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A new feasible moving ball projection algorithm for pseudomonotone variational inequalities

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Abstract

The projection is often used in solving variational inequalities. When projection onto the feasible set is not easy to calculate, the projection algorithms are replaced by the relaxed projection algorithms. However, these relaxed projection algorithms are not feasible, and to ensure the convergence of these relaxed projection algorithms, in addition to assuming some basic conditions, such as the Slater condition holds for the feasible set, the mapping is pseudomonotone and Lipschitz continuous, but also need to assume some additional conditions, which require some relationship between the mapping and the feasible set. In this paper, by replacing the projection onto the feasible set with the projection onto a ball (which changes from iteration) contained in the feasible set, a new feasible moving ball projection algorithm for pseudomonotone variational inequalities is obtained. Since the projection onto a ball has an explicit expression, this algorithm is easy to implement. At the same time, all the balls are contained in the feasible set, so the iteration points generated by this algorithm are all in the feasible set, which ensures the feasibility of this algorithm. The convergence of this algorithm is proved when the Slater condition holds for the feasible set, and the mapping is pseudomonotone and Lipschitz continuous. The fundamental difference between this moving ball projection algorithm and the previous relaxed projection algorithms lie in that the previous relaxed projection algorithms are all projected onto the half-space containing the feasible set, and this moving ball projection algorithm is projected onto a ball contained in the feasible set. In particular, this algorithm does not need to assume any additional conditions between the mapping and the feasible set. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

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References

  1. Anh, P.N.: New outer proximal methods for solving variational inequality problems. J. Optim. Theory Appl. (2023). https://doi.org/10.1007/s10957-023-02202-7

    Article  MathSciNet  Google Scholar 

  2. Anceschi, F., Barbagallo, A., Guarino Lo Bianco, S.: Inverse tensor variational inequalities and applications. J. Optim. Theory Appl. 196(2), 570–589 (2023)

    Article  MathSciNet  Google Scholar 

  3. Auslender, A., Shefi, R., Teboulle, M.: A moving balls approximation method for a class of smooth constrained minimization problems. SIAM J. Optim. 20(6), 3232–3259 (2010)

    Article  MathSciNet  Google Scholar 

  4. Barbu, V., Röckner, M.: Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise. Arch. Ration. Mech. Anal. 209(3), 797–834 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, vol. 408. Springer, New York (2011)

    Book  Google Scholar 

  6. Cruz, J.B., Iusem, A.N.: An explicit algorithm for monotone variational inequalities. Optimization 61(7), 855–871 (2012)

    Article  MathSciNet  Google Scholar 

  7. Cruz, J.B., Iusem, A.N.: Convergence of direct methods for paramonotone variational inequalities. Comput. Optim. Appl. 46(2), 247–263 (2010)

    Article  MathSciNet  Google Scholar 

  8. Cruz, J.B., Iusem, A.N.: Full convergence of an approximate projection method for nonsmooth variational inequalities. Math. Comput. Simul. 114, 2–13 (2015)

    Article  MathSciNet  Google Scholar 

  9. Cao, Y., Guo, K.: On the convergence of inertial two-subgradient extragradient method for variational inequality problems. Optimization 69(6), 1237–1253 (2020)

    Article  MathSciNet  Google Scholar 

  10. Cegielski, A., Gibali, A., Reich, S., Zalas, R.: An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space. Numer. Funct. Anal. Optim. 34(10), 1067–1096 (2013)

    Article  MathSciNet  Google Scholar 

  11. Cegielski, A., Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities defined over the solution set of a split convex feasibility problem. Numer. Funct. Anal. Optim. 41(9), 1089–1108 (2020)

    Article  MathSciNet  Google Scholar 

  12. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, New York (2012)

    Google Scholar 

  13. Censor, Y., Gibali, A.: Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional space. J. Nonlinear Convex Anal. 9(3), 461–475 (2008)

    MathSciNet  Google Scholar 

  14. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26(4–5), 827–845 (2011)

    Article  MathSciNet  Google Scholar 

  15. Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148(2), 318–335 (2011)

    Article  MathSciNet  Google Scholar 

  16. Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space. Optimization 61(9), 1119–1132 (2012)

    Article  MathSciNet  Google Scholar 

  17. Chadli, O., Gwinner, J., Nashed, M.Z.: Noncoercive variational-hemivariational inequalities: existence, approximation by double regularization, and application to nonmonotone contact problems. J. Optim. Theory Appl. 193(1–3), 42–65 (2022)

    Article  MathSciNet  Google Scholar 

  18. Chen, J.X., Ye, M.L.: A new modified two-subgradient extragradient algorithm for solving variational inequality problems. J. Math. Res. Appl. 42(4), 402–412 (2022)

    MathSciNet  Google Scholar 

  19. Eslamian, M.: Variational inequality over the set of common solutions of a system of bilevel variational inequality problem with applications. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturale. Serie A. Matematicas 116, 47 (2022)

    Article  MathSciNet  Google Scholar 

  20. Freund, R.M., Grigas, P., Mazumder, R.: An extended Frank-Wolfe method with in-face directions, and its application to low-rank matrix completion. SIAM J. Optim. 27, 319–346 (2017)

    Article  MathSciNet  Google Scholar 

  21. Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35(1), 58–70 (1986)

    Article  MathSciNet  Google Scholar 

  22. Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66(3), 417–437 (2017)

    Article  MathSciNet  Google Scholar 

  23. Gibali, A., Reich, S., Zalas, R.: Iterative methods for solving variational inequalities in Euclidean space. J. Fixed Point Theory Appl. 17, 775–811 (2015)

    Article  MathSciNet  Google Scholar 

  24. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Non-expansive Mappings. Marcel Dekker, New York and Basel (1984)

    Google Scholar 

  25. Goldstein, A.A.: Convex programming in Hilbert space. Bull. Am. Math. Soc. 70(5), 709–710 (1964)

    Article  MathSciNet  Google Scholar 

  26. Gwinner, J., Jadamba, B., Khan, A.A., Raciti, F.: Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications. CRC Press (2021)

  27. Harchaoui, Z., Juditsky, A., Nemirovski, A.: Conditional gradient algorithms for norm-regularized smooth convex optimization. Math. Program. 152, 75–112 (2015)

    Article  MathSciNet  Google Scholar 

  28. He, S.N., Dong, Q.L., Tian, H.L.: Relaxed projection and contraction methods for solving Lipschitz continuous monotone variational inequalities. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 113(3), 2773–2791 (2019)

    Article  MathSciNet  Google Scholar 

  29. He, S.N., Wu, T.: A modified subgradient extragradient method for solving monotone variational inequalities. J. Inequal. Appl. 2017, 89 (2017)

    Article  MathSciNet  Google Scholar 

  30. He, S.N., Wu, T., Gibali, A., Dong, Q.L.: Totally relaxed, self-adaptive algorithm for solving variational inequalities over the intersection of sub-level sets. Optimization 67, 1487–1504 (2018)

    Article  MathSciNet  Google Scholar 

  31. He, S.N., Xu, H.K.: Uniqueness of supporting hyperplanes and an alternative to solutions of variational inequalities. J. Global Optim. 57(4), 1375–1384 (2013)

    Article  MathSciNet  Google Scholar 

  32. He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185(1), 166–173 (2006)

    Article  MathSciNet  Google Scholar 

  33. Heinemann, C., Sturm, K.: Shape optimization for a class of semilinear variational inequalities with applications to damage models. SIAM J. Math. Anal. 48(5), 3579–3617 (2016)

    Article  MathSciNet  Google Scholar 

  34. Hung, N.V., Tam, V.M.: Error bound analysis of the D-gap functions for a class of elliptic variational inequalities with applications to frictional contact mechanics. Zeitschrift fur angewandte Mathematik und Physik 72, 173 (2021)

    Article  MathSciNet  Google Scholar 

  35. Kolobov, V.I., Reich, S., Zalas, R.: Finitely convergent iterative methods with overrelaxations revisited. J. Fixed Point Theory Appl. 23, 57 (2021)

    Article  MathSciNet  Google Scholar 

  36. Kolobov, V.I., Reich, S., Zalas, R.: Finitely convergent deterministic and stochastic iterative methods for solving convex feasibility problems. Math. Program. 194, 1163–1183 (2022)

    Article  MathSciNet  Google Scholar 

  37. Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)

    MathSciNet  Google Scholar 

  38. Levitin, E., Polyak, B.: Constrained minimization methods. USSR Comput. Math. Math. Phys. 6, 1–50 (1966)

    Article  Google Scholar 

  39. Mathiesen, L.: An algorithm based on a sequence of linear complementarity problems applied to a Walrasian equilibrium model: an example. Math. Program. 37(1), 1–18 (1987)

    Article  MathSciNet  Google Scholar 

  40. Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. Society for Industrial and Applied Mathematics (2000)

  41. Pang, J.S., Gabriel, S.A.: NE/SQP: a robust algorithm for the nonlinear complementarity problem. Math. Program. 60(1), 295–337 (1993)

    Article  MathSciNet  Google Scholar 

  42. Rockafellar, R.T., Sun, J.: Solving Lagrangian variational inequalities with applications to stochastic programming. Math. Program. 181(2), 435–451 (2020)

    Article  MathSciNet  Google Scholar 

  43. 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  Google Scholar 

  44. Tan, B., Cho, S.Y.: Inertial extragradient algorithms with non-monotone stepsizes for pseudomonotone variational inequalities and applications. Comput. Appl. Math. 41, 121 (2022)

    Article  MathSciNet  Google Scholar 

  45. Thong, D.V., Phan, T.V.: Improved subgradient extragradient methods for solving pseudomonotone variational inequalities in Hilbert spaces. Appl. Numer. Math. 163, 221–238 (2021)

    Article  MathSciNet  Google Scholar 

  46. Thong, D.V., Shehu, Y., Iyiola, O.S., Van Thang, H.: New hybrid projection methods for variational inequalities involving pseudomonotone mappings. Optim. Eng. 22, 363–386 (2021)

    Article  MathSciNet  Google Scholar 

  47. Thong, D.V., Van Hieu, D., Rassias, T.M.: Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim. Lett. 14, 115–144 (2020)

    Article  MathSciNet  Google Scholar 

  48. Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38(2), 431–446 (2000)

    Article  MathSciNet  Google Scholar 

  49. Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algorithms 81(1), 269–291 (2019)

    Article  MathSciNet  Google Scholar 

  50. Yang, Q.: On variable-step relaxed projection algorithm for variational inequalities. J. Math. Anal. Appl. 302(1), 166–179 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author was supported partly by the National Natural Science Foundation of China (11901414) and (11871359). The third author was supported partly by the National Natural Science Foundation of China (11871359).

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  1. Department of Mathematics, Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, 610066, Sichuan, People’s Republic of China

    Limei Feng, Yongle Zhang & Yiran He

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  1. Limei Feng
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Correspondence to Yongle Zhang.

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Feng, L., Zhang, Y. & He, Y. A new feasible moving ball projection algorithm for pseudomonotone variational inequalities. Optim Lett 18, 1437–1455 (2024). https://doi.org/10.1007/s11590-023-02053-1

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