Copy(lmbda, lmbda0) blas.Copy(lmbdasq, lmbdasq0) // ??? mCopy(rx, rx0) mCopy(ry, ry0) blas.Copy(rznl, rznl0) blas.Copy(rzl, rzl0) dsdz0 = dsdz sigma0, eta0 = sigma, eta relaxed_iters = 1 } backtrack = false //fmt.Printf("break 3 : newphi=%.7f\n", newphi) } else if relaxed_iters >= 0 && relaxed_iters < MAX_RELAXED_ITERS && MAX_RELAXED_ITERS > 0 { if newphi <= phi0+ALPHA*step0*dphi0 { // Relaxed l.s. gives sufficient decrease. relaxed_iters = 0 } else { // Relaxed line search relaxed_iters += 1 } backtrack = false //fmt.Printf("break 4 : newphi=%.7f\n", newphi) } else if relaxed_iters == MAX_RELAXED_ITERS && MAX_RELAXED_ITERS > 0 { if newphi <= phi0+ALPHA*step0*dphi0 { // Series of relaxed line searches ends // with sufficient decrease w.r.t. phi0. backtrack = false relaxed_iters = 0 //fmt.Printf("break 5 : newphi=%.7f\n", newphi) } else if newphi >= phi0 { // Resume last saved line search phi, dphi, gap = phi0, dphi0, gap0 step = step0 blas.Copy(W0.At("dnl")[0], W.At("dnl")[0]) blas.Copy(W0.At("dnli")[0], W.At("dnli")[0]) blas.Copy(W0.At("d")[0], W.At("d")[0]) blas.Copy(W0.At("di")[0], W.At("di")[0]) blas.Copy(W0.At("beta")[0], W.At("beta")[0]) for k, _ := range dims.At("q") { blas.Copy(W0.At("v")[k], W.At("v")[k]) } for k, _ := range dims.At("s") { blas.Copy(W0.At("r")[k], W.At("r")[k]) blas.Copy(W0.At("rti")[k], W.At("rti")[k]) } mCopy(x, x0) mCopy(y, y0) mCopy(dx, dx0) mCopy(dy, dy0) blas.Copy(s, s0) blas.Copy(z, z0) blas.Copy(ds2, ds20) blas.Copy(dz2, dz20) blas.Copy(lmbda, lmbda0) blas.Copy(lmbdasq, lmbdasq0) // ??? mCopy(rx, rx0) mCopy(ry, ry0) blas.Copy(rznl, rznl0) blas.Copy(rzl, rzl0) dsdz = dsdz0 sigma, eta = sigma0, eta0 relaxed_iters = -1 } else if newphi <= phi+ALPHA*step*dphi { // Series of relaxed line searches ends // with sufficient decrease w.r.t. phi0. backtrack = false relaxed_iters = -1 //fmt.Printf("break 6 : newphi=%.7f\n", newphi) } } } } // end of line search checkpnt.Check("eol", minor+900) } // end for [0,1] // Update x, y dx.Axpy(x, step) dy.Axpy(y, step) checkpnt.Check("updatexy", 5000) // Replace nonlinear, 'l' and 'q' blocks of ds and dz with the // updated variables in the current scaling. // Replace 's' blocks of ds and dz with the factors Ls, Lz in a // factorization Ls*Ls', Lz*Lz' of the updated variables in the // current scaling. // ds := e + step*ds for nonlinear, 'l' and 'q' blocks. // dz := e + step*dz for nonlinear, 'l' and 'q' blocks. blas.ScalFloat(ds, step, &la.IOpt{"n", mnl + dims.Sum("l", "q")}) blas.ScalFloat(dz, step, &la.IOpt{"n", mnl + dims.Sum("l", "q")}) ind := mnl + dims.At("l")[0] is := matrix.MakeIndexSet(0, ind, 1) ds.Add(1.0, is...) dz.Add(1.0, is...) for _, m := range dims.At("q") { ds.SetIndex(ind, 1.0+ds.GetIndex(ind)) dz.SetIndex(ind, 1.0+dz.GetIndex(ind)) ind += m } checkpnt.Check("updatedsdz", 5100) // ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz. // // This replaces the 'l' and 'q' components of ds and dz with the // updated variables in the current scaling. // The 's' components of ds and dz are replaced with // // diag(lmbda_k)^{1/2} * Qs * diag(lmbda_k)^{1/2} // diag(lmbda_k)^{1/2} * Qz * diag(lmbda_k)^{1/2} scale2(lmbda, ds, dims, mnl, true) scale2(lmbda, dz, dims, mnl, true) checkpnt.Check("scale2", 5200) // sigs := ( e + step*sigs ) ./ lambda for 's' blocks. // sigz := ( e + step*sigz ) ./ lambda for 's' blocks. blas.ScalFloat(sigs, step) blas.ScalFloat(sigz, step) sigs.Add(1.0) sigz.Add(1.0) sdimsum := dims.Sum("s") qdimsum := dims.Sum("l", "q") blas.TbsvFloat(lmbda, sigs, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0}, &la.IOpt{"lda", 1}, &la.IOpt{"offseta", mnl + qdimsum}) blas.TbsvFloat(lmbda, sigz, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0}, &la.IOpt{"lda", 1}, &la.IOpt{"offseta", mnl + qdimsum}) checkpnt.Check("sigs", 5300) ind2 := mnl + qdimsum ind3 := 0 sdims := dims.At("s") for k := 0; k < len(sdims); k++ { m := sdims[k] for i := 0; i < m; i++ { a := math.Sqrt(sigs.GetIndex(ind3 + i)) blas.ScalFloat(ds, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m}) a = math.Sqrt(sigz.GetIndex(ind3 + i)) blas.ScalFloat(dz, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m}) } ind2 += m * m ind3 += m } checkpnt.Check("scaling", 5400) err = updateScaling(W, lmbda, ds, dz) checkpnt.Check("postscaling", 5500) // Unscale s, z, tau, kappa (unscaled variables are used only to // compute feasibility residuals). ind = mnl + dims.Sum("l", "q") ind2 = ind blas.Copy(lmbda, s, &la.IOpt{"n", ind}) for _, m := range dims.At("s") { blas.ScalFloat(s, 0.0, &la.IOpt{"offset", ind2}) blas.Copy(lmbda, s, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2}, &la.IOpt{"n", m}, &la.IOpt{"incy", m + 1}) ind += m ind2 += m * m } scale(s, W, true, false) checkpnt.Check("unscale_s", 5600) ind = mnl + dims.Sum("l", "q") ind2 = ind blas.Copy(lmbda, z, &la.IOpt{"n", ind}) for _, m := range dims.At("s") { blas.ScalFloat(z, 0.0, &la.IOpt{"offset", ind2}) blas.Copy(lmbda, z, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2}, &la.IOpt{"n", m}, &la.IOpt{"incy", m + 1}) ind += m ind2 += m * m } scale(z, W, false, true) checkpnt.Check("unscale_z", 5700) gap = blas.DotFloat(lmbda, lmbda) } return }
func 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) }
/* Returns the Nesterov-Todd scaling W at points s and z, and stores the scaled variable in lmbda. W * z = W^{-T} * s = lmbda. W is a MatrixSet with entries: - W['dnl']: positive vector - W['dnli']: componentwise inverse of W['dnl'] - W['d']: positive vector - W['di']: componentwise inverse of W['d'] - W['v']: lists of 2nd order cone vectors with unit hyperbolic norms - W['beta']: list of positive numbers - W['r']: list of square matrices - W['rti']: list of square matrices. rti[k] is the inverse transpose of r[k]. */ func computeScaling(s, z, lmbda *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int) (W *sets.FloatMatrixSet, err error) { /*DEBUGGED*/ err = nil W = sets.NewFloatSet("dnl", "dnli", "d", "di", "v", "beta", "r", "rti") // For the nonlinear block: // // W['dnl'] = sqrt( s[:mnl] ./ z[:mnl] ) // W['dnli'] = sqrt( z[:mnl] ./ s[:mnl] ) // lambda[:mnl] = sqrt( s[:mnl] .* z[:mnl] ) var stmp, ztmp, lmd *matrix.FloatMatrix if mnl > 0 { stmp = matrix.FloatVector(s.FloatArray()[:mnl]) ztmp = matrix.FloatVector(z.FloatArray()[:mnl]) //dnl := stmp.Div(ztmp) //dnl.Apply(dnl, math.Sqrt) dnl := matrix.Sqrt(matrix.Div(stmp, ztmp)) //dnli := dnl.Copy() //dnli.Apply(dnli, func(a float64)float64 { return 1.0/a }) dnli := matrix.Inv(dnl) W.Set("dnl", dnl) W.Set("dnli", dnli) //lmd = stmp.Mul(ztmp) //lmd.Apply(lmd, math.Sqrt) lmd = matrix.Sqrt(matrix.Mul(stmp, ztmp)) lmbda.SetIndexesFromArray(lmd.FloatArray(), matrix.MakeIndexSet(0, mnl, 1)...) } else { // set for empty matrices //W.Set("dnl", matrix.FloatZeros(0, 1)) //W.Set("dnli", matrix.FloatZeros(0, 1)) mnl = 0 } // For the 'l' block: // // W['d'] = sqrt( sk ./ zk ) // W['di'] = sqrt( zk ./ sk ) // lambdak = sqrt( sk .