Article Contents
Article Contents

# An extension of hybrid method without extrapolation step to equilibrium problems

• 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.

Mathematics Subject Classification: Primary: 65K10, 65K15; Secondary: 90C33.

 Citation:

• Table 1.  Results for given starting points in Example 1

 $x_0$ Iter. Time Alg. 1 Alg. 3 Alg. 4 Alg. 5 Alg. 1 Alg. 3 Alg. 4 Alg. 5 (2, 5) 241 111 221 122 1.928 2.331 1.768 4.636 (5, 5) 192 108 215 122 2.880 2.700 2.580 3.294 (4, 4.5) 194 108 215 122 2.910 2.700 2.795 3.294 (-0.75, 0) 196 108 215 122 3.724 3.132 2.795 3.172

Table 2.  Results for given starting points in Example 2

 $x_0$ Iter. Time Alg. 1 Alg. 3 Alg. 4 Alg. 5 Alg. 1 Alg. 3 Alg. 4 Alg. 5 $x_0^1$ 1918 1022 225 812 15.344 10.220 7.875 334.544 $x_0^2$ 1470 524 173 924 10.290 8.384 5.709 370.524 $x_0^3$ 1421 980 296 1043 12.789 32.34 10.360 617.456

Table 3.  Results for different given parameter $\lambda$ in Example 2

 $\tau$ Iter. Time Alg. 1 Alg. 3 Alg. 4 Alg. 1 Alg. 3 Alg. 4 $10^{-2}$ 1423 572 351 25.614 20.020 15.093 $10^{-4}$ 1341 655 373 22.797 20.960 16.039 $10^{-6}$ 1093 635 367 18.581 17.145 15.414

Table 4.  Numerical results for Example 3

 $m$ Iter. Time Alg. 1 Alg. 3 Alg. 4 Alg. 5 Alg. 1 Alg. 3 Alg. 4 Alg. 5 $10$ 521 212 183 312 17.193 13.356 17.934 37.440 $15$ 674 406 366 467 31.004 32.074 25.254 135.897 $20$ 912 388 348 Slow 127.680 111.356 98.136 - $50$ Slow - - - - - - -

Table 5.  The parameters of the functions $c_j$ in Example 4

 $j$ $\hat{\alpha}_j$ $\hat{\beta}_j$ $\hat{\gamma}_j$ $\tilde{\alpha}_j$ $\tilde{\beta}_j$ $\tilde{\gamma}_j$ 1 0.0400 2.00 0.00 2.0000 1.0000 25.0000 2 0.0350 1.75 0.00 1.7500 1.0000 28.5714 3 0.1250 1.00 0.00 1.0000 1.0000 8.0000 4 0.0116 3.25 0.00 3.2500 1.0000 86.2069 5 0.0500 3.00 0.00 3.0000 1.0000 20.0000 6 0.0500 3.00 0.00 3.0000 1.0000 20.0000

Table 6.  Results for given starting points in Example 4

 $x_0$ Iter. Time Alg. 1 Alg. 3 Alg. 4 Alg. 5 Alg. 1 Alg. 3 Alg. 4 Alg. 5 $x_0^1$ 1965 934 557 805 47.160 36.426 23.394 79.695 $x_0^2$ 2001 1023 573 928 58.029 31.713 26.358 144.768 $x_0^3$ 1877 1279 567 1011 46.925 48.602 24.948 105.144 $x_0^4$ 1574 1089 598 873 39.350 34.848 24.518 139.680

Table 7.  Results for given starting points in Example 5

 $x_0$ Iter. Time Alg. 1 Alg. 3 Alg. 1 Alg. 3 $x_0^1$ 2365 1565 2059.915 3428.915 $x_0^2$ 2417 1437 2259.895 3463.170 $x_0^3$ 2877 1729 2557.653 3577.301
•  [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. Ceng, N. 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. Censor, A. 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. Censor, A. 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. Hieu, P. 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. Hieu, L. 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. Lyashko, V. 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. Mordukhovich, B. Panicucci, M. 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. Nguyen, J. 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. Nguyen, J. J. Strodiot, V. 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. Quoc, L. D. Muu and N. V. Hien, Extragradient algorithms extended to equilibrium problems, Optimization, 57 (2008), 749-776.  doi: 10.1080/02331930601122876. [30] T. D. Quoc, P. 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. Strodiot, T. 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. Vuong, J. 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.

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