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On the Tikhonov regularization of affine pseudomonotone mappings

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Abstract

The pseudomonotonicity of affine mappings on polyhedral convex sets is characterized in the one-dimensional case and in a higher-dimensional setting. The obtained results allow us to investigate the pseudomonotonicity of the regularized mappings (in the sense of Tikhonov regularization). Among other things, it is shown that there exists a pseudomonotone affine variational inequality problem VI(\(K,F\)) with a nonempty solution set for which the regularized problem VI(\(K,F_\varepsilon \)) is not pseudomonotone for every \(\varepsilon \in (0,\frac{1}{2})\). In addition, we prove that the feasibility of a pseudomonotone linear complementarity problem implies the solution uniqueness of the regularized problem.

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References

  1. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, New York (1992)

  2. Crouzeix, J.-P., Hassouni, A., Lahlou, A., Schaible, S.: Positive subdefinite matrices, generalized monotonicity, and linear complementarity problems. SIAM J. Matrix Anal. Appl. 22, 66–85 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. El-Farouq, N.: Pseudomonotone variational inequalities: convergence of proximal methods. J. Optim. Theory Appl. 109, 311–326 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I and II. Springer, New York (2003)

  5. Gowda, M.S.: Affine pseudo-monotone mappings and the linear complementarity problems. SIAM J. Matrix Anal. Appl. 11, 373–380 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Huong, N.T.T., Khanh, P.D., Yen, N.D.: Multivalued Tikhonov trajectories of general affine variational inequalities. J. Optim. Theory Appl. (2012, online first). doi:10.1007/s10957-012-0226-z

  7. Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18, 445–454 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Khanh, P.D.: Partial solution for an open question on pseudomonotone variational inequalities. Appl. Anal. 91, 1691–1698 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Langenberg, N.: Pseudomonotone operators and the Bregman proximal point algorithm. J. Glob. Optim. 47, 537–555 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tam, N.N., Yao, J.-C., Yen, N.D.: Solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008)

    Google Scholar 

  11. Thanh Hao, N.: Tikhonov regularization algorithm for pseudomonotone variational inequalities. Acta Math. Vietnam. 31, 283–289 (2006)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2011.01. P.D. Khanh would like to thank Prof. Nguyen Dong Yen and Dr. Trinh Cong Dieu for helpful discussions on the subject. The detailed comments and suggestions of the two anonymous referees are gratefully acknowledged.

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  1. Department of Mathematics, University of Pedagogy of Ho Chi Minh City, 280 An Duong Vuong, Ho Chi Minh City, Vietnam

    Pham Duy Khanh

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  1. Pham Duy Khanh
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Khanh, P.D. On the Tikhonov regularization of affine pseudomonotone mappings. Optim Lett 8, 1325–1336 (2014). https://doi.org/10.1007/s11590-013-0659-9

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  • DOI: https://doi.org/10.1007/s11590-013-0659-9

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