A projection-like method for quasimonotone variational inequalities without Lipschitz continuity. (English) Zbl 1503.90145
Summary: For most projection methods, the operator of a variational inequality problem is assumed to be monotone (or pseudomonotone) and Lipschitz continuous. In this paper, we present a projection-like method to solve quasimonotone variational inequality problems without Lipschitz continuity. Under some mild assumptions, we prove that the sequence generated by the proposed algorithm converges to a solution. Numerical experiments are provided to show the effectiveness of the method.
MSC:
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
Keywords:
projection-like method; variational inequality problem; quasimonotone operator; convergenceReferences:
| [1] | Korpelevich, GM, The extragradient method for finding saddle points and other problems, Ekonomika i Mat Metody., 12, 747-756 (1976) · Zbl 0342.90044 |
| [2] | Censor, Y.; Gibali, A.; Reich, S., The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148, 318-335 (2011) · Zbl 1229.58018 · doi:10.1007/s10957-010-9757-3 |
| [3] | Censor, Y.; Gibali, A.; Reich, S., Strong convergence of subgradient extragradient methods for the variational inequality problems in Hilbert space, Optim., 26, 827-845 (2011) · Zbl 1232.58008 |
| [4] | Censor, Y.; Gibali, A.; Reich, S., Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optim., 61, 1119-1132 (2012) · Zbl 1260.65056 · doi:10.1080/02331934.2010.539689 |
| [5] | Gibali, A., A new Bregman projection method for solving variational inequalities in Hilbert spaces, Pure Appl. Funct. Anal., 3, 403-415 (2018) · Zbl 1474.47125 |
| [6] | Hieu, DV; Thong, DV, New extragradient-like algorithms for strongly pseudomonotone variational inequalities, J. Glob. Optim., 70, 385-399 (2018) · Zbl 1384.65041 · doi:10.1007/s10898-017-0564-3 |
| [7] | Malitsky, YV; Semenov, VV, An extragradient algorithm for monotone variational inequalities, Cybern. Syst. Anal., 50, 271-277 (2014) · Zbl 1311.49024 · doi:10.1007/s10559-014-9614-8 |
| [8] | Rockfellar, RT, Monotone operators and proximal point algorithm, SIAM J. Control. Optim., 14, 877-898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056 |
| [9] | He, YR, A new double projection algorithm for variational inequalities, J. Comput. Appl. Math., 185, 166-173 (2006) · Zbl 1081.65066 · doi:10.1016/j.cam.2005.01.031 |
| [10] | Zheng, L.; Gou, QM, The subgradient double projection algorithm for solving variational inequalities, Acta Math. Appl. Sin., 37, 968-975 (2014) · Zbl 1324.90177 |
| [11] | Han, DR; Lo, HK, Two new self-adaptive projection methods for variational inequality problems, Comput. Math. Appl., 43, 1529-1537 (2002) · Zbl 1012.65064 · doi:10.1016/S0898-1221(02)00116-5 |
| [12] | Langenberg, N., An interior proximal method for a class of quasimonotone variational inequalities, J. Optim. Theory Appl., 155, 902-922 (2012) · Zbl 1257.90104 · doi:10.1007/s10957-012-0111-9 |
| [13] | Brito, AS; Da Cruz Neto, JX; Lopes, JO; Oliveira, PR, Interior proximal method for quasiconvex programming problems and variational inequalities with linear constraints, J. Optim. Theory Appl., 154, 217-234 (2012) · Zbl 1273.90238 · doi:10.1007/s10957-012-0002-0 |
| [14] | Yekini, S.; Qiao-Li, D.; Dan, J., Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optim., 68, 1385-409 (2019) · Zbl 1431.49009 |
| [15] | Morrey, CB, Quasiconvexity and lower semicontinuity of multiple intergrals, Pacific J. Math., 2, 25-53 (1952) · Zbl 0046.10803 · doi:10.2140/pjm.1952.2.25 |
| [16] | Morrey, CB, Multiple Integrals in the Calculus of Variations (1952), Berlin: Springer, Berlin · Zbl 1213.49002 |
| [17] | Ball, JM, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 86, 337-403 (1971) · Zbl 0368.73040 |
| [18] | Landes, R., Quasiminotone versus pseudomonotone, Proc. R. Soc. Edinburgh Sect. A Math., 126, 4, 705-717 (1996) · Zbl 0863.35033 · doi:10.1017/S0308210500023015 |
| [19] | Danilidis, A., Hadjisavvas: characterization of nonsmooth semistrictly quasiconvex and Strictly quasiconvex functions, J. Optim. Theory App., 102, 3, 525-536 (1999) · Zbl 1010.49013 · doi:10.1023/A:1022693822102 |
| [20] | Hadjisavvas, N.; Schaible, S., Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90, 1, 95-111 (1996) · Zbl 0904.49005 · doi:10.1007/BF02192248 |
| [21] | Ye, ML; He, YR, A double projection method for solving variational inequalities without monotonicity, Comput. Optim. Appl., 60, 141-150 (2015) · Zbl 1308.90184 · doi:10.1007/s10589-014-9659-7 |
| [22] | Hu, X.; Wang, J., Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17, 1487-1499 (2006) · doi:10.1109/TNN.2006.879774 |
| [23] | Sun, DF, A new step-size skill for solving a class of nonlinear equations, J. Comput. Math., 13, 357-368 (1995) · Zbl 0854.65048 |
| [24] | Liu, HW; Yang, J., Weak convergence of iterative methods for solving quasimonotone variational inequalities, Comput. Optim. Appl., 77, 491-508 (2020) · Zbl 07342388 · doi:10.1007/s10589-020-00217-8 |
| [25] | Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequality and Complementarity Problems, Volume I and II. Springer Series in Operations Research, (2003) · Zbl 1062.90002 |
| [26] | Zhu, T.; Yu, ZG, A simple proof for some important properties of the projection mapping, Math. Inequal. Appl., 7, 453-456 (2004) · Zbl 1086.49007 |
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