/* 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 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) }
// Computes analytic center of A*x <= b with A m by n of rank n. // We assume that b > 0 and the feasible set is bounded. func acent(A, b *matrix.FloatMatrix, niters int) (x *matrix.FloatMatrix, ntdecrs []float64, err error) { err = nil if niters <= 0 { niters = MAXITERS } ntdecrs = make([]float64, 0, niters) if A.Rows() != b.Rows() { return nil, nil, errors.New("A.Rows() != b.Rows()") } m, n := A.Size() x = matrix.FloatZeros(n, 1) H := matrix.FloatZeros(n, n) // Helper m*n matrix Dmn := matrix.FloatZeros(m, n) for i := 0; i < niters; i++ { // Gradient is g = A^T * (1.0/(b - A*x)). d = 1.0/(b - A*x) // d is m*1 matrix, g is n*1 matrix d := matrix.Minus(b, matrix.Times(A, x)).Inv() g := matrix.Times(A.Transpose(), d) // Hessian is H = A^T * diag(1./(b-A*x))^2 * A. // in the original python code expression d[:,n*[0]] creates // a m*n matrix where each column is copy of column 0. // We do it here manually. for i := 0; i < n; i++ { Dmn.SetColumn(i, d) } // Function mul creates element wise product of matrices. Asc := matrix.Mul(Dmn, A) blas.SyrkFloat(Asc, H, 1.0, 0.0, linalg.OptTrans) // Newton step is v = H^-1 * g. v := matrix.Scale(g, -1.0) PosvFloat(H, v) // Directional derivative and Newton decrement. lam := blas.DotFloat(g, v) ntdecrs = append(ntdecrs, math.Sqrt(-lam)) if ntdecrs[len(ntdecrs)-1] < TOL { return x, ntdecrs, err } // Backtracking line search. // y = d .* A*v y := matrix.Mul(d, matrix.Times(A, v)) step := 1.0 for 1-step*y.Max() < 0 { step *= BETA } search: for { // t = -step*y + 1 [e.g. t = 1 - step*y] t := matrix.Scale(y, -step).Add(1.0) // ts = sum(log(1-step*y)) ts := t.Log().Sum() if -ts < ALPHA*step*lam { break search } step *= BETA } v.Scale(step) x.Plus(v) } // no solution !! err = errors.New(fmt.Sprintf("Iteration %d exhausted\n", niters)) return x, ntdecrs, err }