* zk ) // // where sk and zk are the first dims['l'] entries of s and z. // lambda_k is stored in the first dims['l'] positions of lmbda. m := dims.At("l")[0] //td := s.FloatArray() stmp = matrix.FloatVector(s.FloatArray()[mnl : mnl+m]) //zd := z.FloatArray() ztmp = matrix.FloatVector(z.FloatArray()[mnl : mnl+m]) //fmt.Printf(".Sqrt()=\n%v\n", matrix.Div(stmp, ztmp).Sqrt().ToString("%.17f")) //d := stmp.Div(ztmp) //d.Apply(d, math.Sqrt) d := matrix.Div(stmp, ztmp).Sqrt() //di := d.Copy() //di.Apply(di, func(a float64)float64 { return 1.0/a }) di := matrix.Inv(d) //fmt.Printf("d:\n%v\n", d) //fmt.Printf("di:\n%v\n", di) W.Set("d", d) W.Set("di", di) //lmd = stmp.Mul(ztmp) //lmd.Apply(lmd, math.Sqrt) lmd = matrix.Mul(stmp, ztmp).Sqrt() // lmd has indexes mnl:mnl+m and length of m lmbda.SetIndexesFromArray(lmd.FloatArray(), matrix.MakeIndexSet(mnl, mnl+m, 1)...) //fmt.Printf("after l:\n%v\n", lmbda) /* For the 'q' blocks, compute lists 'v', 'beta'. The vector v[k] has unit hyperbolic norm: (sqrt( v[k]' * J * v[k] ) = 1 with J = [1, 0; 0, -I]). beta[k] is a positive scalar. The hyperbolic Householder matrix H = 2*v[k]*v[k]' - J defined by v[k] satisfies (beta[k] * H) * zk = (beta[k] * H) \ sk = lambda_k where sk = s[indq[k]:indq[k+1]], zk = z[indq[k]:indq[k+1]]. lambda_k is stored in lmbda[indq[k]:indq[k+1]]. */ ind := mnl + dims.At("l")[0] var beta *matrix.FloatMatrix for _, k := range dims.At("q") { W.Append("v", matrix.FloatZeros(k, 1)) } beta = matrix.FloatZeros(len(dims.At("q")), 1) W.Set("beta", beta) vset := W.At("v") for k, m := range dims.At("q") { v := vset[k] // a = sqrt( sk' * J * sk ) where J = [1, 0; 0, -I] aa := jnrm2(s, m, ind) // b = sqrt( zk' * J * zk ) bb := jnrm2(z, m, ind) // beta[k] = ( a / b )**1/2 beta.SetIndex(k, math.Sqrt(aa/bb)) // c = sqrt( (sk/a)' * (zk/b) + 1 ) / sqrt(2) c0 := blas.DotFloat(s, z, &la_.IOpt{"n", m}, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind}) cc := math.Sqrt((c0/aa/bb + 1.0) / 2.0) // vk = 1/(2*c) * ( (sk/a) + J * (zk/b) ) blas.CopyFloat(z, v, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m}) blas.ScalFloat(v, -1.0/bb) v.SetIndex(0, -1.0*v.GetIndex(0)) blas.AxpyFloat(s, v, 1.0/aa, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m}) blas.ScalFloat(v, 1.0/2.0/cc) // v[k] = 1/sqrt(2*(vk0 + 1)) * ( vk + e ), e = [1; 0] v.SetIndex(0, v.GetIndex(0)+1.0) blas.ScalFloat(v, (1.0 / math.Sqrt(2.0*v.GetIndex(0)))) /* To get the scaled variable lambda_k d = sk0/a + zk0/b + 2*c lambda_k = [ c; (c + zk0/b)/d * sk1/a + (c + sk0/a)/d * zk1/b ] lambda_k *= sqrt(a * b) */ lmbda.SetIndex(ind, cc) dd := 2*cc + s.GetIndex(ind)/aa + z.GetIndex(ind)/bb blas.CopyFloat(s, lmbda, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) zz := (cc + z.GetIndex(ind)/bb) / dd / aa ss := (cc + s.GetIndex(ind)/aa) / dd / bb blas.ScalFloat(lmbda, zz, &la_.IOpt{"offset", ind + 1}, &la_.IOpt{"n", m - 1}) blas.AxpyFloat(z, lmbda, ss, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) blas.ScalFloat(lmbda, math.Sqrt(aa*bb), &la_.IOpt{"offset", ind}, &la_.IOpt{"n", m}) ind += m //fmt.Printf("after q[%d]:\n%v\n", k, lmbda) } /* For the 's' blocks: compute two lists 'r' and 'rti'. r[k]' * sk^{-1} * r[k] = diag(lambda_k)^{-1} r[k]' * zk * r[k] = diag(lambda_k) where sk and zk are the entries inds[k] : inds[k+1] of s and z, reshaped into symmetric matrices. rti[k] is the inverse of r[k]', so rti[k]' * sk * rti[k] = diag(lambda_k)^{-1} rti[k]' * zk^{-1} * rti[k] = diag(lambda_k). The vectors lambda_k are stored in lmbda[ dims['l'] + sum(dims['q']) : -1 ] */ for _, k := range dims.At("s") { W.Append("r", matrix.FloatZeros(k, k)) W.Append("rti", matrix.FloatZeros(k, k)) } maxs := maxdim(dims.At("s")) work := matrix.FloatZeros(maxs*maxs, 1) Ls := matrix.FloatZeros(maxs*maxs, 1) Lz := matrix.FloatZeros(maxs*maxs, 1) ind2 := ind for k, m := range dims.At("s") { r := W.At("r")[k] rti := W.At("rti")[k] // Factor sk = Ls*Ls'; store Ls in ds[inds[k]:inds[k+1]]. blas.CopyFloat(s, Ls, &la_.IOpt{"offsetx", ind2}, &la_.IOpt{"n", m * m}) lapack.PotrfFloat(Ls, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m}) // Factor zs[k] = Lz*Lz'; store Lz in dz[inds[k]:inds[k+1]]. blas.CopyFloat(z, Lz, &la_.IOpt{"offsetx", ind2}, &la_.IOpt{"n", m * m}) lapack.PotrfFloat(Lz, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m}) // SVD Lz'*Ls = U*diag(lambda_k)*V'. Keep U in work. for i := 0; i < m; i++ { blas.ScalFloat(Ls, 0.0, &la_.IOpt{"offset", i * m}, &la_.IOpt{"n", i}) } blas.CopyFloat(Ls, work, &la_.IOpt{"n", m * m}) blas.TrmmFloat(Lz, work, 1.0, la_.OptTransA, &la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m}) lapack.GesvdFloat(work, lmbda, nil, nil, la_.OptJobuO, &la_.IOpt{"lda", m}, &la_.IOpt{"offsetS", ind}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m}) // r = Lz^{-T} * U blas.CopyFloat(work, r, &la_.IOpt{"n", m * m}) blas.TrsmFloat(Lz, r, 1.0, la_.OptTransA, &la_.IOpt{"lda", m}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m}) // rti = Lz * U blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m}) blas.TrmmFloat(Lz, rti, 1.0, &la_.IOpt{"lda", m}, &la_.IOpt{"n", m}, &la_.IOpt{"m", m}) // r := r * diag(sqrt(lambda_k)) // rti := rti * diag(1 ./ sqrt(lambda_k)) for i := 0; i < m; i++ { a := math.Sqrt(lmbda.GetIndex(ind + i)) blas.ScalFloat(r, a, &la_.IOpt{"offset", m * i}, &la_.IOpt{"n", m}) blas.ScalFloat(rti, 1.0/a, &la_.IOpt{"offset", m * i}, &la_.IOpt{"n", m}) } ind += m ind2 += m * m } return }
func updateScaling(W *sets.FloatMatrixSet, lmbda, s, z *matrix.FloatMatrix) (err error) { err = nil var stmp, ztmp *matrix.FloatMatrix /* Nonlinear and 'l' blocks d := d .* sqrt( s ./ z ) lmbda := lmbda .* sqrt(s) .* sqrt(z) */ mnl := 0 dnlset := W.At("dnl") dnliset := W.At("dnli") dset := W.At("d") diset := W.At("di") beta := W.At("beta")[0] if dnlset != nil && dnlset[0].NumElements() > 0 { mnl = dnlset[0].NumElements() } ml := dset[0].NumElements() m := mnl + ml //fmt.Printf("ml=%d, mnl=%d, m=%d'n", ml, mnl, m) stmp = matrix.FloatVector(s.FloatArray()[:m]) stmp.Apply(math.Sqrt) s.SetIndexesFromArray(stmp.FloatArray(), matrix.MakeIndexSet(0, m, 1)...) ztmp = matrix.FloatVector(z.FloatArray()[:m]) ztmp.Apply(math.Sqrt) z.SetIndexesFromArray(ztmp.FloatArray(), matrix.MakeIndexSet(0, m, 1)...) // d := d .* s .