12 research outputs found
Total variation based community detection using a nonlinear optimization approach
Maximizing the modularity of a network is a successful tool to identify an
important community of nodes. However, this combinatorial optimization problem
is known to be NP-complete. Inspired by recent nonlinear modularity eigenvector
approaches, we introduce the modularity total variation and show that
its box-constrained global maximum coincides with the maximum of the original
discrete modularity function. Thus we describe a new nonlinear optimization
approach to solve the equivalent problem leading to a community detection
strategy based on . The proposed approach relies on the use of a fast
first-order method that embeds a tailored active-set strategy. We report
extensive numerical comparisons with standard matrix-based approaches and the
Generalized RatioDCA approach for nonlinear modularity eigenvectors, showing
that our new method compares favourably with state-of-the-art alternatives
Second-order negative-curvature methods for box-constrained and general constrained optimization
A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited
Second-order negative-curvature methods for box-constrained and general constrained optimization
A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited