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October  2017, 13(4): 1723-1741. doi: 10.3934/jimo.2017015

An extension of hybrid method without extrapolation step to equilibrium problems

Department of Mathematics, Vietnam National University, 334 - Nguyen Trai Street, Thanh Xuan, Ha Noi 100000, Viet Nam

* Corresponding author: dv.hieu83@gmail.com

Received  October 2015 Published  December 2016

In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method and the hybrid (outer approximation) method. In this algorithm, only one optimization program is solved at each iteration without any extra-step dealing with the feasible set like as in the hybrid extragradient method and the hybrid Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in the proposed algorithm. The strongly convergent theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.

Citation: Van Hieu Dang. An extension of hybrid method without extrapolation step to equilibrium problems. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1723-1741. doi: 10.3934/jimo.2017015
References:
[1]

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

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Program., 63 (1994), 123-145. 

[3]

L. C. CengN. Hadjisavvas and N. C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646.  doi: 10.1007/s10898-009-9454-7.

[4]

L. C. Ceng and J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., 190 (2007), 205-215.  doi: 10.1016/j.amc.2007.01.021.

[5]

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

[6]

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

[7]

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal, 6 (2005), 117-136. 

[8]

P. Daniele, F. Giannessi and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003. doi: 10.1007/978-1-4613-0239-1.

[9]

K. Fan, A minimax inequality and applications, In: Shisha, O. (ed. ) Inequality, Ⅲ [Academic Press, New York, 1972], 103-113.

[10]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984. doi: MR744194.

[11]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Math., vol. 28 and Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511526152.

[12]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., (2016), 1-22.  doi: 10.1007/s10589-016-9857-6.

[13]

D. V. Hieu, A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space, J. Korean Math. Soc., 52 (2015), 373-388.  doi: 10.4134/JKMS.2015.52.2.373.

[14]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.

[15]

D. V. HieuL. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5.

[16]

D. V. Hieu, Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings, J. Appl. Math. Comput., 73 (2016), 1-24.  doi: 10.1007/s12190-015-0980-9.

[17]

H. Iiduka, Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping, Math. Program. Ser. A, 149 (2015), 131-165.  doi: 10.1007/s10107-013-0741-1.

[18]

I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56886-2.

[19]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747-756. 

[20]

S. I. LyashkoV. V. Semenov and T. A. Voitova, Low-cost modification of Korpelevich's methods for monotone equilibrium problems, Cybernetics and Systems Analysis, 47 (2011), 631-639.  doi: 10.1007/s10559-011-9343-1.

[21]

Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202.  doi: 10.1007/s10898-014-0150-x.

[22]

B. Martinet, R$\rm\acute{e}$gularisation d$\rm\acute{i}$n$\rm\acute{e}$quations variationelles par approximations successives, Rev. Fr. Autom. Inform. Rech. Op$\rm\acute{e}$r., Anal. Num$\acute{e}$r., 4 (1970), 154-158. 

[23]

G. Mastroeni, On auxiliary principle for equilibrium problems, Chapter: Equilibrium Problems and Variational Models, 68 (2003), 289-298.  doi: 10.1007/978-1-4613-0239-1_15.

[24]

B. MordukhovichB. PanicucciM. Pappalardo and M. Passacantando, Hybrid proximal methods for equilibrium problems, Optim. Lett., 6 (2012), 1535-1550.  doi: 10.1007/s11590-011-0348-5.

[25]

L. D. Muu and W. Oettli, Convergence of an adative penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA, 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.

[26]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim, 16 (2006), 1230-1241.  doi: 10.1137/050624315.

[27]

T. T. V. NguyenJ. J. Strodiot and V. H. Nguyen, Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, J. Optim. Theory Appl., 160 (2014), 809-831.  doi: 10.1007/s10957-013-0400-y.

[28]

T. P. D. NguyenJ. J. StrodiotV. H. Nguyen and T. T. V. Nguyen, A family of extragradient methods for solving equilibrium problems, J. Ind. Manag. Optim., 11 (2015), 619-630.  doi: 10.3934/jimo.2015.11.619.

[29]

T. D. QuocL. D. Muu and N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776.  doi: 10.1080/02331930601122876.

[30]

T. D. QuocP. N. Anh and L. D. Muu, Dual extragradient algorithms extended to equilibrium problems, J. Glob. Optim., 52 (2012), 139-159.  doi: 10.1007/s10898-011-9693-2.

[31]

R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, 1970. doi: MR0274683.

[32]

J. J. StrodiotT. T. V. Nguyen and V. H. Nguyen, A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems, J. Glob. Optim., 56 (2013), 373-397.  doi: 10.1007/s10898-011-9814-y.

[33]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2012), 605-627.  doi: 10.1007/s10957-012-0085-7.

[34]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, In: Butnariu, D., Censor, Y., Reich, S. (eds. ) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, Elsevier, Amsterdam, 8 (2001), 473-504. doi: 10.1016/S1570-579X(01)80028-8.

show all references

References:
[1]

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

[2]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Program., 63 (1994), 123-145. 

[3]

L. C. CengN. Hadjisavvas and N. C. Wong, Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems, J. Glob. Optim., 46 (2010), 635-646.  doi: 10.1007/s10898-009-9454-7.

[4]

L. C. Ceng and J. C. Yao, An extragradient-like approximation method for variational inequality problems and fixed point problems, Appl. Math. Comput., 190 (2007), 205-215.  doi: 10.1016/j.amc.2007.01.021.

[5]

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

[6]

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

[7]

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal, 6 (2005), 117-136. 

[8]

P. Daniele, F. Giannessi and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003. doi: 10.1007/978-1-4613-0239-1.

[9]

K. Fan, A minimax inequality and applications, In: Shisha, O. (ed. ) Inequality, Ⅲ [Academic Press, New York, 1972], 103-113.

[10]

K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984. doi: MR744194.

[11]

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Math., vol. 28 and Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511526152.

[12]

D. V. HieuP. K. Anh and L. D. Muu, Modified hybrid projection methods for finding common solutions to variational inequality problems, Comput. Optim. Appl., (2016), 1-22.  doi: 10.1007/s10589-016-9857-6.

[13]

D. V. Hieu, A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space, J. Korean Math. Soc., 52 (2015), 373-388.  doi: 10.4134/JKMS.2015.52.2.373.

[14]

D. V. Hieu, Parallel extragradient-proximal methods for split equilibrium problems, Math. Model. Anal., 21 (2016), 478-501.  doi: 10.3846/13926292.2016.1183527.

[15]

D. V. HieuL. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5.

[16]

D. V. Hieu, Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings, J. Appl. Math. Comput., 73 (2016), 1-24.  doi: 10.1007/s12190-015-0980-9.

[17]

H. Iiduka, Acceleration method for convex optimization over the fixed point set of a nonexpansive mapping, Math. Program. Ser. A, 149 (2015), 131-165.  doi: 10.1007/s10107-013-0741-1.

[18]

I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56886-2.

[19]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Metody, 12 (1976), 747-756. 

[20]

S. I. LyashkoV. V. Semenov and T. A. Voitova, Low-cost modification of Korpelevich's methods for monotone equilibrium problems, Cybernetics and Systems Analysis, 47 (2011), 631-639.  doi: 10.1007/s10559-011-9343-1.

[21]

Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202.  doi: 10.1007/s10898-014-0150-x.

[22]

B. Martinet, R$\rm\acute{e}$gularisation d$\rm\acute{i}$n$\rm\acute{e}$quations variationelles par approximations successives, Rev. Fr. Autom. Inform. Rech. Op$\rm\acute{e}$r., Anal. Num$\acute{e}$r., 4 (1970), 154-158. 

[23]

G. Mastroeni, On auxiliary principle for equilibrium problems, Chapter: Equilibrium Problems and Variational Models, 68 (2003), 289-298.  doi: 10.1007/978-1-4613-0239-1_15.

[24]

B. MordukhovichB. PanicucciM. Pappalardo and M. Passacantando, Hybrid proximal methods for equilibrium problems, Optim. Lett., 6 (2012), 1535-1550.  doi: 10.1007/s11590-011-0348-5.

[25]

L. D. Muu and W. Oettli, Convergence of an adative penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA, 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.

[26]

N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim, 16 (2006), 1230-1241.  doi: 10.1137/050624315.

[27]

T. T. V. NguyenJ. J. Strodiot and V. H. Nguyen, Hybrid methods for solving simultaneously an equilibrium problem and countably many fixed point problems in a Hilbert space, J. Optim. Theory Appl., 160 (2014), 809-831.  doi: 10.1007/s10957-013-0400-y.

[28]

T. P. D. NguyenJ. J. StrodiotV. H. Nguyen and T. T. V. Nguyen, A family of extragradient methods for solving equilibrium problems, J. Ind. Manag. Optim., 11 (2015), 619-630.  doi: 10.3934/jimo.2015.11.619.

[29]

T. D. QuocL. D. Muu and N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776.  doi: 10.1080/02331930601122876.

[30]

T. D. QuocP. N. Anh and L. D. Muu, Dual extragradient algorithms extended to equilibrium problems, J. Glob. Optim., 52 (2012), 139-159.  doi: 10.1007/s10898-011-9693-2.

[31]

R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, 1970. doi: MR0274683.

[32]

J. J. StrodiotT. T. V. Nguyen and V. H. Nguyen, A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems, J. Glob. Optim., 56 (2013), 373-397.  doi: 10.1007/s10898-011-9814-y.

[33]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2012), 605-627.  doi: 10.1007/s10957-012-0085-7.

[34]

I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, In: Butnariu, D., Censor, Y., Reich, S. (eds. ) Inherently Parallel Algorithms for Feasibility and Optimization and Their Applications, Elsevier, Amsterdam, 8 (2001), 473-504. doi: 10.1016/S1570-579X(01)80028-8.

Table 1.  Results for given starting points in Example 1
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
(2, 5)2411112211221.9282.3311.7684.636
(5, 5)1921082151222.8802.7002.5803.294
(4, 4.5)1941082151222.9102.7002.7953.294
(-0.75, 0)1961082151223.7243.1322.7953.172
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
(2, 5)2411112211221.9282.3311.7684.636
(5, 5)1921082151222.8802.7002.5803.294
(4, 4.5)1941082151222.9102.7002.7953.294
(-0.75, 0)1961082151223.7243.1322.7953.172
Table 2.  Results for given starting points in Example 2
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$x_0^1$1918102222581215.34410.2207.875334.544
$x_0^2$147052417392410.2908.3845.709370.524
$x_0^3$1421980296104312.78932.3410.360617.456
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$x_0^1$1918102222581215.34410.2207.875334.544
$x_0^2$147052417392410.2908.3845.709370.524
$x_0^3$1421980296104312.78932.3410.360617.456
Table 3.  Results for different given parameter $\lambda$ in Example 2
$\tau$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 1Alg. 3Alg. 4
$10^{-2}$142357235125.61420.02015.093
$10^{-4}$134165537322.79720.96016.039
$10^{-6}$109363536718.58117.14515.414
$\tau$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 1Alg. 3Alg. 4
$10^{-2}$142357235125.61420.02015.093
$10^{-4}$134165537322.79720.96016.039
$10^{-6}$109363536718.58117.14515.414
Table 4.  Numerical results for Example 3
$m$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$10$52121218331217.19313.35617.93437.440
$15$67440636646731.00432.07425.254135.897
$20$912388348Slow127.680111.35698.136-
$50$Slow-------
$m$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$10$52121218331217.19313.35617.93437.440
$15$67440636646731.00432.07425.254135.897
$20$912388348Slow127.680111.35698.136-
$50$Slow-------
Table 5.  The parameters of the functions $c_j$ in Example 4
$j$$\hat{\alpha}_j$$\hat{\beta}_j$$\hat{\gamma}_j$$\tilde{\alpha}_j$$\tilde{\beta}_j$$\tilde{\gamma}_j$
10.04002.000.002.00001.000025.0000
20.03501.750.001.75001.000028.5714
30.12501.000.001.00001.00008.0000
40.01163.250.003.25001.000086.2069
50.05003.000.003.00001.000020.0000
60.05003.000.003.00001.000020.0000
$j$$\hat{\alpha}_j$$\hat{\beta}_j$$\hat{\gamma}_j$$\tilde{\alpha}_j$$\tilde{\beta}_j$$\tilde{\gamma}_j$
10.04002.000.002.00001.000025.0000
20.03501.750.001.75001.000028.5714
30.12501.000.001.00001.00008.0000
40.01163.250.003.25001.000086.2069
50.05003.000.003.00001.000020.0000
60.05003.000.003.00001.000020.0000
Table 6.  Results for given starting points in Example 4
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$x_0^1$196593455780547.16036.42623.39479.695
$x_0^2$2001102357392858.02931.71326.358144.768
$x_0^3$18771279567101146.92548.60224.948105.144
$x_0^4$1574108959887339.35034.84824.518139.680
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 4Alg. 5Alg. 1Alg. 3Alg. 4Alg. 5
$x_0^1$196593455780547.16036.42623.39479.695
$x_0^2$2001102357392858.02931.71326.358144.768
$x_0^3$18771279567101146.92548.60224.948105.144
$x_0^4$1574108959887339.35034.84824.518139.680
Table 7.  Results for given starting points in Example 5
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 1Alg. 3
$x_0^1$236515652059.9153428.915
$x_0^2$241714372259.8953463.170
$x_0^3$287717292557.6533577.301
$x_0$Iter.Time
Alg. 1Alg. 3Alg. 1Alg. 3
$x_0^1$236515652059.9153428.915
$x_0^2$241714372259.8953463.170
$x_0^3$287717292557.6533577.301
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