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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[25]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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. 

[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.

[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.

[25]

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

[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.

[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.

[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.

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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