* z if len(dnlset) > 0 { blas.TbmvFloat(s, dnlset[0], &la_.IOpt{"n", mnl}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}) blas.TbsvFloat(z, dnlset[0], &la_.IOpt{"n", mnl}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}) //dnliset[0].Apply(dnlset[0], func(a float64)float64 { return 1.0/a}) //--dnliset[0] = matrix.Inv(dnlset[0]) matrix.Set(dnliset[0], dnlset[0]) dnliset[0].Inv() } blas.TbmvFloat(s, dset[0], &la_.IOpt{"n", ml}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}, &la_.IOpt{"offseta", mnl}) blas.TbsvFloat(z, dset[0], &la_.IOpt{"n", ml}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}, &la_.IOpt{"offseta", mnl}) //diset[0].Apply(dset[0], func(a float64)float64 { return 1.0/a}) //--diset[0] = matrix.Inv(dset[0]) matrix.Set(diset[0], dset[0]) diset[0].Inv() // lmbda := s .* z blas.CopyFloat(s, lmbda, &la_.IOpt{"n", m}) blas.TbmvFloat(z, lmbda, &la_.IOpt{"n", m}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}) // 'q' blocks. // Let st and zt be the new variables in the old scaling: // // st = s_k, zt = z_k // // and a = sqrt(st' * J * st), b = sqrt(zt' * J * zt). // // 1. Compute the hyperbolic Householder transformation 2*q*q' - J // that maps st/a to zt/b. // // c = sqrt( (1 + st'*zt/(a*b)) / 2 ) // q = (st/a + J*zt/b) / (2*c). // // The new scaling point is // // wk := betak * sqrt(a/b) * (2*v[k]*v[k]' - J) * q // // with betak = W['beta'][k]. // // 3. The scaled variable: // // lambda_k0 = sqrt(a*b) * c // lambda_k1 = sqrt(a*b) * ( (2vk*vk' - J) * (-d*q + u/2) )_1 // // where // // u = st/a - J*zt/b // d = ( vk0 * (vk'*u) + u0/2 ) / (2*vk0 *(vk'*q) - q0 + 1). // // 4. Update scaling // // v[k] := wk^1/2 // = 1 / sqrt(2*(wk0 + 1)) * (wk + e). // beta[k] *= sqrt(a/b) ind := m for k, v := range W.At("v") { m = v.NumElements() // ln = sqrt( lambda_k' * J * lambda_k ) !! NOT USED!! jnrm2(lmbda, m, ind) // ?? NOT USED ?? // a = sqrt( sk' * J * sk ) = sqrt( st' * J * st ) // s := s / a = st / a aa := jnrm2(s, m, ind) blas.ScalFloat(s, 1.0/aa, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind}) // b = sqrt( zk' * J * zk ) = sqrt( zt' * J * zt ) // z := z / a = zt / b bb := jnrm2(z, m, ind) blas.ScalFloat(z, 1.0/bb, &la_.IOpt{"n", m}, &la_.IOpt{"offset", ind}) // c = sqrt( ( 1 + (st'*zt) / (a*b) ) / 2 ) cc := blas.DotFloat(s, z, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind}, &la_.IOpt{"n", m}) cc = math.Sqrt((1.0 + cc) / 2.0) // vs = v' * st / a vs := blas.DotFloat(v, s, &la_.IOpt{"offsety", ind}, &la_.IOpt{"n", m}) // vz = v' * J *zt / b vz := jdot(v, z, m, 0, ind) // vq = v' * q where q = (st/a + J * zt/b) / (2 * c) vq := (vs + vz) / 2.0 / cc // vq = v' * q where q = (st/a + J * zt/b) / (2 * c) vu := vs - vz // lambda_k0 = c lmbda.SetIndex(ind, cc) // wk0 = 2 * vk0 * (vk' * q) - q0 wk0 := 2.0*v.GetIndex(0)*vq - (s.GetIndex(ind)+z.GetIndex(ind))/2.0/cc // d = (v[0] * (vk' * u) - u0/2) / (wk0 + 1) dd := (v.GetIndex(0)*vu - s.GetIndex(ind)/2.0 + z.GetIndex(ind)/2.0) / (wk0 + 1.0) // lambda_k1 = 2 * v_k1 * vk' * (-d*q + u/2) - d*q1 + u1/2 blas.CopyFloat(v, lmbda, &la_.IOpt{"offsetx", 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) blas.ScalFloat(lmbda, (2.0 * (-dd*vq + 0.5*vu)), &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) blas.AxpyFloat(s, lmbda, 0.5*(1.0-dd/cc), &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) blas.AxpyFloat(z, lmbda, 0.5*(1.0+dd/cc), &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) // Scale so that sqrt(lambda_k' * J * lambda_k) = sqrt(aa*bb). blas.ScalFloat(lmbda, math.Sqrt(aa*bb), &la_.IOpt{"offset", ind}, &la_.IOpt{"n", m}) // v := (2*v*v' - J) * q // = 2 * (v'*q) * v' - (J* st/a + zt/b) / (2*c) blas.ScalFloat(v, 2.0*vq) v.SetIndex(0, v.GetIndex(0)-(s.GetIndex(ind)/2.0/cc)) blas.AxpyFloat(s, v, 0.5/cc, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", 1}, &la_.IOpt{"n", m - 1}) blas.AxpyFloat(z, v, -0.5/cc, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m}) // v := v^{1/2} = 1/sqrt(2 * (v0 + 1)) * (v + e) v0 := v.GetIndex(0) + 1.0 v.SetIndex(0, v0) blas.ScalFloat(v, 1.0/math.Sqrt(2.0*v0)) // beta[k] *= ( aa / bb )**1/2 bk := beta.GetIndex(k) beta.SetIndex(k, bk*math.Sqrt(aa/bb)) ind += m } //fmt.Printf("-- end of q:\nz=\n%v\nlmbda=\n%v\n", z.ConvertToString(), lmbda.ConvertToString()) //fmt.Printf("beta=\n%v\n", beta.ConvertToString()) // 's' blocks // // Let st, zt be the updated variables in the old scaling: // // st = Ls * Ls', zt = Lz * Lz'. // // where Ls and Lz are the 's' components of s, z. // // 1. SVD Lz'*Ls = Uk * lambda_k^+ * Vk'. // // 2. New scaling is // // r[k] := r[k] * Ls * Vk * diag(lambda_k^+)^{-1/2} // rti[k] := r[k] * Lz * Uk * diag(lambda_k^+)^{-1/2}. // maxr := 0 for _, m := range W.At("r") { if m.Rows() > maxr { maxr = m.Rows() } } work := matrix.FloatZeros(maxr*maxr, 1) vlensum := 0 for _, m := range W.At("v") { vlensum += m.NumElements() } ind = mnl + ml + vlensum ind2 := ind ind3 := 0 rset := W.At("r") rtiset := W.At("rti") for k, _ := range rset { r := rset[k] rti := rtiset[k] m = r.Rows() //fmt.Printf("m=%d, r=\n%v\nrti=\n%v\n", m, r.ConvertToString(), rti.ConvertToString()) // r := r*sk = r*Ls blas.GemmFloat(r, s, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offsetb", ind2}) //fmt.Printf("1 work=\n%v\n", work.ConvertToString()) blas.CopyFloat(work, r, &la_.IOpt{"n", m * m}) // rti := rti*zk = rti*Lz blas.GemmFloat(rti, z, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offsetb", ind2}) //fmt.Printf("2 work=\n%v\n", work.ConvertToString()) blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m}) // SVD Lz'*Ls = U * lmbds^+ * V'; store U in sk and V' in zk. ' blas.GemmFloat(z, s, work, 1.0, 0.0, la_.OptTransA, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offseta", ind2}, &la_.IOpt{"offsetb", ind2}) //fmt.Printf("3 work=\n%v\n", work.ConvertToString()) // U = s, Vt = z lapack.GesvdFloat(work, lmbda, s, z, la_.OptJobuAll, la_.OptJobvtAll, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"lda", m}, &la_.IOpt{"ldu", m}, &la_.IOpt{"ldvt", m}, &la_.IOpt{"offsets", ind}, &la_.IOpt{"offsetu", ind2}, &la_.IOpt{"offsetvt", ind2}) // r := r*V blas.GemmFloat(r, z, work, 1.0, 0.0, la_.OptTransB, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offsetb", ind2}) //fmt.Printf("4 work=\n%v\n", work.ConvertToString()) blas.CopyFloat(work, r, &la_.IOpt{"n", m * m}) // rti := rti*U blas.GemmFloat(rti, s, work, 1.0, 0.0, &la_.IOpt{"m", m}, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offsetb", ind2}) //fmt.Printf("5 work=\n%v\n", work.ConvertToString()) blas.CopyFloat(work, rti, &la_.IOpt{"n", m * m}) for i := 0; i < m; i++ { a := 1.0 / math.Sqrt(lmbda.GetIndex(ind+i)) blas.ScalFloat(r, a, &la_.IOpt{"n", m}, &la_.IOpt{"offset", m * i}) blas.ScalFloat(rti, a, &la_.IOpt{"n", m}, &la_.IOpt{"offset", m * i}) } ind += m ind2 += m * m ind3 += m // !!NOT USED: ind3!! } //fmt.Printf("-- end of s:\nz=\n%v\nlmbda=\n%v\n", z.ConvertToString(), lmbda.ConvertToString()) return }
// The product x := (y o x). If diag is 'D', the 's' part of y is // diagonal and only the diagonal is stored. func sprod(x, y *matrix.FloatMatrix, dims *sets.DimensionSet, mnl int, opts ...la_.Option) (err error) { err = nil diag := la_.GetStringOpt("diag", "N", opts...) // For the nonlinear and 'l' blocks: // // yk o xk = yk .* xk. ind := mnl + dims.At("l")[0] err = blas.Tbmv(y, x, &la_.IOpt{"n", ind}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}) if err != nil { return } //fmt.Printf("Sprod l:x=\n%v\n", x) // For 'q' blocks: // // [ l0 l1' ] // yk o xk = [ ] * xk // [ l1 l0*I ] // // where yk = (l0, l1). for _, m := range dims.At("q") { dd := blas.DotFloat(x, y, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"offsety", ind}, &la_.IOpt{"n", m}) //fmt.Printf("dd=%v\n", dd) alpha := y.GetIndex(ind) //fmt.Printf("scal=%v\n", alpha) blas.ScalFloat(x, alpha, &la_.IOpt{"offset", ind + 1}, &la_.IOpt{"n", m - 1}) alpha = x.GetIndex(ind) //fmt.Printf("axpy=%v\n", alpha) blas.AxpyFloat(y, x, alpha, &la_.IOpt{"offsetx", ind + 1}, &la_.IOpt{"offsety", ind + 1}, &la_.IOpt{"n", m - 1}) x.SetIndex(ind, dd) ind += m } //fmt.Printf("Sprod q :x=\n%v\n", x) // For the 's' blocks: // // yk o sk = .5 * ( Yk * mat(xk) + mat(xk) * Yk ) // // where Yk = mat(yk) if diag is 'N' and Yk = diag(yk) if diag is 'D'. if diag[0] == 'N' { // DEBUGGED maxm := maxdim(dims.At("s")) A := matrix.FloatZeros(maxm, maxm) for _, m := range dims.At("s") { blas.Copy(x, A, &la_.IOpt{"offsetx", ind}, &la_.IOpt{"n", m * m}) for i := 0; i < m-1; i++ { // i < m-1 --> i < m symm(A, m, 0) symm(y, m, ind) } err = blas.Syr2kFloat(A, y, x, 0.5, 0.0, &la_.IOpt{"n", m}, &la_.IOpt{"k", m}, &la_.IOpt{"lda", m}, &la_.IOpt{"ldb", m}, &la_.IOpt{"ldc", m}, &la_.IOpt{"offsetb", ind}, &la_.IOpt{"offsetc", ind}) if err != nil { return } ind += m * m } //fmt.Printf("Sprod diag=N s:x=\n%v\n", x) } else { ind2 := ind for _, m := range dims.At("s") { for i := 0; i < m; i++ { // original: u = 0.5 * ( y[ind2+i:ind2+m] + y[ind2+i] ) // creates matrix of elements: [ind2+i ... ind2+m] then // element wisely adds y[ind2+i] and scales by 0.5 iset := matrix.MakeIndexSet(ind2+i, ind2+m, 1) u := matrix.FloatVector(y.GetIndexes(iset...)) u.Add(y.GetIndex(ind2 + i)) u.Scale(0.5) err = blas.Tbmv(u, x, &la_.IOpt{"n", m - i}, &la_.IOpt{"k", 0}, &la_.IOpt{"lda", 1}, &la_.IOpt{"offsetx", ind + i*(m+1)}) if err != nil { return } } ind += m * m ind2 += m } //fmt.Printf("Sprod diag=T s:x=\n%v\n", x) } return }
func conelp_solver(c MatrixVariable, G MatrixVarG, h *matrix.FloatMatrix, A MatrixVarA, b MatrixVariable, dims *sets.DimensionSet, kktsolver KKTConeSolverVar, solopts *SolverOptions, primalstart, dualstart *sets.FloatMatrixSet) (sol *Solution, err error) { err = nil const EXPON = 3 const STEP = 0.99 sol = &Solution{Unknown, nil, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0} var refinement int if solopts.Refinement > 0 { refinement = solopts.Refinement } else { refinement = 0 if len(dims.At("q")) > 0 || len(dims.At("s")) > 0 { refinement = 1 } } feasTolerance := FEASTOL absTolerance := ABSTOL relTolerance := RELTOL maxIter := MAXITERS if solopts.FeasTol > 0.0 { feasTolerance = solopts.FeasTol } if solopts.AbsTol > 0.0 { absTolerance = solopts.AbsTol } if solopts.RelTol > 0.0 { relTolerance = solopts.RelTol } if solopts.MaxIter > 0 { maxIter = solopts.MaxIter } if err = checkConeLpDimensions(dims); err != nil { return } cdim := dims.Sum("l", "q") + dims.SumSquared("s") //cdim_pckd := dims.Sum("l", "q") + dims.SumPacked("s") cdim_diag := dims.Sum("l", "q", "s") if h.Rows() != cdim { err = errors.New(fmt.Sprintf("'h' must be float matrix of size (%d,1)", cdim)) return } // Data for kth 'q' constraint are found in rows indq[k]:indq[k+1] of G. indq := make([]int, 0) indq = append(indq, dims.At("l")[0]) for _, k := range dims.At("q") { indq = append(indq, indq[len(indq)-1]+k) } // Data for kth 's' constraint are found in rows inds[k]:inds[k+1] of G. inds := make([]int, 0) inds = append(inds, indq[len(indq)-1]) for _, k := range dims.At("s") { inds = append(inds, inds[len(inds)-1]+k*k) } Gf := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error { return G.Gf(x, y, alpha, beta, trans) } Af := func(x, y MatrixVariable, alpha, beta float64, trans la.Option) error { return A.Af(x, y, alpha, beta, trans) } // kktsolver(W) returns a routine for solving 3x3 block KKT system // // [ 0 A' G'*W^{-1} ] [ ux ] [ bx ] // [ A 0 0 ] [ uy ] = [ by ]. // [ G 0 -W' ] [ uz ] [ bz ] if kktsolver == nil { err = errors.New("nil kktsolver not allowed.") return } // res() evaluates residual in 5x5 block KKT system // // [ vx ] [ 0 ] [ 0 A' G' c ] [ ux ] // [ vy ] [ 0 ] [-A 0 0 b ] [ uy ] // [ vz ] += [ W'*us ] - [-G 0 0 h ] [ W^{-1}*uz ] // [ vtau ] [ dg*ukappa ] [-c' -b' -h' 0 ] [ utau/dg ] // // vs += lmbda o (dz + ds) // vkappa += lmbdg * (dtau + dkappa). ws3 := matrix.FloatZeros(cdim, 1) wz3 := matrix.FloatZeros(cdim, 1) checkpnt.AddMatrixVar("ws3", ws3) checkpnt.AddMatrixVar("wz3", wz3) // res := func(ux, uy MatrixVariable, uz, utau, us, ukappa *matrix.FloatMatrix, vx, vy MatrixVariable, vz, vtau, vs, vkappa *matrix.FloatMatrix, W *sets.FloatMatrixSet, dg float64, lmbda *matrix.FloatMatrix) (err error) { err = nil // vx := vx - A'*uy - G'*W^{-1}*uz - c*utau/dg Af(uy, vx, -1.0, 1.0, la.OptTrans) //fmt.Printf("post-Af vx=\n%v\n", vx) blas.Copy(uz, wz3) scale(wz3, W, false, true) Gf(&matrixVar{wz3}, vx, -1.0, 1.0, la.OptTrans) //blas.AxpyFloat(c, vx, -utau.Float()/dg) c.Axpy(vx, -utau.Float()/dg) // vy := vy + A*ux - b*utau/dg Af(ux, vy, 1.0, 1.0, la.OptNoTrans) //blas.AxpyFloat(b, vy, -utau.Float()/dg) b.Axpy(vy, -utau.Float()/dg) // vz := vz + G*ux - h*utau/dg + W'*us Gf(ux, &matrixVar{vz}, 1.0, 1.0, la.OptNoTrans) blas.AxpyFloat(h, vz, -utau.Float()/dg) blas.Copy(us, ws3) scale(ws3, W, true, false) blas.AxpyFloat(ws3, vz, 1.0) // vtau := vtau + c'*ux + b'*uy + h'*W^{-1}*uz + dg*ukappa var vtauplus float64 = dg*ukappa.Float() + c.Dot(ux) + b.Dot(uy) + sdot(h, wz3, dims, 0) vtau.SetValue(vtau.Float() + vtauplus) // vs := vs + lmbda o (uz + us) blas.Copy(us, ws3) blas.AxpyFloat(uz, ws3, 1.0) sprod(ws3, lmbda, dims, 0, &la.SOpt{"diag", "D"}) blas.AxpyFloat(ws3, vs, 1.0) // vkappa += vkappa + lmbdag * (utau + ukappa) lscale := lmbda.GetIndex(-1) var vkplus float64 = lscale * (utau.Float() + ukappa.Float()) vkappa.SetValue(vkappa.Float() + vkplus) return } resx0 := math.Max(1.0, math.Sqrt(c.Dot(c))) resy0 := math.Max(1.0, math.Sqrt(b.Dot(b))) resz0 := math.Max(1.0, snrm2(h, dims, 0)) // select initial points //fmt.Printf("** initial resx0=%.4f, resy0=%.4f, resz0=%.4f \n", resx0, resy0, resz0) x := c.Copy() //blas.ScalFloat(x, 0.0) x.Scal(0.0) y := b.Copy() //blas.ScalFloat(y, 0.0) y.Scal(0.0) s := matrix.FloatZeros(cdim, 1) z := matrix.FloatZeros(cdim, 1) dx := c.Copy() dy := b.Copy() ds := matrix.FloatZeros(cdim, 1) dz := matrix.FloatZeros(cdim, 1) // these