doi: 10.3934/naco.2021046
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A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

1. 

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

2. 

DSI-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)

3. 

Mountain Top University, Prayer City, Ogun State, Nigeria

* Corresponding author: Francis Akutsah

Received  February 2021 Revised  September 2021 Accepted  October 2021 Early access November 2021

In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.

Citation: Francis Akutsah, Akindele Adebayo Mebawondu, Hammed Anuoluwapo Abass, Ojen Kumar Narain. A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021046
References:
[1]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar

[2]

P. N. Anh and N. X. Phuong, A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems, J. Fixed Point Theory Appl., 20 (2018), 1-17.  doi: 10.1007/s11784-018-0554-1.  Google Scholar

[3]

H. AttouchX. Goudon and P. Redont, The heavy ball with friction. I. the continuous dynamical system, Commun. Contemp Math., 21 (2000), 1-34.  doi: 10.1142/S0219199700000025.  Google Scholar

[4]

H. Attouch and M. O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Diff. Eq., 179 (2002), 278-310.  doi: 10.1006/jdeq.2001.4034.  Google Scholar

[5]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[6]

G. CaiQ-.L. Dong and Y. Peng, Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-lipschitz operators, J. Optim. Theory Appl., 188 (2020), 447-472.  doi: 10.1007/s10957-020-01792-w.  Google Scholar

[7]

L. C. CengQ. H. Ansari and Q. H. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Analysis, 75 (2012), 2116-2125.  doi: 10.1016/j.na.2011.10.012.  Google Scholar

[8]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[9]

Y. CensorX. A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-set split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1224-1256.  doi: 10.1016/j.jmaa.2006.05.010.  Google Scholar

[10]

Y. CensorT. ElfvingN. Kopt and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Prob., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[11]

G. Ficher, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Rend., Cl. Sci. Fis. Mat. Natur, 34 (1963), 138-142.   Google Scholar

[12]

G. Ficher, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincci, Cl. Sci. Fis. Mat. Nat., Sez., 7 (1964), 91-140.   Google Scholar

[13]

Y. Hao, Some results of variational inclusion problems and fixed point problems with applications, Appl. Math. Mech., 30 (2009), 1589-1596.  doi: 10.1007/s10483-009-1210-x.  Google Scholar

[14]

Z. HeC. Chen and F. Gu, Viscosity approximation method for nonexpansive nonself-nonexpansive mappings and variational inequality, J. Nonlinear Sci. Appl., 1 (2008), 169-178.  doi: 10.22436/jnsa.001.03.05.  Google Scholar

[15]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, 1$^st$ edition, Marcel Dekker, New York, 1984.  Google Scholar

[16]

E. Kopeck and S. Reich, A note on alternating projections in Hilbert space, J. Fixed Point Theory Appl., 12 (2012), 41-47.  doi: 10.1007/s11784-013-0097-4.  Google Scholar

[17]

H. Liu and J. Yang, Weak convergence of iterative methods for solving quasimonotone variational inequalities, Computation Optimization and Applications, 77 (2020), 491-508.  doi: 10.1007/s10589-020-00217-8.  Google Scholar

[18]

P. E. Mainge, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl., 34 (2008), 876-887.  doi: 10.1016/j.jmaa.2008.03.028.  Google Scholar

[19]

P. E. Mainge, A hybrid extragradient viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.  Google Scholar

[20]

N. H. MinhL. H. M. Van and T. V. Anh, An algorithm for a class of bilevel variational inequality with split variational inequality and fixed point problem constraints, Acta Mathematica Vietnamica, 46 (2021), 515-530.  doi: 10.1007/s40306-020-00389-9.  Google Scholar

[21]

Y. Nesterov, A method of solving a convex programming problem with convergence rate O($1/k^2$), Soviet Math. Doklady, 27 (1983), 372-376.   Google Scholar

[22]

B. T. Polyak, Some methods of speeding up the convergence of iterates methods, U.S.S.R Comput. Math. Phys., 4 (1964), 1-17.   Google Scholar

[23]

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350.  doi: 10.1016/j.na.2003.07.023.  Google Scholar

[24]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 57 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.  Google Scholar

[25]

G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes,, C. R. Math. Acad. Sci., 258 (1964), 4413-4416.   Google Scholar

[26]

D. V. ThongD. V. Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving psedomonotone variational inequality problems, Optim. Lett., 14 (2020), 115-144.  doi: 10.1007/s11590-019-01511-z.  Google Scholar

[27]

D. V. Thong and D. V. Hieu, Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algor., 82 (2019), 761-789.  doi: 10.1007/s11075-018-0626-8.  Google Scholar

[28]

J. Yang and H. Liu, A modified projected gradient method for monotone variational inequalities, J. Optim Theory Appl., 179 (2018), 197-211.  doi: 10.1007/s10957-018-1351-0.  Google Scholar

show all references

References:
[1]

F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), 3-11.  doi: 10.1023/A:1011253113155.  Google Scholar

[2]

P. N. Anh and N. X. Phuong, A parallel extragradient-like projection method for unrelated variational inequalities and fixed point problems, J. Fixed Point Theory Appl., 20 (2018), 1-17.  doi: 10.1007/s11784-018-0554-1.  Google Scholar

