A structure-preserving pivotal method for affine variational inequalities
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Affine variational inequalities (AVI) are an important problem class that subsumes systems of linear equations, linear complementarity problems and optimality conditions for quadratic programs. This paper describes PathAVI, a structure-preserving pivotal approach, that can efficiently process (solve or determine infeasible) large-scale sparse instances of the problem with theoretical guarantees and at high accuracy. PathAVI implements a strategy known to process models with good theoretical properties without reducing the problem to specialized forms, since such reductions may destroy sparsity in the models and can lead to very long computational times. We demonstrate formally that PathAVI implicitly follows the theoretically sound iteration paths, and can be implemented in a large scale setting using existing sparse linear algebra and linear programming techniques without employing a reduction. We also extend the class of problems that PathAVI can process. The paper illustrates the effectiveness of our approach by comparison to the Path solver used on a complementarity reformulation of the AVI in the context of applications in friction contact and Nash Equilibria. PathAVI is a general purpose solver, and freely available under the same conditions as Path.
KeywordsAffine variational inequality Normal map Path-following algorithm
Mathematics Subject Classification90C33 90C49 65K10 65K15
This work is supported in part by the Air Force Office of Scientific Research and the Department of Energy. The authors are grateful to Steven Dirkse and Todd Munson for comments and suggestions leading to improved computational performance.
- 1.Acary, V., Brémond, M., Koziara, T., Pérignon, F.: FCLIB: a collection of discrete 3D Frictional Contact problems. Technical Report RT-0444, INRIA. https://hal.inria.fr/hal-00945820 (2014)
- 3.Acary V., Brémond M., Huber O., Pérignon F., Sinclair, S.: An introduction to SICONOS, Rapport Technique, INRIA (2017)Google Scholar
- 8.Davis, T.A.: UMFPACK. http://faculty.cse.tamu.edu/davis/suitesparse.html (2007)
- 11.Dirkse, S.P., Ferris, M.C.: Crash techniques for large-scale complementarity problems. In: Ferris, M.C., Pang J.S. (eds.) Complementarity and Variational Problems: State of the Art. SIAM, Philadelphia (1997)Google Scholar
- 12.Dubois, F., Jean, M., Renouf, M., Mozul, R., Martin, A., Bagneris, M.: LMGC90. In: 10e colloque national en calcul des structures. Giens. https://hal.archives-ouvertes.fr/hal-00596875 (2011)
- 13.Eaves, B.C.: A short course in solving equations with PL homotopies. Nonlinear Programming. In: SIAM-AMS Proceedings, vol. 9, pp. 73–143 (1976)Google Scholar
- 17.Ferris, M.C., Munson, T.S.: PATH 4.7. http://www.gams.com/dd/docs/solvers/path (2013)
- 18.Huber, O., Ferris, M.C.: Friction contact problems from a variational inequality perspective (2016) (in preparation)Google Scholar
- 22.Maros, I.: QP examples. http://www.doc.ic.ac.uk/~im/00README.QP (1998)
- 26.Robinson, S.M.: Convexity in Finite-Dimensional Spaces (2015) (unpublished manuscript)Google Scholar