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Equilibria and Weighted Complementarity Problems

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Numerical Analysis and Optimization (NAO 2017)

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Abstract

We review some results about market equilibria and weighted complementarity problems (wCP). The latter problem consists in finding a pair of vectors (x, s) belonging to the intersection of a manifold with a cone, such that their product in a certain algebra, \(x\circ s\), equals a given weight vector w. When w is the zero vector, then wCP reduces to a Complementarity Problem (CP). The motivation for introducing the more general notion of a wCP lies in the fact that several equilibrium problems in economics can be formulated in a natural way as wCP. Moreover, those formulations lend themselves to the development of highly efficient algorithms for solving the corresponding equilibrium problems. For example, Fisher’s competitive market equilibrium model can be formulated as a wCP that can be efficiently solved by interior-point methods. The Quadratic Programming and Weighted Centering problem, which generalizes the notion of a Linear Programming and Weighted Centering problem proposed by Anstreicher, can also be formulated as a special linear monotone wCP. The paper summarizes some previous results about the convergence of two interior-point methods for general monotone linear wCP and a corrector-predictor method for sufficient linear wCP. Some recent smoothing Newton methods for monotone linear wCP are also presented.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1311923.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 22150, USA

    Florian A. Potra

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  1. Florian A. Potra
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Correspondence to Florian A. Potra .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  2. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Calabria University, Arcavacata, Italy

    Lucio Grandinetti

  3. Department Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

    Anton Purnama

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Potra, F.A. (2018). Equilibria and Weighted Complementarity Problems. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. NAO 2017. Springer Proceedings in Mathematics & Statistics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-319-90026-1_12

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