def P(u, v, alpha=1.0, beta=0.0): """ v := alpha * [ Qc^-1, 0, 0; 0, 0, 0; 0, 0, 0 ] * u + beta * v """ v *= beta base.symv(Qinv, u, v, alpha=alpha, beta=1.0)
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
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
def softmargin_completion(Q, d, gamma): """ Solves the QP minimize (1/2)*y'*Qc^{-1}*y + gamma*sum(v) subject to diag(d)*(y + b*ones) + v >= 1 v >= 0 (with variables y, b, v) and its dual, the 'soft-margin' SVM problem, maximize -(1/2)*z'*Qc*z + d'*z subject to 0 <= diag(d)*z <= gamma*ones sum(z) = 0 (with variables z). Qc is the max determinant completion of Q. Input arguments. Q is a sparse N x N sparse matrix with chordal sparsity pattern and a positive definite completion d is an N-vector of labels -1 or 1. gamma is a positive parameter. F is the chompack pattern corresponding to Q. If F is None, the pattern is computed. Output. z, y, b, v, optval, L, iters """ if verbose: solvers.options['show_progress'] = True else: solvers.options['show_progress'] = False N = Q.size[0] p = chompack.maxcardsearch(Q) symb = chompack.symbolic(Q, p) Qc = chompack.cspmatrix(symb) + Q # Qinv is the inverse of the max. determinant p.d. completion of Q Lc = Qc.copy() chompack.completion(Lc) Qinv = Lc.copy() chompack.llt(Qinv) Qinv = Qinv.spmatrix(reordered=False) Qinv = chompack.symmetrize(Qinv) def P(u, v, alpha=1.0, beta=0.0): """ v := alpha * [ Qc^-1, 0, 0; 0, 0, 0; 0, 0, 0 ] * u + beta * v """ v *= beta base.symv(Qinv, u, v, alpha=alpha, beta=1.0) def G(u, v, alpha=1.0, beta=0.0, trans='N'): """ If trans is 'N': v := alpha * [ -diag(d), -d, -I; 0, 0, -I ] * u + beta * v. If trans is 'T': v := alpha * [ -diag(d), 0; -d', 0; -I, -I ] * u + beta * v. """ v *= beta if trans is 'N': v[:N] -= alpha * (base.mul(d, u[:N] + u[N]) + u[-N:]) v[-N:] -= alpha * u[-N:] else: v[:N] -= alpha * base.mul(d, u[:N]) v[N] -= alpha * blas.dot(d, u, n=N) v[-N:] -= alpha * (u[:N] + u[N:]) K = spmatrix(0.0, Qinv.I, Qinv.J) dy1, dy2 = matrix(0.0, (N, 1)), matrix(0.0, (N, 1)) def Fkkt(W): """ Custom KKT solver for [ Qinv 0 0 -D 0 ] [ ux_y ] [ bx_y ] [ 0 0 0 -d' 0 ] [ ux_b ] [ bx_b ] [ 0 0 0 -I -I ] [ ux_v ] = [ bx_v ] [ -D -d -I -D1 0 ] [ uz_z ] [ bz_z ] [ 0 0 -I 0 -D2 ] [ uz_w ] [ bz_w ] with D1 = diag(d1), D2 = diag(d2), d1 = W['d'][:N]**2, d2 = W['d'][N:])**2. """ d1, d2 = W['d'][:N]**2, W['d'][N:]**2 d3, d4 = (d1 + d2)**-1, (d1**-1 + d2**-1)**-1 # Factor the chordal matrix K = Qinv + (D_1+D_2)^-1. K.V = Qinv.V K[::N + 1] = K[::N + 1] + d3 L = chompack.cspmatrix(symb) + K chompack.cholesky(L) # Solve (Qinv + (D1+D2)^-1) * dy2 = (D1 + D2)^{-1} * 1 blas.copy(d3, dy2) chompack.trsm(L, dy2, trans='N') chompack.trsm(L, dy2, trans='T') def g(x, y, z): # Solve # # [ K d3 ] [ ux_y ] # [ ] [ ] = # [ d3' 1'*d3 ] [ ux_b ] # # [ bx_y ] [ D ] # [ ] - [ ] * D3 * (D2 * bx_v + bx_z - bx_w). # [ bx_b ] [ d' ] x[:N] -= mul(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:])) x[N] -= blas.dot(d, mul(d3, mul(d2, x[-N:]) + z[:N] - z[-N:])) # Solve dy1 := K^-1 * x[:N] blas.copy(x, dy1, n=N) chompack.trsm(L, dy1, trans='N') chompack.trsm(L, dy1, trans='T') # Find ux_y = dy1 - ux_b * dy2 s.t # # d3' * ( dy1 - ux_b * dy2 + ux_b ) = x[N] # # i.e. x[N] := ( x[N] - d3'* dy1 ) / ( d3'* ( 1 - dy2 ) ). x[N] = ( x[N] - blas.dot(d3, dy1) ) / \ ( blas.asum(d3) - blas.dot(d3, dy2) ) x[:N] = dy1 - x[N] * dy2 # ux_v = D4 * ( bx_v - D1^-1 (bz_z + D * (ux_y + ux_b)) # - D2^-1 * bz_w ) x[-N:] = mul( d4, x[-N:] - div(z[:N] + mul(d, x[:N] + x[N]), d1) - div(z[N:], d2)) # uz_z = - D1^-1 * ( bx_z - D * ( ux_y + ux_b ) - ux_v ) # uz_w = - D2^-1 * ( bx_w - uz_w ) z[:N] += base.mul(d, x[:N] + x[N]) + x[-N:] z[-N:] += x[-N:] blas.scal(-1.0, z) # Return W['di'] * uz blas.tbmv(W['di'], z, n=2 * N, k=0, ldA=1) return g q = matrix(0.0, (2 * N + 1, 1)) if weights is 'proportional': dlist = list(d) C1 = 0.5 * N * gamma / dlist.count(1) C2 = 0.5 * N * gamma / dlist.count(-1) gvec = matrix([C1 if w == 1 else C2 for w in dlist], (N, 1)) del dlist q[-N:] = gvec elif weights is 'equal': q[-N:] = gamma h = matrix(0.0, (2 * N, 1)) h[:N] = -1.0 sol = solvers.coneqp(P, q, G, h, kktsolver=Fkkt) u = matrix(0.0, (N, 1)) y, b, v = sol['x'][:N], sol['x'][N], sol['x'][N + 1:] z = mul(d, sol['z'][:N]) base.symv(Qinv, y, u) optval = 0.5 * blas.dot(y, u) + gamma * sum(v) return y, b, v, z, optval, Lc, sol['iterations']
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])
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])
def fP(x, y, alpha=1.0, beta=0.0): base.symv(P, x, y, alpha=alpha, beta=beta)