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Interior point methods for equilibrium problems

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Abstract

In the present paper we discuss three methods for solving equilibrium-type fixed point problems. Concentrating on problems whose solutions possess some stability property, we establish convergence of these three proximal-like algorithms that promise a very high numerical tractability and efficiency. For example, due to the implemented application of zone coercive Bregman functions, all these methods allow to treat the generated subproblems as unconstrained and, partly, explicitly solvable ones.

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Algorithm 1
Algorithm 2
Algorithm 3

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Notes

  1. We do not distinguish between “solution”, “fixed point” and “equilibrium” unless this is of importance.

  2. A definition of such functions including some remarks is given below.

  3. The parameters δ k will allow some inexactness in the solution of the subproblems, and ε k will be the parameter for some ε-subdifferential. These issues will get more clear in the introduction of the methods below, but for consistency reasons, the related assumptions are listed here.

  4. The parameters χ k will play the role of some regularization parameters which will be more clear in the introduction of the methods below, but, again, for consistency reasons, the related assumptions are listed here.

  5. \(\mathcal{F}(x,y) + \mathcal{F}(y,x) \leq\mathcal {F}(x,x) + \mathcal{F}(y,y)\) for all x,y∈K.

  6. To be more precise, we shall give two remarks. The first one is on stability of the equilibrium: The variational inequality does not provide a stable solution, but as ∇h is Lipschitz continuous, this does not constitute a problem here. The second one is on the type of additional assumption: It is known for variational inequalities that a stronger monotonicity condition is needed to ensure convergence of Algorithm 2, but obviously condition (S) is exclusively related to smoothness. In turn, Korpelevich’s extragradient method does not necessarily require a stronger monotonicity assumption, but a kind of additional smoothness.

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Acknowledgements

The author is grateful to one reviewer for his or her constructive remarks.

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  1. Department of Mathematics, University of Trier, 54286, Trier, Germany

    Nils Langenberg

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Langenberg, N. Interior point methods for equilibrium problems. Comput Optim Appl 53, 453–483 (2012). https://doi.org/10.1007/s10589-011-9450-y

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