From sparsity to block-sparsity: direct solution of linear systems of dimension 10^9

20 October 2005
Prof Jacek Gondzio

We discuss a method for solving very large structured symmetric indefinite equation systems arising in optimization with interior point methods.

Many real-life economic models involve system dynamics, spatial distribution or uncertainty and lead to large-scale optimization problems. Such problems usually have a hidden structure: they are constructed by replication of some small generic block. The linear algebra subproblems which arise in optimization algorithms for such problems involve matrices which are not only sparse, but they additionally display a block-structure with many smaller blocks sparsely distributed in the large matrix.

We have developed a structure-exploiting parallel interior point solver for optimization problems. Its design uses object-orientated programming techniques. The progress OOPS (Object-Orientated Parallel Solver: on a number of different computing platforms and achieves scalability on a number of different computing platforms. We illustrate its performance on a collection of problems with sizes reaching 109 variables arising from asset liability management and portfolio optimization.

This is a joint work with Andreas Grothey.

  • Computational Mathematics and Applications Seminar