// Kinetic energy integral
double kinetic(int N, Eigen::MatrixXd& A, Eigen::MatrixXd& A_p,
Eigen::MatrixXd& A_dp, Eigen::MatrixXd& A_dp_inv,
Eigen::MatrixXd& k, Eigen::MatrixXd& k_p,
Eigen::MatrixXd& k_dp, double S, Eigen::MatrixXd& L2)
{
// Calculate the T factor
// Start with A''^{-1} A' L2 A A''^{-1}
Eigen::MatrixXd tempMat = A_dp_inv*A_p*L2*A*A_dp_inv;
// Determine k''. A''^{-1} A' L2 A A''^{-1} . k''
double T = 0.0;
for (int i = 0; i < N; i++){
for (int j = 0; j < N; j++){
double temp = k_dp(i, 0) * k_dp(j, 0) + k_dp(i, 1) * k_dp(j, 1);
temp += k_dp(i, 2) * k_dp(j, 2);
T += tempMat(i, j) * temp;
}
}
// Then k . L2 . k'
for (int i = 0; i < N; i++){
T += k(i, 0) * L2(i, i) * k_p(i, 0);
T += k(i, 1) * L2(i, i) * k_p(i, 1);
T += k(i, 2) * L2(i, i) * k_p(i, 2);
}
// And -k'.A L2 A''^{-1}.k''
tempMat = A*L2*A_dp_inv;
for (int i = 0; i < N; i++){
for (int j = 0; j < N; j++){
double temp = k_p(i, 0) * k_dp(j, 0);
temp += k_p(i, 1) * k_dp(j, 1) + k_p(i, 2) * k_dp(j, 2);
T -= tempMat(i, j) * temp;
}
}
// And -k''.A''^-1 L2 A'.k
tempMat = A_dp_inv * L2 * A_p;
for (int i = 0; i < N; i++){
for (int j = 0; j < N; j++){
double temp = k_dp(i, 0) * k(j, 0);
temp += k_dp(i, 1) * k(j, 1) + k_dp(i, 2) * k(j, 2);
T -= tempMat(i, j) * temp;
}
}
// This just leaves 6Tr{A A''^-1 A' L2}
tempMat = A * A_dp_inv * A_p*L2;
T += 6.0*tempMat.trace();
T = 0.5*T*S;
return T;
}
double weight_unsymkl_gauss(PCObject &o1, PCObject &o2)
{
int dim = o1.gaussian.dim;
Eigen::MatrixXd multicov = Eigen::MatrixXd(3,3);
multicov = o2.gaussian.cov_inverse * o1.gaussian.covariance;
Eigen::VectorXd mean = o2.gaussian.mean-o1.gaussian.mean;
double unsymkl_12 = (multicov.trace()
+ mean.transpose()*o2.gaussian.cov_inverse*mean
+ log(o1.gaussian.cov_determinant/o2.gaussian.cov_determinant)-dim) / 2.;
// cout<<"kl: "<<unsymkl_12<<endl;
return unsymkl_12;
}
IGL_INLINE bool igl::copyleft::quadprog(
const Eigen::MatrixXd & G,
const Eigen::VectorXd & g0,
const Eigen::MatrixXd & CE,
const Eigen::VectorXd & ce0,
const Eigen::MatrixXd & CI,
const Eigen::VectorXd & ci0,
Eigen::VectorXd& x)
{
using namespace Eigen;
typedef double Scalar;
const auto distance = [](Scalar a, Scalar b)->Scalar
{
Scalar a1, b1, t;
a1 = std::abs(a);
b1 = std::abs(b);
if (a1 > b1)
{
t = (b1 / a1);
return a1 * std::sqrt(1.0 + t * t);
}
else
if (b1 > a1)
{
t = (a1 / b1);
return b1 * std::sqrt(1.0 + t * t);
}
return a1 * std::sqrt(2.0);
};
const auto compute_d = [](VectorXd &d, const MatrixXd& J, const VectorXd& np)
{
d = J.adjoint() * np;
};
const auto update_z =
[](VectorXd& z, const MatrixXd& J, const VectorXd& d, int iq)
{
z = J.rightCols(z.size()-iq) * d.tail(d.size()-iq);
};
const auto update_r =
[](const MatrixXd& R, VectorXd& r, const VectorXd& d, int iq)
{
r.head(iq) =
R.topLeftCorner(iq,iq).triangularView<Upper>().solve(d.head(iq));
};
const auto add_constraint = [&distance](
MatrixXd& R,
MatrixXd& J,
VectorXd& d,
int& iq,
double& R_norm)->bool
{
int n=J.rows();
#ifdef TRACE_SOLVER
std::cerr << "Add constraint " << iq << '/';
#endif
int i, j, k;
double cc, ss, h, t1, t2, xny;
/* we have to find the Givens rotation which will reduce the element
d(j) to zero.
if it is already zero we don't have to do anything, except of
decreasing j */
for (j = n - 1; j >= iq + 1; j--)
{
/* The Givens rotation is done with the matrix (cc cs, cs -cc).
If cc is one, then element (j) of d is zero compared with element
(j - 1). Hence we don't have to do anything.
If cc is zero, then we just have to switch column (j) and column (j - 1)
of J. Since we only switch columns in J, we have to be careful how we
update d depending on the sign of gs.
Otherwise we have to apply the Givens rotation to these columns.
The i - 1 element of d has to be updated to h. */
cc = d(j - 1);
ss = d(j);
h = distance(cc, ss);
if (h == 0.0)
continue;
d(j) = 0.0;
ss = ss / h;
cc = cc / h;
if (cc < 0.0)
{
cc = -cc;
ss = -ss;
d(j - 1) = -h;
}
else
d(j - 1) = h;
xny = ss / (1.0 + cc);
for (k = 0; k < n; k++)
{
t1 = J(k,j - 1);
t2 = J(k,j);
J(k,j - 1) = t1 * cc + t2 * ss;
J(k,j) = xny * (t1 + J(k,j - 1)) - t2;
}
}
/* update the number of constraints added*/
iq++;
/* To update R we have to put the iq components of the d vector
into column iq - 1 of R
*/
R.col(iq-1).head(iq) = d.head(iq);
#ifdef TRACE_SOLVER
std::cerr << iq << std::endl;
#endif
if (std::abs(d(iq - 1)) <= std::numeric_limits<double>::epsilon() * R_norm)
{
// problem degenerate
return false;
}
R_norm = std::max<double>(R_norm, std::abs(d(iq - 1)));
return true;
};
const auto delete_constraint = [&distance](
MatrixXd& R,
MatrixXd& J,
VectorXi& A,
VectorXd& u,
int p,
int& iq,
int l)
{
int n = R.rows();
#ifdef TRACE_SOLVER
std::cerr << "Delete constraint " << l << ' ' << iq;
#endif
int i, j, k, qq;
double cc, ss, h, xny, t1, t2;
/* Find the index qq for active constraint l to be removed */
for (i = p; i < iq; i++)
if (A(i) == l)
{
qq = i;
break;
}
/* remove the constraint from the active set and the duals */
for (i = qq; i < iq - 1; i++)
{
A(i) = A(i + 1);
u(i) = u(i + 1);
R.col(i) = R.col(i+1);
}
A(iq - 1) = A(iq);
u(iq - 1) = u(iq);
A(iq) = 0;
u(iq) = 0.0;
for (j = 0; j < iq; j++)
R(j,iq - 1) = 0.0;
/* constraint has been fully removed */
iq--;
#ifdef TRACE_SOLVER
std::cerr << '/' << iq << std::endl;
#endif
if (iq == 0)
return;
for (j = qq; j < iq; j++)
{
cc = R(j,j);
ss = R(j + 1,j);
h = distance(cc, ss);
if (h == 0.0)
continue;
cc = cc / h;
ss = ss / h;
R(j + 1,j) = 0.0;
if (cc < 0.0)
{
R(j,j) = -h;
cc = -cc;
ss = -ss;
}
else
R(j,j) = h;
xny = ss / (1.0 + cc);
for (k = j + 1; k < iq; k++)
{
t1 = R(j,k);
t2 = R(j + 1,k);
R(j,k) = t1 * cc + t2 * ss;
R(j + 1,k) = xny * (t1 + R(j,k)) - t2;
}
for (k = 0; k < n; k++)
{
t1 = J(k,j);
t2 = J(k,j + 1);
J(k,j) = t1 * cc + t2 * ss;
J(k,j + 1) = xny * (J(k,j) + t1) - t2;
}
}
};
int i, j, k, l; /* indices */
int ip, me, mi;
int n=g0.size(); int p=ce0.size(); int m=ci0.size();
MatrixXd R(G.rows(),G.cols()), J(G.rows(),G.cols());
LLT<MatrixXd,Lower> chol(G.cols());
VectorXd s(m+p), z(n), r(m + p), d(n), np(n), u(m + p);
VectorXd x_old(n), u_old(m + p);
double f_value, psi, c1, c2, sum, ss, R_norm;
const double inf = std::numeric_limits<double>::infinity();
double t, t1, t2; /* t is the step length, which is the minimum of the partial step length t1
* and the full step length t2 */
VectorXi A(m + p), A_old(m + p), iai(m + p);
int q;
int iq, iter = 0;
std::vector<bool> iaexcl(m + p);
me = p; /* number of equality constraints */
mi = m; /* number of inequality constraints */
q = 0; /* size of the active set A (containing the indices of the active constraints) */
/*
* Preprocessing phase
*/
/* compute the trace of the original matrix G */
c1 = G.trace();
/* decompose the matrix G in the form LL^T */
chol.compute(G);
/* initialize the matrix R */
d.setZero();
R.setZero();
R_norm = 1.0; /* this variable will hold the norm of the matrix R */
/* compute the inverse of the factorized matrix G^-1, this is the initial value for H */
// J = L^-T
J.setIdentity();
J = chol.matrixU().solve(J);
c2 = J.trace();
#ifdef TRACE_SOLVER
print_matrix("J", J, n);
#endif
/* c1 * c2 is an estimate for cond(G) */
/*
* Find the unconstrained minimizer of the quadratic form 0.5 * x G x + g0 x
* this is a feasible point in the dual space
* x = G^-1 * g0
*/
x = chol.solve(g0);
x = -x;
/* and compute the current solution value */
f_value = 0.5 * g0.dot(x);
#ifdef TRACE_SOLVER
std::cerr << "Unconstrained solution: " << f_value << std::endl;
print_vector("x", x, n);
#endif
/* Add equality constraints to the working set A */
iq = 0;
for (i = 0; i < me; i++)
{
np = CE.col(i);
compute_d(d, J, np);
update_z(z, J, d, iq);
update_r(R, r, d, iq);
#ifdef TRACE_SOLVER
print_matrix("R", R, iq);
print_vector("z", z, n);
print_vector("r", r, iq);
print_vector("d", d, n);
#endif
/* compute full step length t2: i.e., the minimum step in primal space s.t. the contraint
becomes feasible */
t2 = 0.0;
if (std::abs(z.dot(z)) > std::numeric_limits<double>::epsilon()) // i.e. z != 0
t2 = (-np.dot(x) - ce0(i)) / z.dot(np);
x += t2 * z;
/* set u = u+ */
u(iq) = t2;
u.head(iq) -= t2 * r.head(iq);
/* compute the new solution value */
f_value += 0.5 * (t2 * t2) * z.dot(np);
A(i) = -i - 1;
if (!add_constraint(R, J, d, iq, R_norm))
{
// FIXME: it should raise an error
// Equality constraints are linearly dependent
return false;
}
}
/* set iai = K \ A */
for (i = 0; i < mi; i++)
iai(i) = i;
l1: iter++;
#ifdef TRACE_SOLVER
print_vector("x", x, n);
#endif
/* step 1: choose a violated constraint */
for (i = me; i < iq; i++)
{
ip = A(i);
iai(ip) = -1;
}
/* compute s(x) = ci^T * x + ci0 for all elements of K \ A */
ss = 0.0;
psi = 0.0; /* this value will contain the sum of all infeasibilities */
ip = 0; /* ip will be the index of the chosen violated constraint */
for (i = 0; i < mi; i++)
{
iaexcl[i] = true;
sum = CI.col(i).dot(x) + ci0(i);
s(i) = sum;
psi += std::min(0.0, sum);
}
#ifdef TRACE_SOLVER
print_vector("s", s, mi);
#endif
if (std::abs(psi) <= mi * std::numeric_limits<double>::epsilon() * c1 * c2* 100.0)
{
/* numerically there are not infeasibilities anymore */
q = iq;
return true;
}
/* save old values for u, x and A */
u_old.head(iq) = u.head(iq);
A_old.head(iq) = A.head(iq);
x_old = x;
l2: /* Step 2: check for feasibility and determine a new S-pair */
for (i = 0; i < mi; i++)
{
if (s(i) < ss && iai(i) != -1 && iaexcl[i])
{
ss = s(i);
ip = i;
}
}
if (ss >= 0.0)
{
q = iq;
return true;
}
/* set np = n(ip) */
np = CI.col(ip);
/* set u = (u 0)^T */
u(iq) = 0.0;
/* add ip to the active set A */
A(iq) = ip;
#ifdef TRACE_SOLVER
std::cerr << "Trying with constraint " << ip << std::endl;
print_vector("np", np, n);
#endif
l2a:/* Step 2a: determine step direction */