are singleton matrix dkappa := matrix.FloatValue(0.0) dtau := matrix.FloatValue(0.0) checkpnt.AddVerifiable("x", x) checkpnt.AddMatrixVar("s", s) checkpnt.AddMatrixVar("z", z) checkpnt.AddVerifiable("dx", dx) checkpnt.AddMatrixVar("ds", ds) checkpnt.AddMatrixVar("dz", dz) checkpnt.Check("00init", 1) var W *sets.FloatMatrixSet var f KKTFuncVar if primalstart == nil || dualstart == nil { // Factor // // [ 0 A' G' ] // [ A 0 0 ]. // [ G 0 -I ] // W = sets.NewFloatSet("d", "di", "v", "beta", "r", "rti") dd := dims.At("l")[0] mat := matrix.FloatOnes(dd, 1) W.Set("d", mat) mat = matrix.FloatOnes(dd, 1) W.Set("di", mat) dq := len(dims.At("q")) W.Set("beta", matrix.FloatOnes(dq, 1)) for _, n := range dims.At("q") { vm := matrix.FloatZeros(n, 1) vm.SetIndex(0, 1.0) W.Append("v", vm) } for _, n := range dims.At("s") { W.Append("r", matrix.FloatIdentity(n)) W.Append("rti", matrix.FloatIdentity(n)) } f, err = kktsolver(W) if err != nil { fmt.Printf("kktsolver error: %s\n", err) return } checkpnt.AddScaleVar(W) } checkpnt.Check("05init", 5) if primalstart == nil { // minimize || G * x - h ||^2 // subject to A * x = b // // by solving // // [ 0 A' G' ] [ x ] [ 0 ] // [ A 0 0 ] * [ dy ] = [ b ]. // [ G 0 -I ] [ -s ] [ h ] //blas.ScalFloat(x, 0.0) //blas.CopyFloat(b, dy) checkpnt.MinorPush(5) x.Scal(0.0) mCopy(b, dy) blas.CopyFloat(h, s) err = f(x, dy, s) if err != nil { fmt.Printf("f(x,dy,s): %s\n", err) return } blas.ScalFloat(s, -1.0) //fmt.Printf("initial s=\n%v\n", s.ToString("%.5f")) checkpnt.MinorPop() } else { mCopy(&matrixVar{primalstart.At("x")[0]}, x) blas.Copy(primalstart.At("s")[0], s) } // ts = min{ t | s + t*e >= 0 } ts, _ := maxStep(s, dims, 0, nil) if ts >= 0 && primalstart != nil { err = errors.New("initial s is not positive") return } //fmt.Printf("initial ts=%.5f\n", ts) checkpnt.Check("10init", 10) if dualstart == nil { // minimize || z ||^2 // subject to G'*z + A'*y + c = 0 // // by solving // // [ 0 A' G' ] [ dx ] [ -c ] // [ A 0 0 ] [ y ] = [ 0 ]. // [ G 0 -I ] [ z ] [ 0 ] //blas.Copy(c, dx) //blas.ScalFloat(dx, -1.0) //blas.ScalFloat(y, 0.0) checkpnt.MinorPush(10) mCopy(c, dx) dx.Scal(-1.0) y.Scal(0.0) blas.ScalFloat(z, 0.0) err = f(dx, y, z) if err != nil { fmt.Printf("f(dx,y,z): %s\n", err) return } //fmt.Printf("initial z=\n%v\n", z.ToString("%.5f")) checkpnt.MinorPop() } else { if len(dualstart.At("y")) > 0 { mCopy(&matrixVar{dualstart.At("y")[0]}, y) } blas.Copy(dualstart.At("z")[0], z) } // ts = min{ t | z + t*e >= 0 } tz, _ := maxStep(z, dims, 0, nil) if tz >= 0 && dualstart != nil { err = errors.New("initial z is not positive") return } //fmt.Printf("initial tz=%.5f\n", tz) nrms := snrm2(s, dims, 0) nrmz := snrm2(z, dims, 0) gap := 0.0 pcost := 0.0 dcost := 0.0 relgap := 0.0 checkpnt.Check("20init", 0) if primalstart == nil && dualstart == nil { gap = sdot(s, z, dims, 0) pcost = c.Dot(x) dcost = -b.Dot(y) - sdot(h, z, dims, 0) if pcost < 0.0 { relgap = gap / -pcost } else if dcost > 0.0 { relgap = gap / dcost } else { relgap = math.NaN() } if ts <= 0 && tz < 0 && (gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance)) { // Constructed initial points happen to be feasible and optimal ind := dims.At("l")[0] + dims.Sum("q") for _, m := range dims.At("s") { symm(s, m, ind) symm(z, m, ind) ind += m * m } // rx = A'*y + G'*z + c rx := c.Copy() Af(y, rx, 1.0, 1.0, la.OptTrans) Gf(&matrixVar{z}, rx, 1.0, 1.0, la.OptTrans) resx := math.Sqrt(rx.Dot(rx)) // ry = b - A*x ry := b.Copy() Af(x, ry, -1.0, -1.0, la.OptNoTrans) resy := math.Sqrt(ry.Dot(ry)) // rz = s + G*x - h rz := matrix.FloatZeros(cdim, 1) Gf(x, &matrixVar{rz}, 1.0, 0.0, la.OptNoTrans) blas.AxpyFloat(s, rz, 1.0) blas.AxpyFloat(h, rz, -1.0) resz := snrm2(rz, dims, 0) pres := math.Max(resy/resy0, resz/resz0) dres := resx / resx0 cx := c.Dot(x) by := b.Dot(y) hz := sdot(h, z, dims, 0) //sol.X = x; sol.Y = y; sol.S = s; sol.Z = z sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", x.Matrix()) sol.Result.Append("y", y.Matrix()) sol.Result.Append("s", s) sol.Result.Append("z", z) sol.Status = Optimal sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = cx sol.DualObjective = -(by + hz) sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz return } if ts >= -1e-8*math.Max(nrms, 1.0) { a := 1.0 + ts is := make([]int, 0) // indexes s[:dims['l']] if dims.At("l")[0] > 0 { is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) } // indexes s[indq[:-1]] if len(indq) > 1 { is = append(is, indq[:len(indq)-1]...) } // indexes s[ind:ind+m*m:m+1] (diagonal) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m } for _, k := range is { s.SetIndex(k, a+s.GetIndex(k)) } } if tz >= -1e-8*math.Max(nrmz, 1.0) { a := 1.0 + tz is := make([]int, 0) // indexes z[:dims['l']] if dims.At("l")[0] > 0 { is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) } // indexes z[indq[:-1]] if len(indq) > 1 { is = append(is, indq[:len(indq)-1]...) } // indexes z[ind:ind+m*m:m+1] (diagonal) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m } for _, k := range is { z.SetIndex(k, a+z.GetIndex(k)) } } } else if primalstart == nil && dualstart != nil { if ts >= -1e-8*math.Max(nrms, 1.0) { a := 1.0 + ts is := make([]int, 0) if dims.At("l")[0] > 0 { is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) } if len(indq) > 1 { is = append(is, indq[:len(indq)-1]...) } ind := dims.Sum("l", "q") for _, m := range dims.At("s") { is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m } for _, k := range is { s.SetIndex(k, a+s.GetIndex(k)) } } } else if primalstart != nil && dualstart == nil { if tz >= -1e-8*math.Max(nrmz, 1.0) { a := 1.0 + tz is := make([]int, 0) if dims.At("l")[0] > 0 { is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) } if len(indq) > 1 { is = append(is, indq[:len(indq)-1]...) } ind := dims.Sum("l", "q") for _, m := range dims.At("s") { is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m } for _, k := range is { z.SetIndex(k, a+z.GetIndex(k)) } } } tau := matrix.FloatValue(1.0) kappa := matrix.FloatValue(1.0) wkappa3 := matrix.FloatValue(0.0) rx := c.Copy() hrx := c.Copy() ry := b.Copy() hry := b.Copy() rz := matrix.FloatZeros(cdim, 1) hrz := matrix.FloatZeros(cdim, 1) sigs := matrix.FloatZeros(dims.Sum("s"), 1) sigz := matrix.FloatZeros(dims.Sum("s"), 1) lmbda := matrix.FloatZeros(cdim_diag+1, 1) lmbdasq := matrix.FloatZeros(cdim_diag+1, 1) gap = sdot(s, z, dims, 0) var x1, y1 MatrixVariable var z1 *matrix.FloatMatrix var dg, dgi float64 var th *matrix.FloatMatrix var WS fVarClosure var f3 KKTFuncVar var cx, by, hz, rt float64 var hresx, resx, hresy, resy, hresz, resz float64 var dres, pres, dinfres, pinfres float64 // check point variables