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Error Bounds of Regularized Gap Functions for Polynomial Variational Inequalities

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Abstract

This paper is devoted to presenting new error bounds of regularized gap functions for polynomial variational inequalities with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The main techniques are based on semialgebraic geometry and variational analysis, which allow us to establish a nonsmooth extension of the seminal Łojasiewicz gradient inequality to regularized gap functions with explicitly calculated exponents.

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References

  1. Auchmuty, G.: Variational principles for variational inequalities. Numer. Funct. Anal. Optim. 10, 863–874 (1989)

    Article  MathSciNet  Google Scholar 

  2. Bochnak, J., Coste, M., Roy, M.F.: Real Algebraic Geometry. Springer, Berlin (1998)

    Book  Google Scholar 

  3. Chen, T., Zou, S., Zhang, Y.: New existence theorems for vector equilibrium problems with set-valued mappings. J. Nonlinear Funct. Anal. 2019, Article ID 45 (2019)

  4. Clarke, F.H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  Google Scholar 

  5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  6. D’Acunto, D., Kurdyka, K.: Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. Ann. Pol. Math. 87, 51–61 (2005)

  7. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  8. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problem, vol. I. II. Springer, New-York (2003)

    MATH  Google Scholar 

  9. Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program 53, 99–110 (1992)

    Article  MathSciNet  Google Scholar 

  10. Hà, H.V., Phạm, T.S.: Genericity in Polynomial Optimization, vol. 3. World Scientific Publishing (2017)

  11. Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms, and applications. Math. Program. 48, 161–220 (1990)

    Article  MathSciNet  Google Scholar 

  12. Hebestreit, N.: Vector variational inequalities and related topics: A survey of theory and applications. Appl. Set-Valued Anal. Optim. 1, 231–305 (2019)

    Google Scholar 

  13. Hebestreit, N.: Algorithms for monotone vector variational inequalities. J. Nonlinear Var. Anal. 4, 107–125 (2019)

    MATH  Google Scholar 

  14. Hu, S., Wang, J., Huang, Z.H.: Error bounds for the solution sets of quadratic complementarity problems. J. Optim. Theory Appl. 179, 983–1000 (2018)

    Article  MathSciNet  Google Scholar 

  15. Huang, L.R., Ng, K.F.: Equivalent optimization formulations and error bounds for variational inequality problems. J. Optim. Theory Appl. 125, 299–314 (2005)

    Article  MathSciNet  Google Scholar 

  16. Kurdyka, K., Spodzieja, S.: Separation of real algebraic sets and the Łojasiewicz exponent. Proc. Amer. Math. Soc. 142, 3089–3102 (2014)

    Article  MathSciNet  Google Scholar 

  17. Li, G., Ng, K.F.: Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM J. Optim. 20(2), 667–690 (2009)

    Article  MathSciNet  Google Scholar 

  18. Li, G., Tang, C., Wei, Z.: Error bound results for generalized D-gap functions of nonsmooth variational inequality problems. J. Comput. Appl. Math. 233(11), 2795–2806 (2010)

    Article  MathSciNet  Google Scholar 

  19. Li, G.: On the asymptotic well behaved functions and global error bound for convex polynomials. SIAM J. Optim. 20, 1923–1943 (2010)

    Article  MathSciNet  Google Scholar 

  20. Li, G., Mordukhovich, B.S., Phạm, T.S.: New fractional error bounds for polynomial systems with applications to Höderian stability in optimization and spectral theory of tensors. Math. Program. Ser. A 153, 333–362 (2015)

    Article  Google Scholar 

  21. Li, G., Mordukhovich, B.S., Nghia, T.T.A., Phạm, T.S.: Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. Math. Program. Ser. B 168(1–2), 313–346 (2018)

    Article  MathSciNet  Google Scholar 

  22. Liu, L., Tan, B., Cho, S.Y.: On the resolution of variational inequality problems with a double-hierarchical structure. J. Nonlinear Convex Anal. 21, 377–386 (2020)

    MathSciNet  MATH  Google Scholar 

  23. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  24. Mordukhovich, B.: Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Springer, Berlin (2006)

    Google Scholar 

  25. Muu, L.D., Quy, N.V.: DC-gap function and proximal methods for solving Nash-Cournot oligopolistic equilibrium models involving concave cost. J. Appl. Numer. Optim. 1, 13–24 (2019)

    Google Scholar 

  26. Ng, K.F., Tan, L.L.: Error bounds of regularized gap functions for nonsmooth variational inequality problems. Math. Program 110, 405–429 (2007)

    Article  MathSciNet  Google Scholar 

  27. Phạm, T.S., Truong, X.D.H., Yao, J.C.: The global weak sharp minima with explicit exponents in polynomial vector optimization problems. Positivity 22(1), 219–244 (2018)

    Article  MathSciNet  Google Scholar 

  28. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  29. Solodov, M.V.: Merit functions and error bounds for generalized variational inequalities. J. Math. Anal. Appl. 287, 405–414 (2003)

    Article  MathSciNet  Google Scholar 

  30. Taji, K., Fukushima, M., Ibaraki, T.: A globally convergent Newton method for solving monotone variational inequalities. Math. Program. 58, 369–383 (1993)

    Article  MathSciNet  Google Scholar 

  31. Wu, J.H., Florian, M., Marcotte, P.: A general descent framework for the monotone variational inequality problem. Math. Program. 61, 281–300 (1993)

    Article  MathSciNet  Google Scholar 

  32. Yamashita, N., Fukushima, M.: Equivalent unconstrained minimization and global error bounds for variational inequality problems. SIAM J. Control Optim. 35(1), 273–284 (1997)

    Article  MathSciNet  Google Scholar 

  33. Yamashita, N., Taji, K., Fukushima, M.: Unconstrained optimization reformulations of variational inequality problems. J. Optim. Theory Appl. 92, 439–456 (1997)

    Article  MathSciNet  Google Scholar 

  34. Zhang, J., Wan, C., Xiu, N.: The dual gap function for variational inequalities. Appl. Math. Optim. 48(2), 129–148 (2003)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank Guoyin Li for his helpful comments on the early version of this manuscript. The authors also thank the anonymous referees for their constructive suggestions. The first author is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Numbers 4/2020/STS02 and 101.04-2019.302. The second author is supported by the Vietnam Academy of Science and Technology under Grant Number ĐLTE00.01/21-22 and Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.04-2019.302.

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Authors and Affiliations

  1. Department of Mathematics, Le Quy Don Technical University, Hanoi, Vietnam

    Bui Van Dinh

  2. Department of Optimization and Control Theory, Institue of Mathematics, VAST, Hanoi, Vietnam

    Bui Van Dinh

  3. Department of Mathematics, Dalat University, 1 Phu Dong Thien Vuong, Dalat, Vietnam

    Tien-Son Pham

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  1. Bui Van Dinh
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  2. Tien-Son Pham
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Correspondence to Bui Van Dinh.

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Communicated by Jen-Chih Yao.

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Van Dinh, B., Pham, TS. Error Bounds of Regularized Gap Functions for Polynomial Variational Inequalities. J Optim Theory Appl 192, 226–247 (2022). https://doi.org/10.1007/s10957-021-01960-6

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