This is a small modification of [http://cvxopt.org/applications/nucnrm/] for Python3 & CVXOPT. As the comment says, this program was originally written by Z. Liu and L. Vandenberghe, and it distributed under GPL.
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Rank minimization using Nuclear norm optimization for python3 using CVXOPT
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| """ | |
| Interior-point code for nuclear norm minimization | |
| """ | |
| # Version 1.0. Copyright 2009 Z. Liu and L. Vandenberghe. | |
| # | |
| # This program is free software; you can redistribute it and/or modify | |
| # it under the terms of the GNU General Public License as published by | |
| # the Free Software Foundation; either version 3 of the License, or | |
| # (at your option) any later version. | |
| # | |
| # This program is distributed in the hope that it will be useful, | |
| # but WITHOUT ANY WARRANTY; without even the implied warranty of | |
| # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
| # GNU General Public License for more details. | |
| # | |
| # | |
| # You should have received a copy of the GNU General Public License | |
| # along with this program. If not, see <http://www.gnu.org/licenses/>. | |
| from cvxopt import matrix, spmatrix, sqrt, sparse, mul | |
| from cvxopt import base, blas, lapack, solvers, misc | |
| __all__ = [ 'nrmapp' ] | |
| def nrmapp(A, B, C = None, d = None, G = None, h = None): | |
| """ | |
| Solves the regularized nuclear norm approximation problem | |
| minimize || A(x) + B ||_* + 1/2 x'*C*x + d'*x | |
| subject to G*x <= h | |
| and its dual | |
| maximize -h'*z + tr(B'*Z) - 1/2 v'*C*v | |
| subject to d + G'*z + A'(Z) = C*v | |
| z >= 0 | |
| || Z || <= 1. | |
| A(x) is a linear mapping that maps n-vectors x to (p x q)-matrices A(x). | |
| ||.||_* is the nuclear norm (sum of singular values). | |
| A'(Z) is the adjoint mapping of A(x). | |
| ||.|| is the maximum singular value norm. | |
| INPUT | |
| A real dense or sparse matrix of size (p*q, n). Its columns are | |
| the coefficients A_i of the mapping | |
| A: reals^n --> reals^pxq, A(x) = sum_i=1^n x_i * A_i, | |
| stored in column-major order, as p*q-vectors. | |
| B real dense or sparse matrix of size (p, q), with p >= q. | |
| C real symmetric positive semidefinite dense or sparse matrix of | |
| order n. Only the lower triangular part of C is accessed. | |
| The default value is a zero matrix. | |
| d real dense matrix of size (n, 1). The default value is a zero | |
| vector. | |
| G real dense or sparse matrix of size (m, n), with m >= 0. | |
| The default value is a matrix of size (0, n). | |
| h real dense matrix of size (m, 1). The default value is a | |
| matrix of size (0, 1). | |
| OUTPUT | |
| status 'optimal', 'primal infeasible', or 'unknown'. | |
| x 'd' matrix of size (n, 1) if status is 'optimal'; | |
| None otherwise. | |
| z 'd' matrix of size (m, 1) if status is 'optimal' or 'primal | |