checkpnt.AddMatrixVar("lmbda", lmbda) checkpnt.AddMatrixVar("lmbdasq", lmbdasq) checkpnt.AddVerifiable("rx", rx) checkpnt.AddVerifiable("ry", ry) checkpnt.AddMatrixVar("rz", rz) checkpnt.AddFloatVar("resx", &resx) checkpnt.AddFloatVar("resy", &resy) checkpnt.AddFloatVar("resz", &resz) checkpnt.AddFloatVar("hresx", &hresx) checkpnt.AddFloatVar("hresy", &hresy) checkpnt.AddFloatVar("hresz", &hresz) checkpnt.AddFloatVar("cx", &cx) checkpnt.AddFloatVar("by", &by) checkpnt.AddFloatVar("hz", &hz) checkpnt.AddFloatVar("gap", &gap) checkpnt.AddFloatVar("pres", &pres) checkpnt.AddFloatVar("dres", &dres) for iter := 0; iter < maxIter+1; iter++ { checkpnt.MajorNext() checkpnt.Check("loop-start", 100) // hrx = -A'*y - G'*z Af(y, hrx, -1.0, 0.0, la.OptTrans) Gf(&matrixVar{z}, hrx, -1.0, 1.0, la.OptTrans) hresx = math.Sqrt(hrx.Dot(hrx)) // rx = hrx - c*tau // = -A'*y - G'*z - c*tau mCopy(hrx, rx) c.Axpy(rx, -tau.Float()) resx = math.Sqrt(rx.Dot(rx)) / tau.Float() // hry = A*x Af(x, hry, 1.0, 0.0, la.OptNoTrans) hresy = math.Sqrt(hry.Dot(hry)) // ry = hry - b*tau // = A*x - b*tau mCopy(hry, ry) b.Axpy(ry, -tau.Float()) resy = math.Sqrt(ry.Dot(ry)) / tau.Float() // hrz = s + G*x Gf(x, &matrixVar{hrz}, 1.0, 0.0, la.OptNoTrans) blas.AxpyFloat(s, hrz, 1.0) hresz = snrm2(hrz, dims, 0) // rz = hrz - h*tau // = s + G*x - h*tau blas.ScalFloat(rz, 0.0) blas.AxpyFloat(hrz, rz, 1.0) blas.AxpyFloat(h, rz, -tau.Float()) resz = snrm2(rz, dims, 0) / tau.Float() // rt = kappa + c'*x + b'*y + h'*z ' cx = c.Dot(x) by = b.Dot(y) hz = sdot(h, z, dims, 0) rt = kappa.Float() + cx + by + hz // Statistics for stopping criteria pcost = cx / tau.Float() dcost = -(by + hz) / tau.Float() if pcost < 0.0 { relgap = gap / -pcost } else if dcost > 0.0 { relgap = gap / dcost } else { relgap = math.NaN() } pres = math.Max(resy/resy0, resz/resz0) dres = resx / resx0 pinfres = math.NaN() if hz+by < 0.0 { pinfres = hresx / resx0 / (-hz - by) } dinfres = math.NaN() if cx < 0.0 { dinfres = math.Max(hresy/resy0, hresz/resz0) / (-cx) } if solopts.ShowProgress { if iter == 0 { // show headers of something fmt.Printf("% 10s% 12s% 10s% 8s% 7s % 5s\n", "pcost", "dcost", "gap", "pres", "dres", "k/t") } // show something fmt.Printf("%2d: % 8.4e % 8.4e % 4.0e% 7.0e% 7.0e% 7.0e\n", iter, pcost, dcost, gap, pres, dres, kappa.GetIndex(0)/tau.GetIndex(0)) } checkpnt.Check("isready", 200) if (pres <= feasTolerance && dres <= feasTolerance && (gap <= absTolerance || (!math.IsNaN(relgap) && relgap <= relTolerance))) || iter == maxIter { // done x.Scal(1.0 / tau.Float()) y.Scal(1.0 / tau.Float()) blas.ScalFloat(s, 1.0/tau.Float()) blas.ScalFloat(z, 1.0/tau.Float()) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(s, m, ind) symm(z, m, ind) ind += m * m } ts, _ = maxStep(s, dims, 0, nil) tz, _ = maxStep(z, dims, 0, nil) if iter == maxIter { // MaxIterations exceeded if solopts.ShowProgress { fmt.Printf("No solution. Max iterations exceeded\n") } err = errors.New("No solution. Max iterations exceeded") //sol.X = x; sol.Y = y; sol.S = s; sol.Z = z sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", x.Matrix()) sol.Result.Append("y", y.Matrix()) sol.Result.Append("s", s) sol.Result.Append("z", z) sol.Status = Unknown sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz sol.PrimalResidualCert = pinfres sol.DualResidualCert = dinfres sol.Iterations = iter return } else { // Optimal if solopts.ShowProgress { fmt.Printf("Optimal solution.\n") } err = nil //sol.X = x; sol.Y = y; sol.S = s; sol.Z = z sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", x.Matrix()) sol.Result.Append("y", y.Matrix()) sol.Result.Append("s", s) sol.Result.Append("z", z) sol.Status = Optimal sol.Gap = gap sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz sol.PrimalResidualCert = math.NaN() sol.DualResidualCert = math.NaN() sol.Iterations = iter return } } else if !math.IsNaN(pinfres) && pinfres <= feasTolerance { // Primal Infeasible if solopts.ShowProgress { fmt.Printf("Primal infeasible.\n") } err = errors.New("Primal infeasible") y.Scal(1.0 / (-hz - by)) blas.ScalFloat(z, 1.0/(-hz-by)) //sol.X = nil; sol.Y = nil; sol.S = nil; sol.Z = nil ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(z, m, ind) ind += m * m } tz, _ = maxStep(z, dims, 0, nil) sol.Status = PrimalInfeasible sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", nil) sol.Result.Append("y", nil) sol.Result.Append("s", nil) sol.Result.Append("z", nil) sol.Gap = math.NaN() sol.RelativeGap = math.NaN() sol.PrimalObjective = math.NaN() sol.DualObjective = 1.0 sol.PrimalInfeasibility = math.NaN() sol.DualInfeasibility = math.NaN() sol.PrimalSlack = math.NaN() sol.DualSlack = -tz sol.PrimalResidualCert = pinfres sol.DualResidualCert = math.NaN() sol.Iterations = iter return } else if !math.IsNaN(dinfres) && dinfres <= feasTolerance { // Dual Infeasible if solopts.ShowProgress { fmt.Printf("Dual infeasible.\n") } err = errors.New("Primal infeasible") x.Scal(1.0 / (-cx)) blas.ScalFloat(s, 1.0/(-cx)) //sol.X = nil; sol.Y = nil; sol.S = nil; sol.Z = nil ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(s, m, ind) ind += m * m } ts, _ = maxStep(s, dims, 0, nil) sol.Status = PrimalInfeasible sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", nil) sol.Result.Append("y", nil) sol.Result.Append("s", nil) sol.Result.Append("z", nil) sol.Gap = math.NaN() sol.RelativeGap = math.NaN() sol.PrimalObjective = 1.0 sol.DualObjective = math.NaN() sol.PrimalInfeasibility = math.NaN() sol.DualInfeasibility = math.NaN() sol.PrimalSlack = -ts sol.DualSlack = math.NaN() sol.PrimalResidualCert = math.NaN() sol.DualResidualCert = dinfres sol.Iterations = iter return } // Compute initial scaling W: // // W * z = W^{-T} * s = lambda // dg * tau = 1/dg * kappa = lambdag. if iter == 0 { //fmt.Printf("compute scaling: lmbda=\n%v\n", lmbda.ToString("%.17f")) //fmt.Printf("s=\n%v\n", s.ToString("%.17f")) //fmt.Printf("z=\n%v\n", z.ToString("%.17f")) W, err = computeScaling(s, z, lmbda, dims, 0) checkpnt.AddScaleVar(W) // dg = sqrt( kappa / tau ) // dgi = sqrt( tau / kappa ) // lambda_g = sqrt( tau * kappa ) // // lambda_g is stored in the last position of lmbda. dg = math.Sqrt(kappa.Float() / tau.Float()) dgi = math.Sqrt(float64(tau.Float() / kappa.Float())) lmbda.SetIndex(-1, math.Sqrt(float64(tau.Float()*kappa.Float()))) //fmt.Printf("lmbda=\n%v\n", lmbda.ToString("%.17f")) //W.Print() checkpnt.Check("compute_scaling", 300) } // lmbdasq := lmbda o lmbda ssqr(lmbdasq, lmbda, dims, 0) lmbdasq.SetIndex(-1, lmbda.GetIndex(-1)*lmbda.GetIndex(-1)) // f3(x, y, z) solves // // [ 0 A' G' ] [ ux ] [ bx ] // [ A 0 0 ] [ uy ] = [ by ]. // [ G 0 -W'*W ] [ W^{-1}*uz ] [ bz ] // // On entry, x, y, z contain bx, by, bz. // On exit, they contain ux, uy, uz. // // Also solve // // [ 0 A' G' ] [ x1 ] [ c ] // [-A 0 0 ]*[ y1 ] = -dgi * [ b ]. // [-G 0 W'*W ] [ W^{-1}*z1 ] [ h ] f3, err = kktsolver(W) if