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Solvability of the Minty Variational Inequality

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Abstract

We consider the existence of solutions to the Minty variational inequality, as it plays a key role in a projection-type algorithm for solving the variational inequality. It is shown that, if the underlying mapping has a separable structure with each component of the mapping being quasimonotone, then the Minty variational inequality has a solution. An example shows that the underlying mapping itself is not necessarily quasimonotone, although each of its components is.

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References

  1. Giannessi, F.: On Minty variational principle. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds.) New Trends in Mathematical Programming, pp. 93–99. Kluwer Academic Publishers, Dordrecht (1998)

    Chapter  Google Scholar 

  2. Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86(2, Ser. A), 433–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aussel, D., Hadjisavvas, N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121(2), 445–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bianchi, M., Hadjisavvas, N., Schaible, S.: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122(1), 1–17 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. He, Y.: Stable pseudomonotone variational inequality in reflexive Banach spaces. J. Math. Anal. Appl. 330(1), 352–363 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. He, Y., Ng, K.F.: Strict feasibility of generalized complementarity problems. J. Aust. Math. Soc. 81(1), 15–20 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wang, Y.J., Xiu, N.H., Wang, C.Y.: Unified framework of extragradient-type methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 111(3), 641–656 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ye, M., He, Y.: A double projection method for solving variational inequalities without monotonicity. Comput. Optim. Appl. 60, 140–150 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Konnov, I.V.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 99(1), 165–181 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Professor Franco Giannessi and the referees for valuable suggestions. This work was partially supported by National Natural Science Foundation of China under Grant 11271274.

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Authors and Affiliations

  1. Department of Mathematics, Sichuan Normal University, Chengdu, 610066, Sichuan, China

    Yiran He

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  1. Yiran He
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Correspondence to Yiran He.

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Communicated by Antonino Maugeri.

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He, Y. Solvability of the Minty Variational Inequality. J Optim Theory Appl 174, 686–692 (2017). https://doi.org/10.1007/s10957-017-1124-1

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  • DOI: https://doi.org/10.1007/s10957-017-1124-1

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