| infeasible'; None otherwise. | |
| Z 'd' matrix of size (p, q) if status is 'optimal' or 'primal | |
| infeasible'; None otherwise. | |
| If status is 'optimal', then x, z, Z are approximate solutions of the | |
| optimality conditions | |
| C * x + G' * z + A'(Z) + d = 0 | |
| G * x <= h | |
| z >= 0, || Z || < = 1 | |
| z' * (h - G*x) = 0 | |
| tr (Z' * (A(x) + B)) = || A(x) + B ||_*. | |
| The last (complementary slackness) condition can be replaced by the | |
| following. If the singular value decomposition of A(x) + B is | |
| A(x) + B = [ U1 U2 ] * diag(s, 0) * [ V1 V2 ]', | |
| with s > 0, then | |
| Z = U1 * V1' + U2 * W * V2', || W || <= 1. | |
| If status is 'primal infeasible', then Z = 0 and z is a certificate of | |
| infeasibility for the inequalities G * x <= h, i.e., a vector that | |
| satisfies | |
| h' * z = 1, G' * z = 0, z >= 0. | |
| """ | |
| if type(B) not in (matrix, spmatrix) or B.typecode is not 'd': | |
| raise TypeError("B must be a real dense or sparse matrix") | |
| p, q = B.size | |
| if p < q: | |
| raise ValueError("row dimension of B must be greater than or "\ | |
| "equal to column dimension") | |
| if type(A) not in (matrix, spmatrix) or A.typecode is not 'd' or \ | |
| A.size[0] != p*q: | |
| raise TypeError("A must be a real dense or sparse matrix with "\ | |
| "p*q rows if B has size (p, q)") | |
| n = A.size[1] | |
| if G is None: G = spmatrix([], [], [], (0, n)) | |
| if h is None: h = matrix(0.0, (0, 1)) | |
| if type(h) is not matrix or h.typecode is not 'd' or h.size[1] != 1: | |
| raise TypeError("h must be a real dense matrix with one column") | |
| m = h.size[0] | |
| if type(G) not in (matrix, spmatrix) or G.typecode is not 'd' or \ | |
| G.size != (m, n): | |
| raise TypeError("G must be a real dense matrix or sparse matrix "\ | |
| "of size (m, n) if h has length m and A has n columns") | |
| if C is None: C = spmatrix(0.0, [], [], (n,n)) | |
| if d is None: d = matrix(0.0, (n, 1)) | |
| if type(C) not in (matrix, spmatrix) or C.typecode is not 'd' or \ | |
| C.size != (n,n): | |
| raise TypeError("C must be real dense or sparse matrix of size "\ | |
| "(n, n) if A has n columns") | |
| if type(d) is not matrix or d.typecode is not 'd' or d.size != (n,1): | |
| raise TypeError("d must be a real matrix of size (n, 1) if A has "\ | |
| "n columns") | |
| # The problem is solved as a cone program | |
| # | |
| # minimize (1/2) * x'*C*x + d'*x + (1/2) * (tr X1 + tr X2) | |
| # subject to G*x <= h | |
| # [ X1 (A(x) + B)' ] | |
| # [ A(x) + B X2 ] >= 0. | |
| # | |
| # The primal variable is stored as a list [ x, X1, X2 ]. | |
| def xnewcopy(u): | |
| return [ matrix(u[0]), matrix(u[1]), matrix(u[2]) ] | |
| def xdot(u,v): | |
| return blas.dot(u[0], v[0]) + misc.sdot2(u[1], v[1]) + \ | |
| misc.sdot2(u[2], v[2]) | |
| def xscal(alpha, u): | |
| blas.scal(alpha, u[0]) | |
| blas.scal(alpha, u[1]) | |
