This function is the Lovasz-Theta semidefinite program, as implemented in the following paper: More...

Public Member Functions
	LovaszThetaSDP ()

	LovaszThetaSDP (const arma::mat &edges)
	Initialize the Lovasz-Theta SDP with the given set of edges. More...

const arma::mat &	Edges () const

arma::mat &	Edges ()

double	Evaluate (const arma::mat &coordinates)

double	EvaluateConstraint (const size_t index, const arma::mat &coordinates)

const arma::mat &	GetInitialPoint ()

void	Gradient (const arma::mat &coordinates, arma::mat &gradient)

void	GradientConstraint (const size_t index, const arma::mat &coordinates, arma::mat &gradient)

size_t	NumConstraints () const

Private Attributes
arma::mat	edges

arma::mat	initialPoint

size_t	vertices

Detailed Description

This function is the Lovasz-Theta semidefinite program, as implemented in the following paper:

S. Burer, R. Monteiro "A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization." Journal of Mathematical Programming, 2004

Given a simple, undirected graph G = (V, E), the Lovasz-Theta SDP is defined by:

min_X{Tr(-(e e^T)^T X) : Tr(X) = 1, X_ij = 0 for all (i, j) in E, X >= 0}

where e is the vector of all ones and X has dimension |V| x |V|.

In the Monteiro-Burer formulation, we take X = R * R^T, where R is the coordinates given to the Evaluate(), Gradient(), EvaluateConstraint(), and GradientConstraint() functions.

Definition at line 103 of file aug_lagrangian_test_functions.hpp.

Constructor & Destructor Documentation

mlpack::optimization::LovaszThetaSDP::LovaszThetaSDP ( )

mlpack::optimization::LovaszThetaSDP::LovaszThetaSDP ( const arma::mat & edges )

Initialize the Lovasz-Theta SDP with the given set of edges.

The edge matrix should consist of rows of two dimensions, where dimension 0 is the first vertex of the edge and dimension 1 is the second edge (or vice versa, as it doesn't make a difference).

Parameters

edges Matrix of edges.