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On geometrical properties of preconditioners in IPMs for classes of block-angular problems
Castro Pérez, Jordi; Nasini, Stefano
Universitat Politècnica de Catalunya. Departament d'Estadística i Investigació Operativa; Universitat Politècnica de Catalunya. GNOM - Grup d´Optimització Numèrica i Modelització
J. Castro, S. Nasini, On geometrical properties of preconditioners in IPMs for classes of block-angular problems, Research Report DR 2016/03, Dept. of Statistics and Operations Research, Universitat Politècnica de Catalunya, 2016.
One of the most efficient interior-point methods for some classes of block-angular structured problems solves the normal equations by a combination of Cholesky factorizations and preconditioned conjugate gradient for, respectively, the block and linking constraints. In this work we show that the choice of a good preconditioner depends on geometrical properties of the constraints structure. In particular, it is seen that the principal angles between the subspaces generated by the diagonal blocks and the linking constraints can be used to estimate ex-ante the efficiency of the preconditioner. Numerical validation is provided with some generated optimization problems. An application to the solution of multicommodity network flow problems with nodal capacities and equal flows of up to 127 million of variables and up to 7.5 million of constraints is also presented
Àrees temàtiques de la UPC::Matemàtiques i estadística::Investigació operativa::Optimització
interior-point methods
structured problems
preconditioned conjugate gradient
principal angles
large-scale optimization
Classificació AMS::90 Operations research, mathematical programming
Artículo - Borrador

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