| blas.scal(alpha, u[2]) | |
| def xaxpy(u, v, alpha = 1.0): | |
| blas.axpy(u[0], v[0], alpha) | |
| blas.axpy(u[1], v[1], alpha) | |
| blas.axpy(u[2], v[2], alpha) | |
| def Pf(u, v, alpha = 1.0, beta = 0.0): | |
| base.symv(C, u[0], v[0], alpha = alpha, beta = beta) | |
| blas.scal(beta, v[1]) | |
| blas.scal(beta, v[2]) | |
| c = [ d, matrix(0.0, (q,q)), matrix(0.0, (p,p)) ] | |
| c[1][::q+1] = 0.5 | |
| c[2][::p+1] = 0.5 | |
| # If V is a p+q x p+q matrix | |
| # | |
| # [ V11 V12 ] | |
| # V = [ ] | |
| # [ V21 V22 ] | |
| # | |
| # with V11 q x q, V21 p x q, V12 q x p, and V22 p x p, then I11, I21, | |
| # I22 are the index sets defined by | |
| # | |
| # V[I11] = V11[:], V[I21] = V21[:], V[I22] = V22[:]. | |
| # | |
| I11 = matrix([ i + j*(p+q) for j in range(q) for i in range(q) ]) | |
| I21 = matrix([ q + i + j*(p+q) for j in range(q) for i in range(p) ]) | |
| I22 = matrix([ (p+q)*q + q + i + j*(p+q) for j in range(p) for | |
| i in range(p) ]) | |
| dims = {'l': m, 'q': [], 's': [p+q]} | |
| hh = matrix(0.0, (m + (p+q)**2, 1)) | |
| hh[:m] = h | |
| hh[m + I21] = B[:] | |
| def Gf(u, v, alpha = 1.0, beta = 0.0, trans = 'N'): | |
| if trans == 'N': | |
| # v[:m] := alpha * G * u[0] + beta * v[:m] | |
| base.gemv(G, u[0], v, alpha = alpha, beta = beta) | |
| # v[m:] := alpha * [-u[1], -A(u[0])'; -A(u[0]), -u[2]] | |
| # + beta * v[m:] | |
| blas.scal(beta, v, offset = m) | |
| v[m + I11] -= alpha * u[1][:] | |
| v[m + I21] -= alpha * A * u[0] | |
| v[m + I22] -= alpha * u[2][:] | |
| else: | |
| # v[0] := alpha * ( G.T * u[:m] - 2.0 * A.T * u[m + I21] ) | |
| # + beta v[1] | |
| base.gemv(G, u, v[0], trans = 'T', alpha = alpha, beta = beta) | |
| base.gemv(A, u[m + I21], v[0], trans = 'T', alpha = -2.0*alpha, | |
| beta = 1.0) | |
| # v[1] := -alpha * u[m + I11] + beta * v[1] | |
| blas.scal(beta, v[1]) | |
| blas.axpy(u[m + I11], v[1], alpha = -alpha) | |
| # v[2] := -alpha * u[m + I22] + beta * v[2] | |
| blas.scal(beta, v[2]) | |
| blas.axpy(u[m + I22], v[2], alpha = -alpha) | |
| def Af(u, v, alpha = 1.0, beta = 0.0, trans = 'N'): | |
| if trans == 'N': | |
| pass | |
| else: | |
| blas.scal(beta, v[0]) | |
| blas.scal(beta, v[1]) | |
| blas.scal(beta, v[2]) | |
| L1 = matrix(0.0, (q, q)) | |
| L2 = matrix(0.0, (p, p)) | |
| T21 = matrix(0.0, (p, q)) | |
| s = matrix(0.0, (q, 1)) | |
| SS = matrix(0.0, (q, q)) | |
| V1 = matrix(0.0, (q, q)) | |
| V2 = matrix(0.0, (p, p)) | |
| As = matrix(0.0, (p*q, n)) | |
| As2 = matrix(0.0, (p*q, n)) | |
| tmp = matrix(0.0, (p, q)) | |
| a = matrix(0.0, (p+q, p+q)) | |
| H = matrix(0.0, (n,n)) | |
| Gs = matrix(0.0, (m, n)) | |
| Q1 = matrix(0.0, (q, p+q)) | |
| Q2 = matrix(0.0, (p, p+q)) | |
| tau1 = matrix(0.0, (q,1)) | |
| tau2 = matrix(0.0, (p,1)) | |
| bz11 = matrix(0.0, (q,q)) | |
| bz22 = matrix(0.0, (p,p)) | |
| bz21 = matrix(0.0, (p,q)) | |
| # Suppose V = [V1; V2] is p x q with V1 q x q. If v = V[:] then | |
| # v[Itriu] are the strict upper triangular entries of V1 stored | |
| # columnwise. | |
| Itriu = [ i + j*p for j in range(1,q) for i in range(j) ] | |
| # v[Itril] are the strict lower triangular entries of V1 stored rowwise. | |
| Itril = [ j + i*p for j in range(1,q) for i in range(j) ] | |
| # v[Idiag] are the diagonal entries of V1. | |
| Idiag = [ i*(p+1) for i in range(q) ] | |
| # v[Itriu2] are the upper triangular entries of V1, with the diagonal | |
| # entries stored first, followed by the strict upper triangular entries | |
| # stored columnwise. | |
| Itriu2 = Idiag + Itriu | |
| # If V is a q x q matrix and v = V[:], then v[Itril2] are the strict | |
| # lower triangular entries of V stored columnwise and v[Itril3] are | |
| # the strict lower triangular entries stored rowwise. | |
| Itril2 = [ i + j*q for j in range(q) for i in range(j+1,q) ] | |
| Itril3 = [ i + j*q for i in range(q) for j in range(i) ] | |
| P = spmatrix(0.0, Itriu, Itril, (p*q, p*q)) | |
| D = spmatrix(1.0, range(p*q), range(p*q)) | |
| DV = matrix(1.0, (p*q, 1)) | |
| def F(W): | |
| """ | |
| Create a solver for the linear equations | |
| C * ux + G' * uzl - 2*A'(uzs21) = bx | |
| -uzs11 = bX1 | |
| -uzs22 = bX2 | |
| G * ux - Dl^2 * uzl = bzl | |
| [ -uX1 -A(ux)' ] [ uzs11 uzs21' ] | |
| [ ] - r*r' * [ ] * r*r' = bzs | |
| [ -A(ux) -uX2 ] [ uzs21 uzs22 ] | |
| where Dl = diag(W['l']), r = W['r'][0]. | |
| On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ]. | |
| On exit, x = (ux, uX1, uX2) and z = [ Dl*uzl; (r'*uzs*r)[:] ]. | |
| 1. Compute matrices V1, V2 such that (with T = r*r') | |
| [ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ] | |
| [ ] [ ] [ ] = [ ] | |
| [ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ] | |
| and S = [ diag(s); 0 ], s a positive q-vector. | |
| 2. Factor the mapping X -> X + S * X' * S: | |
| X + S * X' * S = L( L'( X )). | |
| 3. Compute scaled mappings: a matrix As with as its columns the | |
| coefficients of the scaled mapping | |
| L^-1( V2' * A() * V1' ) | |
| and the matrix Gs = Dl^-1 * G. | |
| 4. Cholesky factorization of H = C + Gs'*Gs + 2*As'*As. | |
| """ | |
| # 1. Compute V1, V2, s. | |
| r = W['r'][0] | |
| # LQ factorization R[:q, :] = L1 * Q1. | |
| lapack.lacpy(r, Q1, m = q) | |
| lapack.gelqf(Q1, tau1) | |
| lapack.lacpy(Q1, L1, n = q, uplo = 'L') | |
| lapack.orglq(Q1, tau1) | |
| # LQ factorization R[q:, :] = L2 * Q2. | |
| lapack.lacpy(r, Q2, m = p, offsetA = q) | |
| lapack.gelqf(Q2, tau2) | |
| lapack.lacpy(Q2, L2, n = p, uplo = 'L') | |
| lapack.orglq(Q2, tau2) | |
| # V2, V1, s are computed from an SVD: if | |
| # | |
| # Q2 * Q1' = U * diag(s) * V', | |
| # | |
| # then V1 = V' * L1^-1 and V2 = L2^-T * U. | |
| # T21 = Q2 * Q1.T | |
| blas.gemm(Q2, Q1, T21, transB = 'T') | |
| # SVD T21 = U * diag(s) * V'. Store U in V2 and V' in V1. | |
| lapack.gesvd(T21, s, jobu = 'A', jobvt = 'A', U = V2, Vt = V1) | |
