/**
* Prepare an LRSDP object to solve the Lovasz-Theta SDP in the manner detailed
* in Monteiro + Burer 2004. The list of edges in the graph must be given; that
* is all that is necessary to set up the problem. A matrix which will contain
* initial point coordinates should be given also.
*/
void SetupLovaszTheta(const arma::mat& edges,
LRSDP<SDP<arma::mat>>& lovasz)
{
// Get the number of vertices in the problem.
const size_t vertices = max(max(edges)) + 1;
// C = -(e e^T) = -ones().
lovasz.SDP().C().ones(vertices, vertices);
lovasz.SDP().C() *= -1;
// b_0 = 1; else = 0.
lovasz.SDP().SparseB().zeros(edges.n_cols + 1);
lovasz.SDP().SparseB()[0] = 1;
// A_0 = I_n.
lovasz.SDP().SparseA()[0].eye(vertices, vertices);
// A_ij only has ones at (i, j) and (j, i) and 0 elsewhere.
for (size_t i = 0; i < edges.n_cols; ++i)
{
lovasz.SDP().SparseA()[i + 1].zeros(vertices, vertices);
lovasz.SDP().SparseA()[i + 1](edges(0, i), edges(1, i)) = 1.;
lovasz.SDP().SparseA()[i + 1](edges(1, i), edges(0, i)) = 1.;
}
// Set the Lagrange multipliers right.
lovasz.AugLag().Lambda().ones(edges.n_cols + 1);
lovasz.AugLag().Lambda() *= -1;
lovasz.AugLag().Lambda()[0] = -double(vertices);
}
/**
* Prepare an LRSDP object to solve the Lovasz-Theta SDP in the manner detailed
* in Monteiro + Burer 2004. The list of edges in the graph must be given; that
* is all that is necessary to set up the problem. A matrix which will contain
* initial point coordinates should be given also.
*/
void setupLovaszTheta(const arma::mat& edges,
LRSDP& lovasz)
{
// Get the number of vertices in the problem.
const size_t vertices = max(max(edges)) + 1;
// C = -(e e^T) = -ones().
lovasz.C().ones(vertices, vertices);
lovasz.C() *= -1;
// b_0 = 1; else = 0.
lovasz.B().zeros(edges.n_cols);
lovasz.B()[0] = 1;
// All of the matrices will just contain coordinates because they are
// super-sparse (two entries each). Except for A_0, which is I_n.
lovasz.AModes().ones();
lovasz.AModes()[0] = 0;
// A_0 = I_n.
lovasz.A()[0].eye(vertices, vertices);
// A_ij only has ones at (i, j) and (j, i) and 1 elsewhere.
for (size_t i = 0; i < edges.n_cols; ++i)
{
arma::mat a(3, 2);
a(0, 0) = edges(0, i);
a(1, 0) = edges(1, i);
a(2, 0) = 1;
a(0, 1) = edges(1, i);
a(1, 1) = edges(0, i);
a(2, 1) = 1;
lovasz.A()[i + 1] = a;
}
// Set the Lagrange multipliers right.
lovasz.AugLag().Lambda().ones(edges.n_cols);
lovasz.AugLag().Lambda() *= -1;
lovasz.AugLag().Lambda()[0] = -double(vertices);
}