err != nil { fmt.Printf("kktsolver error=%v\n", err) return } if iter == 0 { x1 = c.Copy() y1 = b.Copy() z1 = matrix.FloatZeros(cdim, 1) checkpnt.AddVerifiable("x1", x1) checkpnt.AddMatrixVar("z1", z1) } mCopy(c, x1) x1.Scal(-1.0) mCopy(b, y1) blas.Copy(h, z1) err = f3(x1, y1, z1) //fmt.Printf("f3 result: x1=\n%v\nf3 result: z1=\n%v\n", x1, z1) x1.Scal(dgi) y1.Scal(dgi) blas.ScalFloat(z1, dgi) if err != nil { if iter == 0 && primalstart != nil && dualstart != nil { err = errors.New("Rank(A) < p or Rank([G; A]) < n") return } else { t_ := 1.0 / tau.Float() x.Scal(t_) y.Scal(t_) blas.ScalFloat(s, t_) blas.ScalFloat(z, t_) ind := dims.Sum("l", "q") for _, m := range dims.At("s") { symm(s, m, ind) symm(z, m, ind) ind += m * m } ts, _ = maxStep(s, dims, 0, nil) tz, _ = maxStep(z, dims, 0, nil) err = errors.New("Terminated (singular KKT matrix).") //sol.X = x; sol.Y = y; sol.S = s; sol.Z = z sol.Result = sets.NewFloatSet("x", "y", "s", "x") sol.Result.Append("x", x.Matrix()) sol.Result.Append("y", y.Matrix()) sol.Result.Append("s", s) sol.Result.Append("z", z) sol.Status = Unknown sol.RelativeGap = relgap sol.PrimalObjective = pcost sol.DualObjective = dcost sol.PrimalInfeasibility = pres sol.DualInfeasibility = dres sol.PrimalSlack = -ts sol.DualSlack = -tz sol.Iterations = iter return } } // f6_no_ir(x, y, z, tau, s, kappa) solves // // [ 0 ] [ 0 A' G' c ] [ ux ] [ bx ] // [ 0 ] [ -A 0 0 b ] [ uy ] [ by ] // [ W'*us ] - [ -G 0 0 h ] [ W^{-1}*uz ] = -[ bz ] // [ dg*ukappa ] [ -c' -b' -h' 0 ] [ utau/dg ] [ btau ] // // lmbda o (uz + us) = -bs // lmbdag * (utau + ukappa) = -bkappa. // // On entry, x, y, z, tau, s, kappa contain bx, by, bz, btau, // bkappa. On exit, they contain ux, uy, uz, utau, ukappa. // th = W^{-T} * h if iter == 0 { th = matrix.FloatZeros(cdim, 1) checkpnt.AddMatrixVar("th", th) } blas.Copy(h, th) scale(th, W, true, true) f6_no_ir := func(x, y MatrixVariable, z, tau, s, kappa *matrix.FloatMatrix) (err error) { // Solve // // [ 0 A' G' 0 ] [ ux ] // [ -A 0 0 b ] [ uy ] // [ -G 0 W'*W h ] [ W^{-1}*uz ] // [ -c' -b' -h' k/t ] [ utau/dg ] // // [ bx ] // [ by ] // = [ bz - W'*(lmbda o\ bs) ] // [ btau - bkappa/tau ] // // us = -lmbda o\ bs - uz // ukappa = -bkappa/lmbdag - utau. // First solve // // [ 0 A' G' ] [ ux ] [ bx ] // [ A 0 0 ] [ uy ] = [ -by ] // [ G 0 -W'*W ] [ W^{-1}*uz ] [ -bz + W'*(lmbda o\ bs) ] minor := checkpnt.MinorTop() err = nil // y := -y = -by y.Scal(-1.0) // s := -lmbda o\ s = -lmbda o\ bs err = sinv(s, lmbda, dims, 0) blas.ScalFloat(s, -1.0) // z := -(z + W'*s) = -bz + W'*(lambda o\ bs) blas.Copy(s, ws3) checkpnt.Check("prescale", minor+5) checkpnt.MinorPush(minor + 5) err = scale(ws3, W, true, false) checkpnt.MinorPop() if err != nil { fmt.Printf("scale error: %s\n", err) } blas.AxpyFloat(ws3, z, 1.0) blas.ScalFloat(z, -1.0) checkpnt.Check("f3-call", minor+20) checkpnt.MinorPush(minor + 20) err = f3(x, y, z) checkpnt.MinorPop() checkpnt.Check("f3-return", minor+40) // Combine with solution of // // [ 0 A' G' ] [ x1 ] [ c ] // [-A 0 0 ] [ y1 ] = -dgi * [ b ] // [-G 0 W'*W ] [ W^{-1}*dzl ] [ h ] // // to satisfy // // -c'*x - b'*y - h'*W^{-1}*z + dg*tau = btau - bkappa/tau. ' // , kappa[0] := -kappa[0] / lmbd[-1] = -bkappa / lmbdag kap_ := kappa.Float() tau_ := tau.Float() kap_ = -kap_ / lmbda.GetIndex(-1) // tau[0] = tau[0] + kappa[0] / dgi = btau[0] - bkappa / tau tau_ = tau_ + kap_/dgi //tau[0] = dgi * ( tau[0] + xdot(c,x) + ydot(b,y) + // misc.sdot(th, z, dims) ) / (1.0 + misc.sdot(z1, z1, dims)) //tau_ = tau_ + blas.DotFloat(c, x) + blas.DotFloat(b, y) + sdot(th, z, dims, 0) tau_ += c.Dot(x) tau_ += b.Dot(y) tau_ += sdot(th, z, dims, 0) tau_ = dgi * tau_ / (1.0 + sdot(z1, z1, dims, 0)) tau.SetValue(tau_) x1.Axpy(x, tau_) y1.Axpy(y, tau_) blas.AxpyFloat(z1, z, tau_) blas.AxpyFloat(z, s, -1.0) kap_ = kap_ - tau_ kappa.SetValue(kap_) return } // f6(x, y, z, tau, s, kappa) solves the same system as f6_no_ir, // but applies iterative refinement. Following variables part of f6-closure // and ~ 12 is the limit. We wrap them to a structure. if iter == 0 { if refinement > 0 || solopts.Debug { WS.wx = c.Copy() WS.wy = b.Copy() WS.wz = matrix.FloatZeros(cdim, 1) WS.ws = matrix.FloatZeros(cdim, 1) WS.wtau = matrix.FloatValue(0.0) WS.wkappa = matrix.FloatValue(0.0) checkpnt.AddVerifiable("wx", WS.wx) checkpnt.AddMatrixVar("wz", WS.wz) checkpnt.AddMatrixVar("ws", WS.ws) } if refinement > 0 { WS.wx2 = c.Copy() WS.wy2 = b.Copy() WS.wz2 = matrix.FloatZeros(cdim, 1) WS.ws2 = matrix.FloatZeros(cdim, 1) WS.wtau2 = matrix.FloatValue(0.0) WS.wkappa2 = matrix.FloatValue(0.0) checkpnt.AddVerifiable("wx2", WS.wx2) checkpnt.AddMatrixVar("wz2", WS.wz2) checkpnt.AddMatrixVar("ws2", WS.ws2) } } f6 := func(x, y MatrixVariable, z, tau, s, kappa *matrix.FloatMatrix) error { var err error = nil minor := checkpnt.MinorTop() checkpnt.Check("startf6", minor+100) if refinement > 0 || solopts.Debug { mCopy(x, WS.wx) mCopy(y, WS.wy) blas.Copy(z, WS.wz) blas.Copy(s, WS.ws) WS.wtau.SetValue(tau.Float()) WS.wkappa.SetValue(kappa.Float()) } checkpnt.Check("pref6_no_ir", minor+200) err = f6_no_ir(x, y, z, tau, s, kappa) checkpnt.Check("postf6_no_ir", minor+399) for i := 0; i < refinement; i++ { mCopy(WS.wx, WS.wx2) mCopy(WS.wy, WS.wy2) blas.Copy(WS.wz, WS.wz2) blas.Copy(WS.ws, WS.ws2) WS.wtau2.SetValue(WS.wtau.Float()) WS.wkappa2.SetValue(WS.wkappa.Float()) checkpnt.Check("res-call", minor+400) checkpnt.MinorPush(minor + 400) err = res(x, y, z, tau, s, kappa, WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2, W, dg, lmbda) checkpnt.MinorPop() checkpnt.Check("refine_pref6_no_ir", minor+500) checkpnt.MinorPush(minor + 500) err = f6_no_ir(WS.wx2, WS.wy2, WS.wz2, WS.wtau2, WS.ws2, WS.wkappa2) checkpnt.MinorPop() checkpnt.Check("refine_postf6_no_ir", minor+600) WS.wx2.Axpy(x, 1.0) WS.wy2.Axpy(y, 1.0) blas.AxpyFloat(WS.wz2, z, 1.0) blas.AxpyFloat(WS.ws2, s, 1.0) tau.SetValue(tau.Float() + WS.wtau2.Float()) kappa.SetValue(kappa.Float() + WS.wkappa2.Float()) } if solopts.Debug { checkpnt.MinorPush(minor + 700) res(x, y, z, tau, s, kappa, WS.wx, WS.wy, WS.wz, WS.wtau, WS.ws, WS.wkappa, W, dg, lmbda) checkpnt.MinorPop() fmt.Printf("KKT residuals\n") fmt.Printf(" 'x' : %.6e\n", math.Sqrt(WS.wx.Dot(WS.wx))) fmt.Printf(" 'y' : %.6e\n", math.Sqrt(WS.wy.Dot(WS.wy))) fmt.Printf(" 'z' : %.6e\n", snrm2(WS.wz, dims, 0)) fmt.Printf(" 'tau' : %.6e\n", math.Abs(WS.wtau.Float())) fmt.Printf(" 's' : %.6e\n", snrm2(WS.ws, dims, 0)) fmt.Printf(" 'kappa': %.6e\n", math.Abs(WS.wkappa.Float())) } return err } var nrm float64 = blas.Nrm2Float(lmbda) mu := math.Pow(nrm, 2.0) / (1.0 + float64(cdim_diag)) sigma := 0.0 var step, tt, tk float64 for i := 0; i < 2; i++ { // Solve // // [ 0 ] [ 0 A' G' c ] [ dx ] // [ 0 ] [ -A 0 0 b ] [ dy ] // [ W'*ds ] - [ -G 0 0 h ] [ W^{-1}*dz ] // [ dg*dkappa ] [ -c' -b' -h' 0 ] [ dtau/dg ] // // [ rx ] // [ ry ] // = - (1-sigma) [ rz ] // [ rtau ] // // lmbda o (dz + ds) = -lmbda o lmbda + sigma*mu*e // lmbdag * (dtau + dkappa) = - kappa * tau + sigma*mu // // ds = -lmbdasq if i is 0 // = -lmbdasq - dsa o dza + sigma*mu*e if i is 1 // dkappa = -lambdasq[-1] if i is 0 // = -lambdasq[-1] - dkappaa*dtaua + sigma*mu if i is 1. ind := dims.Sum("l", "q") ind2 := ind blas.Copy(lmbdasq, ds, &la.IOpt{"n", ind}) blas.ScalFloat(ds, 0.0, &la.IOpt{"offset", ind}) for _, m := range dims.At("s") { blas.Copy(lmbdasq, ds, &la.IOpt{"n", m}, &la.IOpt{"offsetx", ind2}, &la.IOpt{"offsety", ind}, &la.IOpt{"incy", m + 1}) ind += m * m ind2 += m } // dkappa[0] = lmbdasq[-1] dkappa.SetValue(lmbdasq.GetIndex(-1)) if i == 1 { blas.AxpyFloat(ws3, ds, 1.0) ind = dims.Sum("l", "q") is := make([]int, 0) // indexes: [:dims['l']] if dims.At("l")[0] > 0 { is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) } // ...