| # # Q2 := Q2 * Q1' without extracting Q1; store T21 in Q2 | |
| # this will requires lapack.ormlq or lapack.unmlq | |
| # V2 = L2^-T * U | |
| blas.trsm(L2, V2, transA = 'T') | |
| # V1 = V' * L1^-1 | |
| blas.trsm(L1, V1, side = 'R') | |
| # 2. Factorization X + S * X' * S = L( L'( X )). | |
| # | |
| # The factor L is stored as a diagonal matrix D and a sparse lower | |
| # triangular matrix P, such that | |
| # | |
| # L(X)[:] = D**-1 * (I + P) * X[:] | |
| # L^-1(X)[:] = D * (I - P) * X[:]. | |
| # SS is q x q with SS[i,j] = si*sj. | |
| blas.scal(0.0, SS) | |
| blas.syr(s, SS) | |
| # For a p x q matrix X, P*X[:] is Y[:] where | |
| # | |
| # Yij = si * sj * Xji if i < j | |
| # = 0 otherwise. | |
| # | |
| P.V = SS[Itril2] | |
| # For a p x q matrix X, D*X[:] is Y[:] where | |
| # | |
| # Yij = Xij / sqrt( 1 - si^2 * sj^2 ) if i < j | |
| # = Xii / sqrt( 1 + si^2 ) if i = j | |
| # = Xij otherwise. | |
| # | |
| DV[Idiag] = sqrt(1.0 + SS[::q+1]) | |
| DV[Itriu] = sqrt(1.0 - SS[Itril3]**2) | |
| D.V = DV**-1 | |
| # 3. Scaled linear mappings | |
| # Ask := V2' * Ask * V1' | |
| blas.scal(0.0, As) | |
| base.axpy(A, As) | |
| for i in range(n): | |
| # tmp := V2' * As[i, :] | |
| blas.gemm(V2, As, tmp, transA = 'T', m = p, n = q, k = p, | |
| ldB = p, offsetB = i*p*q) | |
| # As[:,i] := tmp * V1' | |
| blas.gemm(tmp, V1, As, transB = 'T', m = p, n = q, k = q, | |
| ldC = p, offsetC = i*p*q) | |
| # As := D * (I - P) * As | |
| # = L^-1 * As. | |
| blas.copy(As, As2) | |
| base.gemm(P, As, As2, alpha = -1.0, beta = 1.0) | |
| base.gemm(D, As2, As) | |
| # Gs := Dl^-1 * G | |
| blas.scal(0.0, Gs) | |
| base.axpy(G, Gs) | |
| for k in range(n): | |
| blas.tbmv(W['di'], Gs, n = m, k = 0, ldA = 1, offsetx = k*m) | |
| # 4. Cholesky factorization of H = C + Gs' * Gs + 2 * As' * As. | |
| blas.syrk(As, H, trans = 'T', alpha = 2.0) | |
| blas.syrk(Gs, H, trans = 'T', beta = 1.0) | |
| base.axpy(C, H) | |
| lapack.potrf(H) | |
| def f(x, y, z): | |
| """ | |
| Solve | |
| C * ux + G' * uzl - 2*A'(uzs21) = bx | |
| -uzs11 = bX1 | |
| -uzs22 = bX2 | |
| G * ux - D^2 * uzl = bzl | |
| [ -uX1 -A(ux)' ] [ uzs11 uzs21' ] | |
| [ ] - T * [ ] * T = bzs. | |
| [ -A(ux) -uX2 ] [ uzs21 uzs22 ] | |
| On entry, x = (bx, bX1, bX2) and z = [ bzl; bzs[:] ]. | |
| On exit, x = (ux, uX1, uX2) and z = [ D*uzl; (r'*uzs*r)[:] ]. | |
| Define X = uzs21, Z = T * uzs * T: | |
| C * ux + G' * uzl - 2*A'(X) = bx | |
| [ 0 X' ] [ bX1 0 ] | |
| T * [ ] * T - Z = T * [ ] * T | |
| [ X 0 ] [ 0 bX2 ] | |
| G * ux - D^2 * uzl = bzl | |
| [ -uX1 -A(ux)' ] [ Z11 Z21' ] | |
| [ ] - [ ] = bzs | |
| [ -A(ux) -uX2 ] [ Z21 Z22 ] | |
| Return x = (ux, uX1, uX2), z = [ D*uzl; (rti'*Z*rti)[:] ]. | |
| We use the congruence transformation | |
| [ V1 0 ] [ T11 T21' ] [ V1' 0 ] [ I S' ] | |
| [ ] [ ] [ ] = [ ] | |
| [ 0 V2' ] [ T21 T22 ] [ 0 V2 ] [ S I ] | |
| and the factorization | |
| X + S * X' * S = L( L'(X) ) | |
| to write this as | |
