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Weak Sharp Solutions for Nonsmooth Variational Inequalities

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Abstract

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set.

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References

  1. Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340–1359 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ferris, M.C.: Weak Sharp Minima and Penalty Functions in Mathematical Programming, Ph.D. Thesis, University of Cambridge (1988)

  3. Bertsekas, D.P.: Necessary and sufficient conditions for a penality method to be exact. Math. Program. 9, 8–99 (1975)

    Article  Google Scholar 

  4. Ferris, M.C.: Finite termination of the proximal point algorithm. Math. Program. 50, 359–366 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Polyak, B.T., Tretiyakov, N.V.: Concerning an iterative methods for linear programming and its economic interpretation. Econ. Math. Methods 8(5), 740–751 (1972) [English Translation: Matekon 10(3), 81–100 (1974)]

  6. Marcotte, P., Zhu, D.L.: Weak sharp solutions of variational inequalities. SIAM J. Optim. 9(1), 179–189 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, Z.L., Wu, S.Y.: Weak sharp solutions of variational inequalities in Hilbert spaces. SIAM J. Optim. 14(4), 1011–1027 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Xiu, N., Zhang, J.: On finite convergence of proximal point algorithms for variational inequalities. J. Math. Anal. Appl. 312, 148–158 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhou, J., Wang, C.: A note on finite termination of iterative algorithms in mathematical programming. Oper. Res. Lett. 36, 715–717 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Y., Wu, Z.: Characterization of weakly sharp solutions of a variational inequality by its primal gap function. Optim. Lett. 10(3), 563–576 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities and Nonsmooth Optimization, CRC Press. Taylor & Francis Group, Boca Raton (2014)

    MATH  Google Scholar 

  12. Alshahrani, M., Al-Homidan, S., Ansari, Q.H.: Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities. Optim. Lett. 10(4), 805–819 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Patriksson, M.: A Unified Framework of Descent Algorithms for Nonlinear Programs and Variational Inequalities, Ph.D. thesis, Department of Mathematics, Linkoping Institute of Technology, Linkoping, Sweden (1993)

  14. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

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Acknowledgements

In this research, the first and second author were funded by the National Plan for Science, Technology and Innovation (MAARIFAH) - King Abdulaziz City for Science and Technology - through the Science and Technology Unit at King Fahd University of Petroleum and Minerals (KFUPM) - the Kingdom of Saudi Arabia, award Number 12-MAT3023-24.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Suliman Al-Homidan & Qamrul Hasan Ansari

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

  3. Department of Natural Science, Hong Duc University, Thanh Hoa, Vietnam

    Luong Van Nguyen

Authors
  1. Suliman Al-Homidan
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  2. Qamrul Hasan Ansari
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  3. Luong Van Nguyen
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Correspondence to Qamrul Hasan Ansari.

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Al-Homidan, S., Ansari, Q.H. & Van Nguyen, L. Weak Sharp Solutions for Nonsmooth Variational Inequalities. J Optim Theory Appl 175, 683–701 (2017). https://doi.org/10.1007/s10957-017-1181-5

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  • DOI: https://doi.org/10.1007/s10957-017-1181-5

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