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Algorithms for Equilibria

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Nonlinear Programming Techniques for Equilibria

Part of the book series: EURO Advanced Tutorials on Operational Research ((EUROATOR))

Abstract

Algorithms for solving the Ky Fan inequality EP(f, C) are tackled exploiting the tools developed in the previous chapter. Some assumptions on C and f hold throughout all the chapter in order to provide a unified algorithmic framework. Precisely, C is supposed to be convex and closed, f to be continuous and to satisfy f(x, x) = 0 for any \(x\in \mathbb {R}^n\) while f(x, ⋅) to be τ-convex for any \(x\in \mathbb {R}^n\) for some τ ≥ 0 that does not depend upon the considered point x. Notice that this framework guarantees all the assumptions of the existence Theorems 2.3.4 and 2.3.8 except for the boundedness of C or some kind of monotonicity of f. Indeed, all the algorithms require at least one of them, so that the existence of a solution is always guaranteed.

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Authors and Affiliations

  1. Department of Computer Science, University of Pisa, Pisa, Italy

    Giancarlo Bigi, Massimo Pappalardo & Mauro Passacantando

  2. Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    Marco Castellani

Authors
  1. Giancarlo Bigi
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  2. Marco Castellani
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  3. Massimo Pappalardo
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  4. Mauro Passacantando
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Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M. (2019). Algorithms for Equilibria. In: Nonlinear Programming Techniques for Equilibria. EURO Advanced Tutorials on Operational Research. Springer, Cham. https://doi.org/10.1007/978-3-030-00205-3_3

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