| C * ux + G' * uzl - 2*A'(X) = bx | |
| L'(V2^-1 * X * V1^-1) - L^-1(V2' * Z21 * V1') = bX | |
| G * ux - D^2 * uzl = bzl | |
| [ -uX1 -A(ux)' ] [ Z11 Z21' ] | |
| [ ] - [ ] = bzs, | |
| [ -A(ux) -uX2 ] [ Z21 Z22 ] | |
| or | |
| C * ux + Gs' * uuzl - 2*As'(XX) = bx | |
| XX - ZZ21 = bX | |
| Gs * ux - uuzl = D^-1 * bzl | |
| -As(ux) - ZZ21 = bbzs_21 | |
| -uX1 - Z11 = bzs_11 | |
| -uX2 - Z22 = bzs_22 | |
| if we introduce scaled variables | |
| uuzl = D * uzl | |
| XX = L'(V2^-1 * X * V1^-1) | |
| = L'(V2^-1 * uzs21 * V1^-1) | |
| ZZ21 = L^-1(V2' * Z21 * V1') | |
| and define | |
| bbzs_21 = L^-1(V2' * bzs_21 * V1') | |
| [ bX1 0 ] | |
| bX = L^-1( V2' * (T * [ ] * T)_21 * V1'). | |
| [ 0 bX2 ] | |
| Eliminating Z21 gives | |
| C * ux + Gs' * uuzl - 2*As'(XX) = bx | |
| Gs * ux - uuzl = D^-1 * bzl | |
| -As(ux) - XX = bbzs_21 - bX | |
| -uX1 - Z11 = bzs_11 | |
| -uX2 - Z22 = bzs_22 | |
| and eliminating uuzl and XX gives | |
| H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bX - bbzs_21) | |
| Gs * ux - uuzl = D^-1 * bzl | |
| -As(ux) - XX = bbzs_21 - bX | |
| -uX1 - Z11 = bzs_11 | |
| -uX2 - Z22 = bzs_22. | |
| In summary, we can use the following algorithm: | |
| 1. bXX := bX - bbzs21 | |
| [ bX1 0 ] | |
| = L^-1( V2' * ((T * [ ] * T)_21 - bzs_21) * V1') | |
| [ 0 bX2 ] | |
| 2. Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX). | |
| 3. From ux, compute | |
| uuzl = Gs*ux - D^-1 * bzl and | |
| X = V2 * L^-T(-As(ux) + bXX) * V1. | |
| 4. Return ux, uuzl, | |
| rti' * Z * rti = r' * [ -bX1, X'; X, -bX2 ] * r | |
| and uX1 = -Z11 - bzs_11, uX2 = -Z22 - bzs_22. | |
| """ | |
| # Save bzs_11, bzs_22, bzs_21. | |
| lapack.lacpy(z, bz11, uplo = 'L', m = q, n = q, ldA = p+q, | |
| offsetA = m) | |
| lapack.lacpy(z, bz21, m = p, n = q, ldA = p+q, offsetA = m+q) | |
| lapack.lacpy(z, bz22, uplo = 'L', m = p, n = p, ldA = p+q, | |
| offsetA = m + (p+q+1)*q) | |
| # zl := D^-1 * zl | |
| # = D^-1 * bzl | |
| blas.tbmv(W['di'], z, n = m, k = 0, ldA = 1) | |
| # zs := r' * [ bX1, 0; 0, bX2 ] * r. | |
| # zs := [ bX1, 0; 0, bX2 ] | |
| blas.scal(0.0, z, offset = m) | |
| lapack.lacpy(x[1], z, uplo = 'L', m = q, n = q, ldB = p+q, | |
| offsetB = m) | |
| lapack.lacpy(x[2], z, uplo = 'L', m = p, n = p, ldB = p+q, | |
| offsetB = m + (p+q+1)*q) | |
| # scale diagonal of zs by 1/2 | |
| blas.scal(0.5, z, inc = p+q+1, offset = m) | |
| # a := tril(zs)*r | |
| blas.copy(r, a) | |
| blas.trmm(z, a, side = 'L', m = p+q, n = p+q, ldA = p+q, ldB = | |
| p+q, offsetA = m) | |
| # zs := a'*r + r'*a | |
| blas.syr2k(r, a, z, trans = 'T', n = p+q, k = p+q, ldB = p+q, | |
| ldC = p+q, offsetC = m) | |
| # bz21 := L^-1( V2' * ((r * zs * r')_21 - bz21) * V1') | |
| # | |
| # [ bX1 0 ] | |
| # = L^-1( V2' * ((T * [ ] * T)_21 - bz21) * V1'). | |
| # [ 0 bX2 ] | |
| # a = [ r21 r22 ] * z | |
| # = [ r21 r22 ] * r' * [ bX1, 0; 0, bX2 ] * r | |
| # = [ T21 T22 ] * [ bX1, 0; 0, bX2 ] * r | |