[indq[:-1]] if len(indq) > 1 { is = append(is, indq[:len(indq)-1]...) } // ...[ind : ind+m*m : m+1] (diagonal) for _, m := range dims.At("s") { is = append(is, matrix.MakeIndexSet(ind, ind+m*m, m+1)...) ind += m * m } //ds.Add(-sigma*mu, is...) for _, k := range is { ds.SetIndex(k, ds.GetIndex(k)-sigma*mu) } dk_ := dkappa.Float() wk_ := wkappa3.Float() dkappa.SetValue(dk_ + wk_ - sigma*mu) } // (dx, dy, dz, dtau) = (1-sigma)*(rx, ry, rz, rt) mCopy(rx, dx) dx.Scal(1.0 - sigma) mCopy(ry, dy) dy.Scal(1.0 - sigma) blas.Copy(rz, dz) blas.ScalFloat(dz, 1.0-sigma) // dtau[0] = (1.0 - sigma) * rt dtau.SetValue((1.0 - sigma) * rt) checkpnt.Check("pref6", (1+i)*1000) checkpnt.MinorPush((1 + i) * 1000) err = f6(dx, dy, dz, dtau, ds, dkappa) checkpnt.MinorPop() checkpnt.Check("postf6", (1+i)*1000+800) // Save ds o dz and dkappa * dtau for Mehrotra correction if i == 0 { blas.Copy(ds, ws3) sprod(ws3, dz, dims, 0) wkappa3.SetValue(dtau.Float() * dkappa.Float()) } // Maximum step to boundary. // // If i is 1, also compute eigenvalue decomposition of the 's' // blocks in ds, dz. The eigenvectors Qs, Qz are stored in // dsk, dzk. The eigenvalues are stored in sigs, sigz. var ts, tz float64 checkpnt.MinorPush((1+i)*1000 + 900) scale2(lmbda, ds, dims, 0, false) scale2(lmbda, dz, dims, 0, false) checkpnt.MinorPop() checkpnt.Check("post-scale2", (1+i)*1000+990) if i == 0 { ts, _ = maxStep(ds, dims, 0, nil) tz, _ = maxStep(dz, dims, 0, nil) } else { ts, _ = maxStep(ds, dims, 0, sigs) tz, _ = maxStep(dz, dims, 0, sigz) } dt_ := dtau.Float() dk_ := dkappa.Float() tt = -dt_ / lmbda.GetIndex(-1) tk = -dk_ / lmbda.GetIndex(-1) t := maxvec([]float64{0.0, ts, tz, tt, tk}) if t == 0.0 { step = 1.0 } else { if i == 0 { step = math.Min(1.0, 1.0/t) } else { step = math.Min(1.0, STEP/t) } } if i == 0 { // sigma = (1 - step)^3 sigma = (1.0 - step) * (1.0 - step) * (1.0 - step) //sigma = math.Pow((1.0 - step), EXPON) } } //fmt.Printf("** tau = %.17f, kappa = %.17f\n", tau.Float(), kappa.Float()) //fmt.Printf("** step = %.17f, sigma = %.17f\n", step, sigma) checkpnt.Check("update-xy", 7000) // Update x, y dx.Axpy(x, step) dy.Axpy(y, step) // Replace 'l' and 'q' blocks of ds and dz with the updated // variables in the current scaling. // Replace 's' blocks of ds and dz with the factors Ls, Lz in a // factorization Ls*Ls', Lz*Lz' of the updated variables in the // current scaling. // // ds := e + step*ds for 'l' and 'q' blocks. // dz := e + step*dz for 'l' and 'q' blocks. blas.ScalFloat(ds, step, &la.IOpt{"n", dims.Sum("l", "q")}) blas.ScalFloat(dz, step, &la.IOpt{"n", dims.Sum("l", "q")}) is := make([]int, 0) is = append(is, matrix.MakeIndexSet(0, dims.At("l")[0], 1)...) is = append(is, indq[:len(indq)-1]...) for _, k := range is { ds.SetIndex(k, 1.0+ds.GetIndex(k)) dz.SetIndex(k, 1.0+dz.GetIndex(k)) } checkpnt.Check("update-dsdz", 7500) // ds := H(lambda)^{-1/2} * ds and dz := H(lambda)^{-1/2} * dz. // // This replaces the 'l' and 'q' components of ds and dz with the // updated variables in the current scaling. // The 's' components of ds and dz are replaced with // // diag(lmbda_k)^{1/2} * Qs * diag(lmbda_k)^{1/2} // diag(lmbda_k)^{1/2} * Qz * diag(lmbda_k)^{1/2} checkpnt.MinorPush(7500) scale2(lmbda, ds, dims, 0, true) scale2(lmbda, dz, dims, 0, true) checkpnt.MinorPop() // sigs := ( e + step*sigs ) ./ lambda for 's' blocks. // sigz := ( e + step*sigz ) ./ lambda for 's' blocks. blas.ScalFloat(sigs, step) blas.ScalFloat(sigz, step) sigs.Add(1.0) sigz.Add(1.0) sdimsum := dims.Sum("s") qdimsum := dims.Sum("l", "q") blas.TbsvFloat(lmbda, sigs, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0}, &la.IOpt{"lda", 1}, &la.IOpt{"offseta", qdimsum}) blas.TbsvFloat(lmbda, sigz, &la.IOpt{"n", sdimsum}, &la.IOpt{"k", 0}, &la.IOpt{"lda", 1}, &la.IOpt{"offseta", qdimsum}) ind2 := qdimsum ind3 := 0 sdims := dims.At("s") for k := 0; k < len(sdims); k++ { m := sdims[k] for i := 0; i < m; i++ { a := math.Sqrt(sigs.GetIndex(ind3 + i)) blas.ScalFloat(ds, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m}) a = math.Sqrt(sigz.GetIndex(ind3 + i)) blas.ScalFloat(dz, a, &la.IOpt{"offset", ind2 + m*i}, &la.IOpt{"n", m}) } ind2 += m * m ind3 += m } checkpnt.Check("pre-update-scaling", 7700) err = updateScaling(W, lmbda, ds, dz) checkpnt.Check("post-update-scaling", 7800) // For kappa, tau block: // // dg := sqrt( (kappa + step*dkappa) / (tau + step*dtau) ) // = dg * sqrt( (1 - step*tk) / (1 - step*tt) ) // // lmbda[-1] := sqrt((tau + step*dtau) * (kappa + step*dkappa)) // = lmbda[-1] * sqrt(( 1 - step*tt) * (1 - step*tk)) dg *= math.Sqrt(1.0-step*tk) / math.Sqrt(1.0-step*tt) dgi = 1.0 / dg a := math.Sqrt(1.0-step*tk) * math.Sqrt(1.0-step*tt) lmbda.SetIndex(-1, a*lmbda.GetIndex(-1)) // Unscale s, z, tau, kappa (unscaled variables are used only to // compute feasibility residuals). ind := dims.Sum("l", "q") ind2 = ind blas.Copy(lmbda, s, &la.IOpt{"n", ind}) for _, m := range dims.At("s") { blas.ScalFloat(s, 0.0, &la.IOpt{"offset", ind2}) blas.Copy(lmbda, s, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2}, &la.IOpt{"n", m}, &la.IOpt{"incy", m + 1}) ind += m ind2 += m * m } scale(s, W, true, false) ind = dims.Sum("l", "q") ind2 = ind blas.Copy(lmbda, z, &la.IOpt{"n", ind}) for _, m := range dims.At("s") { blas.ScalFloat(z, 0.0, &la.IOpt{"offset", ind2}) blas.Copy(lmbda, z, &la.IOpt{"offsetx", ind}, &la.IOpt{"offsety", ind2}, &la.IOpt{"n", m}, &la.IOpt{"incy", m + 1}) ind += m ind2 += m * m } scale(z, W, false, true) kappa.SetValue(lmbda.GetIndex(-1) / dgi) tau.SetValue(lmbda.GetIndex(-1) * dgi) g := blas.Nrm2Float(lmbda, &la.IOpt{"n", lmbda.Rows() - 1}) / tau.Float() gap = g * g checkpnt.Check("end-of-loop", 8000) //fmt.Printf(" ** kappa=%.10f, tau=%.10f, gap=%.10f\n", kappa.Float(), tau.Float(), gap) } return }