/* compute z = H np: the step direction in the primal space (through J, see the paper) */
compute_d(d, J, np);
update_z(z, J, d, iq);
/* compute N* np (if q > 0): the negative of the step direction in the dual space */
update_r(R, r, d, iq);
#ifdef TRACE_SOLVER
std::cerr << "Step direction z" << std::endl;
print_vector("z", z, n);
print_vector("r", r, iq + 1);
print_vector("u", u, iq + 1);
print_vector("d", d, n);
print_ivector("A", A, iq + 1);
#endif
/* Step 2b: compute step length */
l = 0;
/* Compute t1: partial step length (maximum step in dual space without violating dual feasibility */
t1 = inf; /* +inf */
/* find the index l s.t. it reaches the minimum of u+(x) / r */
for (k = me; k < iq; k++)
{
double tmp;
if (r(k) > 0.0 && ((tmp = u(k) / r(k)) < t1) )
{
t1 = tmp;
l = A(k);
}
}
/* Compute t2: full step length (minimum step in primal space such that the constraint ip becomes feasible */
if (std::abs(z.dot(z)) > std::numeric_limits<double>::epsilon()) // i.e. z != 0
t2 = -s(ip) / z.dot(np);
else
t2 = inf; /* +inf */
/* the step is chosen as the minimum of t1 and t2 */
t = std::min(t1, t2);
#ifdef TRACE_SOLVER
std::cerr << "Step sizes: " << t << " (t1 = " << t1 << ", t2 = " << t2 << ") ";
#endif
/* Step 2c: determine new S-pair and take step: */
/* case (i): no step in primal or dual space */
if (t >= inf)
{
/* QPP is infeasible */
// FIXME: unbounded to raise
q = iq;
return false;
}
/* case (ii): step in dual space */
if (t2 >= inf)
{
/* set u = u + t * [-r 1) and drop constraint l from the active set A */
u.head(iq) -= t * r.head(iq);
u(iq) += t;
iai(l) = l;
delete_constraint(R, J, A, u, p, iq, l);
#ifdef TRACE_SOLVER
std::cerr << " in dual space: "
<< f_value << std::endl;
print_vector("x", x, n);
print_vector("z", z, n);
print_ivector("A", A, iq + 1);
#endif
goto l2a;
}
/* case (iii): step in primal and dual space */
x += t * z;
/* update the solution value */
f_value += t * z.dot(np) * (0.5 * t + u(iq));
u.head(iq) -= t * r.head(iq);
u(iq) += t;
#ifdef TRACE_SOLVER
std::cerr << " in both spaces: "
<< f_value << std::endl;
print_vector("x", x, n);
print_vector("u", u, iq + 1);
print_vector("r", r, iq + 1);
print_ivector("A", A, iq + 1);
#endif
if (t == t2)
{
#ifdef TRACE_SOLVER
std::cerr << "Full step has taken " << t << std::endl;
print_vector("x", x, n);
#endif
/* full step has taken */
/* add constraint ip to the active set*/
if (!add_constraint(R, J, d, iq, R_norm))
{
iaexcl[ip] = false;
delete_constraint(R, J, A, u, p, iq, ip);
#ifdef TRACE_SOLVER
print_matrix("R", R, n);
print_ivector("A", A, iq);
#endif
for (i = 0; i < m; i++)
iai(i) = i;
for (i = 0; i < iq; i++)
{
A(i) = A_old(i);
iai(A(i)) = -1;
u(i) = u_old(i);
}
x = x_old;
goto l2; /* go to step 2 */
}
else
iai(ip) = -1;
#ifdef TRACE_SOLVER
print_matrix("R", R, n);
print_ivector("A", A, iq);
#endif
goto l1;
}
/* a patial step has taken */
#ifdef TRACE_SOLVER
std::cerr << "Partial step has taken " << t << std::endl;
print_vector("x", x, n);
#endif
/* drop constraint l */
iai(l) = l;
delete_constraint(R, J, A, u, p, iq, l);
#ifdef TRACE_SOLVER
print_matrix("R", R, n);
print_ivector("A", A, iq);
#endif
s(ip) = CI.col(ip).dot(x) + ci0(ip);
#ifdef TRACE_SOLVER
print_vector("s", s, mi);
#endif
goto l2a;
}