| blas.symm(z, r, a, side = 'R', m = p, n = p+q, ldA = p+q, | |
| ldC = p+q, offsetB = q) | |
| # bz21 := -bz21 + a * [ r11, r12 ]' | |
| # = -bz21 + (T * [ bX1, 0; 0, bX2 ] * T)_21 | |
| blas.gemm(a, r, bz21, transB = 'T', m = p, n = q, k = p+q, | |
| beta = -1.0, ldA = p+q, ldC = p) | |
| # bz21 := V2' * bz21 * V1' | |
| # = V2' * (-bz21 + (T*[bX1, 0; 0, bX2]*T)_21) * V1' | |
| blas.gemm(V2, bz21, tmp, transA = 'T', m = p, n = q, k = p, | |
| ldB = p) | |
| blas.gemm(tmp, V1, bz21, transB = 'T', m = p, n = q, k = q, | |
| ldC = p) | |
| # bz21[:] := D * (I-P) * bz21[:] | |
| # = L^-1 * bz21[:] | |
| # = bXX[:] | |
| blas.copy(bz21, tmp) | |
| base.gemv(P, bz21, tmp, alpha = -1.0, beta = 1.0) | |
| base.gemv(D, tmp, bz21) | |
| # Solve H * ux = bx + Gs' * D^-1 * bzl + 2*As'(bXX). | |
| # x[0] := x[0] + Gs'*zl + 2*As'(bz21) | |
| # = bx + G' * D^-1 * bzl + 2 * As'(bXX) | |
| blas.gemv(Gs, z, x[0], trans = 'T', alpha = 1.0, beta = 1.0) | |
| blas.gemv(As, bz21, x[0], trans = 'T', alpha = 2.0, beta = 1.0) | |
| # x[0] := H \ x[0] | |
| # = ux | |
| lapack.potrs(H, x[0]) | |
| # uuzl = Gs*ux - D^-1 * bzl | |
| blas.gemv(Gs, x[0], z, alpha = 1.0, beta = -1.0) | |
| # bz21 := V2 * L^-T(-As(ux) + bz21) * V1 | |
| # = X | |
| blas.gemv(As, x[0], bz21, alpha = -1.0, beta = 1.0) | |
| blas.tbsv(DV, bz21, n = p*q, k = 0, ldA = 1) | |
| blas.copy(bz21, tmp) | |
| base.gemv(P, tmp, bz21, alpha = -1.0, beta = 1.0, trans = 'T') | |
| blas.gemm(V2, bz21, tmp) | |
| blas.gemm(tmp, V1, bz21) | |
| # zs := -zs + r' * [ 0, X'; X, 0 ] * r | |
| # = r' * [ -bX1, X'; X, -bX2 ] * r. | |
| # a := bz21 * [ r11, r12 ] | |
| # = X * [ r11, r12 ] | |
| blas.gemm(bz21, r, a, m = p, n = p+q, k = q, ldA = p, ldC = p+q) | |
| # z := -z + [ r21, r22 ]' * a + a' * [ r21, r22 ] | |
| # = rti' * uzs * rti | |
| blas.syr2k(r, a, z, trans = 'T', beta = -1.0, n = p+q, k = p, | |
| offsetA = q, offsetC = m, ldB = p+q, ldC = p+q) | |
| # uX1 = -Z11 - bzs_11 | |
| # = -(r*zs*r')_11 - bzs_11 | |
| # uX2 = -Z22 - bzs_22 | |
| # = -(r*zs*r')_22 - bzs_22 | |
| blas.copy(bz11, x[1]) | |
| blas.copy(bz22, x[2]) | |
| # scale diagonal of zs by 1/2 | |
| blas.scal(0.5, z, inc = p+q+1, offset = m) | |
| # a := r*tril(zs) | |
| blas.copy(r, a) | |
| blas.trmm(z, a, side = 'R', m = p+q, n = p+q, ldA = p+q, ldB = | |
| p+q, offsetA = m) | |
| # x[1] := -x[1] - a[:q,:] * r[:q, :]' - r[:q,:] * a[:q,:]' | |
| # = -bzs_11 - (r*zs*r')_11 | |
| blas.syr2k(a, r, x[1], n = q, alpha = -1.0, beta = -1.0) | |
| # x[2] := -x[2] - a[q:,:] * r[q:, :]' - r[q:,:] * a[q:,:]' | |
| # = -bzs_22 - (r*zs*r')_22 | |
| blas.syr2k(a, r, x[2], n = p, alpha = -1.0, beta = -1.0, | |
| offsetA = q, offsetB = q) | |
| # scale diagonal of zs by 1/2 | |
| blas.scal(2.0, z, inc = p+q+1, offset = m) | |
| return f | |
| if C: | |
| sol = solvers.coneqp(Pf, c, Gf, hh, dims, Af, kktsolver = F, | |
| xnewcopy = xnewcopy, xdot = xdot, xaxpy = xaxpy, xscal = xscal) | |
| else: | |
