Direct Solution of Linear Systems of Size 109 Arising in Optimization with Interior Point Methods
Solution methods for very large scale optimization problems are addressed in this paper. Interior point methods are demonstrated to provide unequalled efficiency in this context. They need a small (and predictable) number of iterations to solve a problem. A single iteration of interior point method requires the solution of indefinite system of equations. This system is regularized to guarantee the existence of triangular decomposition. Hence the well-understood parallel computing techniques developed for positive definite matrices can be extended to this class of indefinite matrices. A parallel implementation of an interior point method is described in this paper. It uses object-oriented programming techniques and allows for exploiting different block-structures of matrices. Our implementation outperforms the industry-standard optimizer, shows very good parallel efficiency on massively parallel architecture and solves problems of unprecedented sizes reaching 109 variables.
KeywordsPortfolio Selection Interior Point Method Jacobian Matrice Direct Solution Interior Point Algorithm
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- 2.Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philadelphia (1997)Google Scholar
- 6.Gondzio, J., Grothey, A.: Parallel interior point solver for structured quadratic programs: Application to financial planning problems. Technical Report MS-03-001, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK (2003) Annals of Operations Research (accepted for publication)Google Scholar
- 7.Gondzio, J., Grothey, A.: Exploiting structure in parallel implementation of interior point methods for optimization. Technical Report MS-04-004, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK (2004)Google Scholar
- 8.Gondzio, J., Grothey, A.: Solving nonlinear portfolio optimization problems with the primal-dual interior point method. Technical Report MS-04-001, School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK (2004) European Journal of Operational Research (accepted for publication)Google Scholar
- 9.Ziemba, W.T., Mulvey, J.M.: Worldwide Asset and Liability Modeling. Publications of the Newton Institute. Cambridge University Press, Cambridge (1998)Google Scholar
- 14.Markowitz, H.M.: Portfolio selection. Journal of Finance, 77–91 (1952)Google Scholar