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March  2022, 18(2): 773-794. doi: 10.3934/jimo.2020178

Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems

Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, P.O. Box 90 Medunsa 0204, South Africa

* Corresponding author: Lateef Olakunle Jolaoso

Received  March 2020 Revised  September 2020 Published  March 2022 Early access  December 2020

Using the concept of Bregman divergence, we propose a new subgradient extragradient method for approximating a common solution of pseudo-monotone and Lipschitz continuous variational inequalities and fixed point problem in real Hilbert spaces. The algorithm uses a new self-adjustment rule for selecting the stepsize in each iteration and also, we prove a strong convergence result for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. Finally, we provide some numerical examples to illustrate the performance and accuracy of our algorithm in finite and infinite dimensional spaces.

Citation: Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial and Management Optimization, 2022, 18 (2) : 773-794. doi: 10.3934/jimo.2020178
References:
[1]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self-adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, (2020). doi: 10.1080/02331934.2020.1723586.

[2]

Y. I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G.Kartsatos (Ed.), Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, Marcel Dekker, New York, 178 (1996), 15-50.

[3]

A. S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekonomika i Mat. Metody., 12 (1976), 1164-1173.

[4]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264. doi: 10.1287/moor.26.2.248.10558.

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, New York, Springer, 2011. (CMS Books in Mathematics). doi: 10.1007/978-1-4419-9467-7.

[6]

A. Beck, First-Order Methods in Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974997.ch1.

[7]

J. Y. Bello Cruz and A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Numer. Funct. Anal. Optim., 30 (2009), 23-36. doi: 10.1080/01630560902735223.

[8]

L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7 (1967), 200-217.

[9]

Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for variational inequality problems in Euclidean space, Optim., 61 (2012), 1119-1132. doi: 10.1080/02331934.2010.539689.

[10]

Y. Censor, A. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Software, 26 (2011), 827-845. doi: 10.1080/10556788.2010.551536.

[11]

Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 148 (2011), 318-335. doi: 10.1007/s10957-010-9757-3.

[12]

Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl., 34 (1981), 321-353.  doi: 10.1007/BF00934676.

[13]

S. V. Denisov, V. V. Semenov and P. I. Stetsynk, Bregman extragradient method with monotone rule of step adjustment, Cybern. Syst. Analysis, 55 (2019), 377-383.

[14]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. Ⅱ, Springer Series in Operations Research, Springer, New York, 2003.

[15]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34 (1963), 138-142.

[16]

A. Gibali, A new Bregman projection method for solving variational inequalities in Hilbert spaces, Pure and Appl. Funct. Analy., 3 (2018), 403-415.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, (Marcel Dekker, New York, 1984.

[18]

A. Gibali, S. Reich and R. Zalas, Iterative methods for solving variational inequalities in Euclidean space, J. Fixed Point Theory Appl., 17 (2015), 775-811. doi: 10.1007/s11784-015-0256-x.

[19]

B. Halpern, Fixed points of nonexpanding maps, Proc. Amer. Math. Soc., 73 (1967), 957-961. doi: 10.1090/S0002-9904-1967-11864-0.

[20]

P. Hartman and G. Stampacchia, On some non linear elliptic differential-functional equations, Acta Mathematica, 115 (1966), 271-310. doi: 10.1007/BF02392210.

[21]

H. Iiduka, A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping, Optimization, 59 (2010), 873-885. doi: 10.1080/02331930902884158.

[22]

H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim., 19 (2009), 1881-1893. doi: 10.1137/070702497.

[23]

H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications, Optimization, 58 (2009), 251-261. doi: 10.1080/02331930701762829.

[24]

A. N. Iusem and B. F. Svaiter, A variant of Korpelevich?s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321. doi: 10.1080/02331939708844365.

[25]

L. O. Jolaoso and M. Aphane, Weak and strong convergence Bregman extragradient schemes for solving pseudo-monotone and non-Lipschitz variational inequalities, J. Ineq. Appl., (2020), Paper No. 195, 25 pp. doi: 10.1186/s13660-020-02462-1.

[26]

L. O. Jolaoso and I. Karahan, A general alternative regularization method with line search technique for solving split equilibrium and fixed point problems in Hilbert spaces, Comput. Appl. Math., 39 (2020), Article 150, 22pp. doi: 10.1007/s40314-020-01178-8.

[27]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766. doi: 10.1007/s10957-020-01672-3.

[28]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Paper No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.

[29]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412. doi: 10.1007/s10957-013-0494-2.

[30]

E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comput. Math. Math. Phys., 27 (1987), 120-127.

[31]

D. Kinderlehrer and G. Stampachia, An introduction to variational inequalities and Their Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2000. doi: 10.1137/1.9780898719451.

[32]

F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005), 505-523.

[33]

G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekon. Mat. Metody, 12 (1976), 747-756.

[34]

L. J. Lin, M. F. Yang, Q. H. Ansari and G. Kassay, Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps, Nonlinear Analy. Theory Methods and Appl., 61 (2005), 1-19. doi: 10.1016/j.na.2004.07.038.

[35]

J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.

[36]

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.

[37]

P. E. Mainge, Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with the line-search procedure, Comput. Math. Appl., 72 (2016), 720-728. doi: 10.1016/j.camwa.2016.05.028.

[38]

P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912. doi: 10.1007/s11228-008-0102-z.

[39]

P. E. Mainge and M. L. Gobindass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168. doi: 10.1007/s10957-016-0972-4.

[40]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520. doi: 10.1137/14097238X.

[41]

E. Naraghirad and J.-C. yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Appl., 2013 (2013), Article ID: 141, 43pp. doi: 10.1186/1687-1812-2013-141.

[42]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Analy., 72 (2010), 2086-2099. doi: 10.1016/j.na.2009.10.009.

[43]

A. Nemirovski, Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. on Optimization, 15 (2004), 229-251. doi: 10.1137/S1052623403425629.

[44]

D. A. Nomirovskii, B. V. Rublyov and V. V. Semenov, Convergence of two-step method with Bregman divergence for solving variational inequalities, Cybern. Syst. Analysis, 55 (2019), 359-368.

[45]

R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, 2nd Edition, in: Lecture Notes in Mathematics, vol. 1364, Springer Verlag, Berlin, 1993.

[46]

S. Reich and S. Sabach, A strong convergence theorem for proximal type- algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10 (2009), 471-485.

[47]

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

[48]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control and Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.

[49]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

[50]

J. Yang and H. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert space, J. Numer Algor, 80 (2019), 741-752. doi: 10.1007/s11075-018-0504-4.

show all references

References:
[1]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self-adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, (2020). doi: 10.1080/02331934.2020.1723586.

[2]

Y. I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G.Kartsatos (Ed.), Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, Marcel Dekker, New York, 178 (1996), 15-50.

[3]

A. S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Ekonomika i Mat. Metody., 12 (1976), 1164-1173.

[4]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264. doi: 10.1287/moor.26.2.248.10558.

[5]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, New York, Springer, 2011. (CMS Books in Mathematics). doi: 10.1007/978-1-4419-9467-7.

[6]

A. Beck, First-Order Methods in Optimization, Society for Industrial and Applied Mathematics, Philadelphia, 2017. doi: 10.1137/1.9781611974997.ch1.

[7]

J. Y. Bello Cruz and A. N. Iusem, A strongly convergent direct method for monotone variational inequalities in Hilbert spaces, Numer. Funct. Anal. Optim., 30 (2009), 23-36. doi: 10.1080/01630560902735223.

[8]

L. M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7 (1967), 200-217.

[9]

Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for variational inequality problems in Euclidean space, Optim., 61 (2012), 1119-1132. doi: 10.1080/02331934.2010.539689.

[10]

Y. Censor, A. Gibali and S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Software, 26 (2011), 827-845. doi: 10.1080/10556788.2010.551536.

[11]

Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 148 (2011), 318-335. doi: 10.1007/s10957-010-9757-3.

[12]

Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl., 34 (1981), 321-353.  doi: 10.1007/BF00934676.

[13]

S. V. Denisov, V. V. Semenov and P. I. Stetsynk, Bregman extragradient method with monotone rule of step adjustment, Cybern. Syst. Analysis, 55 (2019), 377-383.

[14]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. Ⅱ, Springer Series in Operations Research, Springer, New York, 2003.

[15]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 34 (1963), 138-142.

[16]

A. Gibali, A new Bregman projection method for solving variational inequalities in Hilbert spaces, Pure and Appl. Funct. Analy., 3 (2018), 403-415.

[17]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, (Marcel Dekker, New York, 1984.

[18]

A. Gibali, S. Reich and R. Zalas, Iterative methods for solving variational inequalities in Euclidean space, J. Fixed Point Theory Appl., 17 (2015), 775-811. doi: 10.1007/s11784-015-0256-x.

[19]

B. Halpern, Fixed points of nonexpanding maps, Proc. Amer. Math. Soc., 73 (1967), 957-961. doi: 10.1090/S0002-9904-1967-11864-0.

[20]

P. Hartman and G. Stampacchia, On some non linear elliptic differential-functional equations, Acta Mathematica, 115 (1966), 271-310. doi: 10.1007/BF02392210.

[21]

H. Iiduka, A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping, Optimization, 59 (2010), 873-885. doi: 10.1080/02331930902884158.

[22]

H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim., 19 (2009), 1881-1893. doi: 10.1137/070702497.

[23]

H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications, Optimization, 58 (2009), 251-261. doi: 10.1080/02331930701762829.

[24]

A. N. Iusem and B. F. Svaiter, A variant of Korpelevich?s method for variational inequalities with a new search strategy, Optimization, 42 (1997), 309-321. doi: 10.1080/02331939708844365.

[25]

L. O. Jolaoso and M. Aphane, Weak and strong convergence Bregman extragradient schemes for solving pseudo-monotone and non-Lipschitz variational inequalities, J. Ineq. Appl., (2020), Paper No. 195, 25 pp. doi: 10.1186/s13660-020-02462-1.

[26]

L. O. Jolaoso and I. Karahan, A general alternative regularization method with line search technique for solving split equilibrium and fixed point problems in Hilbert spaces, Comput. Appl. Math., 39 (2020), Article 150, 22pp. doi: 10.1007/s40314-020-01178-8.

[27]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766. doi: 10.1007/s10957-020-01672-3.

[28]

L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), Paper No. 38, 28 pp. doi: 10.1007/s40314-019-1014-2.

[29]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412. doi: 10.1007/s10957-013-0494-2.

[30]

E. N. Khobotov, Modification of the extragradient method for solving variational inequalities and certain optimization problems, USSR Comput. Math. Math. Phys., 27 (1987), 120-127.

[31]

D. Kinderlehrer and G. Stampachia, An introduction to variational inequalities and Their Applications, Society for Industrial and Applied Mathematics, Philadelphia, 2000. doi: 10.1137/1.9780898719451.

[32]

F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005), 505-523.

[33]

G. M. Korpelevich, An extragradient method for finding saddle points and for other problems, Ekon. Mat. Metody, 12 (1976), 747-756.

[34]

L. J. Lin, M. F. Yang, Q. H. Ansari and G. Kassay, Existence results for Stampacchia and Minty type implicit variational inequalities with multivalued maps, Nonlinear Analy. Theory Methods and Appl., 61 (2005), 1-19. doi: 10.1016/j.na.2004.07.038.

[35]

J. L. Lions and G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. doi: 10.1002/cpa.3160200302.

[36]

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.

[37]

P. E. Mainge, Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with the line-search procedure, Comput. Math. Appl., 72 (2016), 720-728. doi: 10.1016/j.camwa.2016.05.028.

[38]

P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912. doi: 10.1007/s11228-008-0102-z.

[39]

P. E. Mainge and M. L. Gobindass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168. doi: 10.1007/s10957-016-0972-4.

[40]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520. doi: 10.1137/14097238X.

[41]

E. Naraghirad and J.-C. yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory and Appl., 2013 (2013), Article ID: 141, 43pp. doi: 10.1186/1687-1812-2013-141.

[42]

J. Mashreghi and M. Nasri, Forcing strong convergence of Korpelevich's method in Banach spaces with its applications in game theory, Nonlinear Analy., 72 (2010), 2086-2099. doi: 10.1016/j.na.2009.10.009.

[43]

A. Nemirovski, Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems, SIAM J. on Optimization, 15 (2004), 229-251. doi: 10.1137/S1052623403425629.

[44]

D. A. Nomirovskii, B. V. Rublyov and V. V. Semenov, Convergence of two-step method with Bregman divergence for solving variational inequalities, Cybern. Syst. Analysis, 55 (2019), 359-368.

[45]

R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, 2nd Edition, in: Lecture Notes in Mathematics, vol. 1364, Springer Verlag, Berlin, 1993.

[46]

S. Reich and S. Sabach, A strong convergence theorem for proximal type- algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal., 10 (2009), 471-485.

[47]

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

[48]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control and Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.

[49]

H. K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

[50]

J. Yang and H. Liu, Strong convergence result for solving monotone variational inequalities in Hilbert space, J. Numer Algor, 80 (2019), 741-752. doi: 10.1007/s11075-018-0504-4.

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: $ m = 5 $; Top Right: $ m = 15 $, Bottom: $ m = 30 $
Table 1.  Computation result for Example 1
Algorithm 4 Algorithm 1 Algorithm 3
Case I Iter. 5 12 29
Time 0.6406 1.0043 0.7661
Case II Iter. 12 45 49
Time 3.0910 9.5282 3.3343
Case III Iter. 10 22 39
Time 1.1391 3.0101 1.7377
Case IV Iter. 13 56 53
Time 0.8596 3.9885 1.8918
Algorithm 4 Algorithm 1 Algorithm 3
Case I Iter. 5 12 29
Time 0.6406 1.0043 0.7661
Case II Iter. 12 45 49
Time 3.0910 9.5282 3.3343
Case III Iter. 10 22 39
Time 1.1391 3.0101 1.7377
Case IV Iter. 13 56 53
Time 0.8596 3.9885 1.8918
Table 2.  Computation result for Example 2
Algorithm 4 Algorithm 3
$ m=5 $ Iter. 7 11
Time 0.0036 0.0050
$ m=15 $ Iter. 8 13
Time 0.0052 0.0099
$ m=30 $ Iter. 8 27
Time 0.0255 0.0884
Algorithm 4 Algorithm 3
$ m=5 $ Iter. 7 11
Time 0.0036 0.0050
$ m=15 $ Iter. 8 13
Time 0.0052 0.0099
$ m=30 $ Iter. 8 27
Time 0.0255 0.0884
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