A primal-dual algorithm for minimizing a sum of Euclidean norms
Journal of Computational and Applied Mathematics
Computing the Shortest Network under a Fixed Topology
IEEE Transactions on Computers
An Improved Extra-Gradient Method for Minimizing a Sum of p-norms--A Variational Inequality Approach
Computational Optimization and Applications
Robust counterparts of errors-in-variables problems
Computational Statistics & Data Analysis
Upper and lower bound limit analysis of plates using FEM and second-order cone programming
Computers and Structures
An entropy regularization technique for minimizing a sum of Tchebycheff norms
Applied Numerical Mathematics
Total variation regularization in electrocardiographic mapping
LSMS/ICSEE'10 Proceedings of the 2010 international conference on Life system modeling and simulation and intelligent computing, and 2010 international conference on Intelligent computing for sustainable energy and environment: Part III
SIAM Journal on Numerical Analysis
An improved algorithm for computing Steiner minimal trees in Euclidean d-space
Discrete Optimization
| Hi-index | 0.01 |
The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primal-dual interior-point algorithm for the problem, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition. The main work at each step consists of solving a system of linear equations (the Schur complement equations). This Schur complement matrix is not symmetric, unlike in linear programming. We incorporate a Mehrotra-type predictor-corrector scheme and present some experimental results comparing several variations of the algorithm, including, as one option, explicit symmetrization of the Schur complement with a skew corrector term. We also present results obtained from a code implemented to solve large sparse problems, using a symmetrized Schur complement. This has been applied to problems arising in plastic collapse analysis, with hundreds of thousands of variables and millions of nonzeros in the constraint matrix. The algorithm typically finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable.