// Yua, Chang, Hsieh and Lin: A Comparison of Optimization Methods and Software for Large-scale L1-regularized Linear Classification; p. 14
double shoot_cdn(int x_i, double lambda) {
// Compute d: (equation 29), i.e the solution to the quadratic approximation of the function
// for weight x_i
if (!active[x_i]) return 0.0;
double violation = 0.0;
double xv = logregprob->x[x_i];
double Ld0, Ldd0;
logreg_cdn_derivandH(lambda, x_i, Ld0, Ldd0);
double Gp = (Ld0+1);
double Gn = (Ld0-1);
// Shrinking (practically copied from LibLinear)
if (xv == 0) {
if (Gp<0) {
violation = -Gp;
} else if (Gn > 0) {
violation = Gn;
} else if(Gp>Gmax_old/logregprob->ny && Gn<-Gmax_old/logregprob->ny) {
// Remove
active[x_i] = false;
return 0.0;
}
} else if(xv > 0)
violation = fabs(Gp);
else
violation = fabs(Gn);
// TODO: should use a reduction here! Or lock.
//if (Gmax < violation)
// Gmax = violation;
logregprob->Gmax.max(0, violation);
// Newton direction d
double rhs = Ldd0*xv;
double d;
if (Gp<= rhs) {
d = -(Gp/Ldd0);
} else if (Gn >= rhs) {
d = -(Gn/Ldd0);
} else {
d = -xv;
}
if (std::abs(d) < 1e-12) {
return 0.0;
}
// Small optimization
d = std::min(std::max(d,-10.0),10.0);
// Backtracking line search (with Armijo-type of condition) (eq. 30)
int iter=0;
int max_num_linesearch=20;
double gamma = 1.0; // Note: Yuan, Chang et al. use lambda instead of gamma
double delta = (Ld0 * d + std::abs(xv+d) - std::abs(xv));
double rhs_c = cdn_sigma * delta;
do {
double change_in_obj = g_xi(d, x_i, lambda);
if (change_in_obj <= gamma * rhs_c) {
// Found ok.
logregprob->x[x_i] += d;
// Update dot products (Ax)
sparse_array &col = logregprob->A_cols[x_i];
#pragma omp parallel for
for(int i=0; i<col.length(); i++) {
logregprob->expAx.mul(col.idxs[i], exp(d * col.values[i]));
}
return std::abs(d);
}
gamma *= 0.5;
d *= 0.5;
} while(++iter < max_num_linesearch);
recompute_expAx();
return 0.0;
}
// Calculate local stiffness matrix. Use reduced integration with hourglass stabilization.
void ElementQuadrangleLin::set_K_isoparametric()
{
double shear = E / (2 + 2 * nu);
double lame = E * nu / ((1 + nu) * (1 - 2 * nu));
Eigen::Matrix3d C;
C << 2 * shear + lame, lame, 0.,
lame, 2 * shear + lame, 0.,
0., 0., shear;
Eigen::Vector4d x, y;
int i;
for (i = 0; i < 4; i++)
{
x(i) = domain->nodes[nodes[i] - 1].x;
y(i) = domain->nodes[nodes[i] - 1].y;
}
const double gp = sqrt(1. / 3.);
double gps[5][2] = { {-gp,-gp},{ gp,-gp },{ gp,gp },{ -gp,gp },{0.,0.} };
Eigen::Vector4d g_xi(-0.25, 0.25, 0.25, -0.25);
Eigen::Vector4d g_eta(-0.25, -0.25, 0.25, 0.25);
Eigen::Vector4d h(0.25, -0.25, 0.25, -0.25);
Eigen::Matrix2d J[5];
Eigen::Matrix2d J_inv[5];
for (i = 0; i < 5; i++)
{
double xi = gps[i][0];
double eta = gps[i][1];
J[i](0, 0) = x.dot(g_xi + eta*h);
J[i](0, 1) = x.dot(g_eta + xi*h);
J[i](1, 0) = y.dot(g_xi + eta*h);
J[i](1, 1) = y.dot(g_eta + xi*h);
J_inv[i] = J[i].inverse();
}
Eigen::MatrixXd xieta(4, 2);
xieta << g_xi, g_eta;
Eigen::MatrixXd b(4, 2);
b = xieta*J_inv[4];
Eigen::MatrixXd B_0T(8,3);
Eigen::Vector4d zers = Eigen::Vector4d::Zero();
B_0T << b.block<4,1>(0,0), zers, b.block<4, 1>(0, 1), zers, b.block<4, 1>(0, 1), b.block<4, 1>(0, 0);
Eigen::MatrixXd xy(4,2);
xy << x,y;
Eigen::Matrix4d m_gamma;
Eigen::Vector4d gamma;
m_gamma = Eigen::Matrix4d::Identity() - b*xy.transpose();
gamma = m_gamma * h;
Eigen::MatrixXd j_0_dev(3, 4);
Eigen::Matrix2d j0iT = J_inv[4].transpose();
j_0_dev << 2 * j0iT.block<1, 2>(0, 0), -1 * j0iT.block<1, 2>(1, 0),
-1 * j0iT.block<1, 2>(0, 0), 2 * j0iT.block<1, 2>(1, 0),
3 * j0iT.block<1, 2>(0, 0), 3 * j0iT.block<1, 2>(1, 0);
j_0_dev *= 1. / 3.;
Eigen::MatrixXd L_hg[4];
for (i = 0; i < 4; i++)
{
double xi = gps[i][0];
double eta = gps[i][1];
L_hg[i].resize(4,2);
L_hg[i] << eta, 0.0,
xi, 0.0,
0.0, eta,
0.0, xi;
}
Eigen::MatrixXd M_hg(8,2);
M_hg << gamma, zers, zers, gamma;
M_hg.transposeInPlace(); // 2x8
Eigen::MatrixXd B_red[4];
Eigen::MatrixXd K_red = Eigen::MatrixXd::Zero(8,8);
Eigen::MatrixXd B[4];
Eigen::MatrixXd Bc[4];
volume = 4 * J[4].determinant()*thickness;
for (i = 0; i < 4; i++)
{
B_red[i].resize(3, 8);
B[i].resize(3, 8);
Bc[i].resize(3, 8);
B_red[i] = j_0_dev * L_hg[i] * M_hg; // 3x4 * 4x2 * 2x8 = 3x8
K_red += B_red[i].transpose()*C*B_red[i]; // 8x3 * 3x3 * 3x8 = 8x8
Bc[i] = B_0T.transpose() + B_red[i]; // 3x8 + 3x8 = 3x8
B[i] << Bc[i].block<3, 1>(0, 0), Bc[i].block<3, 1>(0, 4), Bc[i].block<3, 1>(0, 1), Bc[i].block<3, 1>(0, 5), Bc[i].block<3, 1>(0, 2), Bc[i].block<3, 1>(0, 6), Bc[i].block<3, 1>(0, 3), Bc[i].block<3, 1>(0, 7); // 3x8
}
K_red *= volume / 4.0;
Eigen::MatrixXd K(8, 8);
Eigen::MatrixXd Kc(8, 8);
Eigen::MatrixXd Kc2(8, 8);
Kc = K_red + volume*B_0T*C*B_0T.transpose(); // 8x8 + 8x3 * 3x3 * 3x8
Kc2 << Kc.block<8, 1>(0, 0), Kc.block<8, 1>(0, 4), Kc.block<8, 1>(0, 1), Kc.block<8, 1>(0, 5), Kc.block<8, 1>(0, 2), Kc.block<8, 1>(0, 6), Kc.block<8, 1>(0, 3), Kc.block<8, 1>(0, 7);
K << Kc2.block<1, 8>(0, 0), Kc2.block<1, 8>(4, 0), Kc2.block<1, 8>(1, 0), Kc2.block<1, 8>(5, 0), Kc2.block<1, 8>(2, 0), Kc2.block<1, 8>(6, 0), Kc2.block<1, 8>(3, 0), Kc2.block<1, 8>(7, 0);
// Don't forget to swap rows and/or columns of B and K matrix and store it in the object.
K_loc = K;
B_matrices = B;
}