| sol = solvers.conelp(c, Gf, hh, dims, Af, kktsolver = F, | |
| xnewcopy = xnewcopy, xdot = xdot, xaxpy = xaxpy, xscal = xscal) | |
| if sol['status'] is 'optimal': | |
| x = sol['x'][0] | |
| z = sol['z'][:m] | |
| Z = sol['z'][m:] | |
| Z.size = (p + q, p + q) | |
| Z = -2.0 * Z[-p:, :q] | |
| elif sol['status'] is 'primal infeasible': | |
| x = None | |
| z = sol['z'][:m] | |
| Z = sol['z'][m:] | |
| Z.size = (p + q, p + q) | |
| Z = -2.0 * Z[-p:, :q] | |
| else: | |
| x, z, Z = None, None, None | |
| return {'status': sol['status'], 'x': x, 'z': z, 'Z': Z } | |
| def checksol(sol, A, B, C = None, d = None, G = None, h = None): | |
| """ | |
| Check optimality conditions | |
| C * x + G' * z + A'(Z) + d = 0 | |
| G * x <= h | |
| z >= 0, || Z || < = 1 | |
| z' * (h - G*x) = 0 | |
| tr (Z' * (A(x) + B)) = || A(x) + B ||_*. | |
| """ | |
| p, q = B.size | |
| n = A.size[1] | |
| if G is None: G = spmatrix([], [], [], (0, n)) | |
| if h is None: h = matrix(0.0, (0, 1)) | |
| m = h.size[0] | |
| if C is None: C = spmatrix(0.0, [], [], (n,n)) | |
| if d is None: d = matrix(0.0, (n, 1)) | |
| if sol['status'] is 'optimal': | |
| res = +d | |
| base.symv(C, sol['x'], res, beta = 1.0) | |
| base.gemv(G, sol['z'], res, beta = 1.0, trans = 'T') | |
| base.gemv(A, sol['Z'], res, beta = 1.0, trans = 'T') | |
| print("Dual residual: %e" %blas.nrm2(res)) | |
| if m: | |
| print("Minimum primal slack (scalar inequalities): %e" \ | |
| %min(h - G*sol['x'])) | |
| print("Minimum dual slack (scalar inequalities): %e" \ | |
| %min(sol['z'])) | |
| if p: | |
| s = matrix(0.0, (p,1)) | |
| X = matrix(A*sol['x'], (p, q)) + B | |
| lapack.gesvd(+X, s) | |
| nrmX = sum(s) | |
| lapack.gesvd(+sol['Z'], s) | |
| nrmZ = max(s) | |
| print("Norm of Z: %e" %nrmZ) | |
| print("Nuclear norm of A(x) + B: %e" %nrmX) | |
| print("Inner product of Z and A(x) + B: %e" \ | |
| %blas.dot(sol['Z'], X)) | |
| elif sol['status'] is 'primal infeasible': | |
| res = matrix(0.0, (n,1)) | |
| base.gemv(G, sol['z'], res, beta = 1.0, trans = 'T') | |
| print("Dual residual: %e" %blas.nrm2(res)) | |
| print("h' * z = %e" %blas.dot(h, sol['z'])) | |
| print("Minimum dual slack (scalar inequalities): %e" \ | |
| %min(sol['z'])) | |
| else: | |
| pass | |
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| import cvxopt | |
| import sys | |
| from nucnrm import nucnrm | |
| import numpy as np | |
| G = np.array(([[0.0]*4 for i in range(8)])) | |
| G[0, 0] = 1 | |
| G[1, 0] = -1 | |
| G[2, 1] = 1 | |
| G[3, 1] = -1 | |
| G[4, 2] = 1 | |
| G[5, 2] = -1 | |
| G[6, 3] = 1 | |
| G[7, 3] = -1 | |
| A = np.eye(4) | |
| B = np.zeros((2, 2)) | |
| h = np.array([[1., -0, 1, -0, 2, -1, 3, -2]]).transpose() | |
| res = nucnrm.nrmapp(cvxopt.matrix(A), cvxopt.matrix(B), G=cvxopt.matrix(G), h=cvxopt.matrix(h)) | |
| x = res["x"] | |
| X = x.reshape((2, 2)) | |
| print(np.linalg.matrix_rank(X, tol = 1e-5)) |
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