[3]

H. AttouchX. Goudon and P. Redont, The heavy ball with friction. I. the continuous dynamical system, Commun. Contemp Math., 21 (2000), 1-34.  doi: 10.1142/S0219199700000025.  Google Scholar

[4]

H. Attouch and M. O. Czarnecki, Asymptotic control and stabilization of nonlinear oscillators with non-isolated equilibria, J. Diff. Eq., 179 (2002), 278-310.  doi: 10.1006/jdeq.2001.4034.  Google Scholar

[5]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar

[6]

G. CaiQ-.L. Dong and Y. Peng, Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-lipschitz operators, J. Optim. Theory Appl., 188 (2020), 447-472.  doi: 10.1007/s10957-020-01792-w.  Google Scholar

[7]

L. C. CengQ. H. Ansari and Q. H. Yao, Relaxed extragradient methods for finding minimum-norm solutions of the split feasibility problem, Nonlinear Analysis, 75 (2012), 2116-2125.  doi: 10.1016/j.na.2011.10.012.  Google Scholar

[8]

Y. CensorA. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[9]

Y. CensorX. A. Motova and A. Segal, Perturbed projections and subgradient projections for the multiple-set split feasibility problem, J. Math. Anal. Appl., 327 (2007), 1224-1256.  doi: 10.1016/j.jmaa.2006.05.010.  Google Scholar

[10]

Y. CensorT. ElfvingN. Kopt and T. Bortfeld, The multiple-sets split feasibility problem and its applications, Inverse Prob., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[11]

G. Ficher, Sul pproblem elastostatico di signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Rend., Cl. Sci. Fis. Mat. Natur, 34 (1963), 138-142.   Google Scholar

[12]

G. Ficher, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincci, Cl. Sci. Fis. Mat. Nat., Sez., 7 (1964), 91-140.   Google Scholar

[13]

Y. Hao, Some results of variational inclusion problems and fixed point problems with applications, Appl. Math. Mech., 30 (2009), 1589-1596.  doi: 10.1007/s10483-009-1210-x.  Google Scholar

[14]

Z. HeC. Chen and F. Gu, Viscosity approximation method for nonexpansive nonself-nonexpansive mappings and variational inequality, J. Nonlinear Sci. Appl., 1 (2008), 169-178.  doi: 10.22436/jnsa.001.03.05.  Google Scholar

[15]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, 1$^st$ edition, Marcel Dekker, New York, 1984.  Google Scholar

[16]

E. Kopeck and S. Reich, A note on alternating projections in Hilbert space, J. Fixed Point Theory Appl., 12 (2012), 41-47.  doi: 10.1007/s11784-013-0097-4.  Google Scholar

[17]

H. Liu and J. Yang, Weak convergence of iterative methods for solving quasimonotone variational inequalities, Computation Optimization and Applications, 77 (2020), 491-508.  doi: 10.1007/s10589-020-00217-8.  Google Scholar

[18]

P. E. Mainge, Regularized and inertial algorithms for common fixed points of nonlinear operators, J. Math. Anal. Appl., 34 (2008), 876-887.  doi: 10.1016/j.jmaa.2008.03.028.  Google Scholar

[19]

P. E. Mainge, A hybrid extragradient viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515.  doi: 10.1137/060675319.  Google Scholar

[20]

N. H. MinhL. H. M. Van and T. V. Anh, An algorithm for a class of bilevel variational inequality with split variational inequality and fixed point problem constraints, Acta Mathematica Vietnamica, 46 (2021), 515-530.  doi: 10.1007/s40306-020-00389-9.  Google Scholar

[21]

Y. Nesterov, A method of solving a convex programming problem with convergence rate O($1/k^2$), Soviet Math. Doklady, 27 (1983), 372-376.   Google Scholar

[22]

B. T. Polyak, Some methods of speeding up the convergence of iterates methods, U.S.S.R Comput. Math. Phys., 4 (1964), 1-17.   Google Scholar

[23]

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341-350.  doi: 10.1016/j.na.2003.07.023.  Google Scholar

[24]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 57 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.  Google Scholar

[25]

G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes,, C. R. Math. Acad. Sci., 258 (1964), 4413-4416.   Google Scholar

[26]

D. V. ThongD. V. Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving psedomonotone variational inequality problems, Optim. Lett., 14 (2020), 115-144.  doi: 10.1007/s11590-019-01511-z.  Google Scholar

[27]

D. V. Thong and D. V. Hieu, Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems, Numer. Algor., 82 (2019), 761-789.  doi: 10.1007/s11075-018-0626-8.  Google Scholar

[28]

J. Yang and H. Liu, A modified projected gradient method for monotone variational inequalities, J. Optim Theory Appl., 179 (2018), 197-211.  doi: 10.1007/s10957-018-1351-0.  Google Scholar

Figure 1.  Example 1, Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: case Ⅲ; Bottom Right: Case Ⅳ
Figure 2.  Example 2, Top Left: Case A; Top Right: Case B; Bottom Left: Case C; Bottom Right: Case D
Figure 3.  Example 3, Top Left: Case Ⅰ; Top Right: Case Ⅱ; Bottom Left: case Ⅲ; Bottom Right: Case Ⅳ
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