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Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

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  • Songnian He
  • Caiping Yang

Abstract

Consider the variational inequality VI(C, F) of finding a point x* ∈ C satisfying the property 〈Fx*, x − x*〉≥0, for all x ∈ C, where C is the intersection of finite level sets of convex functions defined on a real Hilbert space H and F : H → H is an L‐Lipschitzian and η‐strongly monotone operator. Relaxed and self‐adaptive iterative algorithms are devised for computing the unique solution of VI(C, F). Since our algorithm avoids calculating the projection PC (calculating PC by computing several sequences of projections onto half‐spaces containing the original domain C) directly and has no need to know any information of the constants L and η, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

Suggested Citation

  • Songnian He & Caiping Yang, 2013. "Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:942315
    DOI: 10.1155/2013/942315
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    References listed on IDEAS

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    1. Yang, Hai & Bell, Michael G. H., 1997. "Traffic restraint, road pricing and network equilibrium," Transportation Research Part B: Methodological, Elsevier, vol. 31(4), pages 303-314, August.
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    Cited by:

    1. Caiping Yang & Songnian He, 2015. "Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators," Journal of Applied Mathematics, John Wiley & Sons, vol. 2015(1).
    2. Jing Zhao & Hang Zhang, 2014. "Solving Split Common Fixed‐Point Problem of Firmly Quasi‐Nonexpansive Mappings without Prior Knowledge of Operators Norms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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