{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T09:31:50Z","timestamp":1744709510339},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,2,20]],"date-time":"2018-02-20T00:00:00Z","timestamp":1519084800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s11075-018-0499-x","type":"journal-article","created":{"date-parts":[[2018,2,20]],"date-time":"2018-02-20T17:41:35Z","timestamp":1519148495000},"page":"617-634","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Selective projection methods for solving a class of variational inequalities"],"prefix":"10.1007","volume":"80","author":[{"given":"Songnian","family":"He","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hanlin","family":"Tian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,2,20]]},"reference":[{"key":"499_CR1","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1186\/1687-1812-2014-17","volume":"2014","author":"K Aoyama","year":"2014","unstructured":"Aoyama, K., Kohsaka, F.: Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl. 2014, 17 (2014)","journal-title":"Fixed Point Theory Appl."},{"key":"499_CR2","volume-title":"Variational and Quasi Variational Inequalities","author":"C Baiocchi","year":"1984","unstructured":"Baiocchi, C., Capelo, A.: Variational and Quasi Variational Inequalities. Wiley, New York (1984)"},{"key":"499_CR3","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1016\/j.jmaa.2004.12.023","volume":"309","author":"A Bnouhachem","year":"2005","unstructured":"Bnouhachem, A.: A self-adaptive method for solving general mixed variational inequalities. J. Math. Anal. Appl. 309, 136\u2013150 (2005)","journal-title":"J. Math. Anal. Appl."},{"key":"499_CR4","doi-asserted-by":"publisher","DOI":"10.1201\/b16485","volume-title":"Iterative Optimization in Inverse Problems","author":"CL Byrne","year":"2014","unstructured":"Byrne, C.L.: Iterative Optimization in Inverse Problems. CRC Press, Boca Raton (2014)"},{"key":"499_CR5","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/s10589-013-9599-7","volume":"57","author":"XJ Cai","year":"2014","unstructured":"Cai, X.J., Gu, G.Y., He, B.S.: On the O ( 1 t ) $O(\\frac {1}{t})$ convergence rate of the projection and contraction methods for varitional inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. 57, 339\u2013363 (2014)","journal-title":"Comput. Optim. Appl."},{"key":"499_CR6","doi-asserted-by":"publisher","first-page":"2165","DOI":"10.1137\/15M1012438","volume":"25","author":"A Cegielski","year":"2015","unstructured":"Cegielski, A.: Application of quasi-nonexpansive operators to an iterative method for variational inequality. SIAM J. Optim. 25, 2165\u20132181 (2015)","journal-title":"SIAM J. Optim."},{"key":"499_CR7","first-page":"461","volume":"9","author":"Y Censor","year":"2008","unstructured":"Censor, Y., Gibali, A.: Projections onto super-half-spaces for monotone variational inequality problems in finite dimensional space. J. Nonlinear Convex Anal. 9, 461\u2013475 (2008)","journal-title":"J. Nonlinear Convex Anal."},{"key":"499_CR8","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1080\/01630563.2013.771656","volume":"34","author":"A Cegielski","year":"2013","unstructured":"Cegielski, A., Gibali, A., Reich, S., Zalas, R.: An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space. Numer. Funct. Anal. Optim. 34, 1067\u20131096 (2013)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"499_CR9","doi-asserted-by":"publisher","first-page":"1463","DOI":"10.1080\/02331934.2016.1147038","volume":"65","author":"A Cegielski","year":"2016","unstructured":"Cegielski, A., Al-Musallam, F.: Strong convergence of a hybrid steepest descent method for the split common fixed point problem. Optimization 65, 1463\u20131476 (2016)","journal-title":"Optimization"},{"key":"499_CR10","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1080\/01630563.2012.716807","volume":"34","author":"A Cegielski","year":"2013","unstructured":"Cegielski, A., Zalas, R.: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 34, 255\u2013283 (2013)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"499_CR11","first-page":"399","volume":"15","author":"A Cegielski","year":"2014","unstructured":"Cegielski, A., Zalas, R.: Properties of a class of approximately shrinking operators and their applications. Fixed Point Theory 15, 399\u2013426 (2014)","journal-title":"Fixed Point Theory"},{"key":"499_CR12","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1080\/01630560802418391","volume":"29","author":"LC Ceng","year":"2008","unstructured":"Ceng, L.C., Ansari, Q.H., Yao, J.C.: Mann type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29, 987\u20131033 (2008)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"499_CR13","volume-title":"Variational Inequalities and Complementarity Problems. Theory and Applications","author":"RW Cottle","year":"1980","unstructured":"Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems. Theory and Applications. Wiley, New York (1980)"},{"issue":"1\u20132","key":"499_CR14","first-page":"33","volume":"19","author":"F Deutsch","year":"1998","unstructured":"Deutsch, F., Yamada, I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1\u20132), 33\u201356 (1998)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"499_CR15","doi-asserted-by":"crossref","unstructured":"Facchinei, F., Pang, J.S.: Finite-dimentional Variational Inequalities and Complementarity Problems, vols. I and II. Springer Series in Operations Research, Springer (2003)","DOI":"10.1007\/b97543"},{"key":"499_CR16","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1007\/BF01589441","volume":"35","author":"M Fukushima","year":"1986","unstructured":"Fukushima, M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58\u201370 (1986)","journal-title":"Math. Program."},{"key":"499_CR17","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/BF01585696","volume":"53","author":"M Fukushima","year":"1992","unstructured":"Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99\u2013110 (1992)","journal-title":"Math. Program."},{"key":"499_CR18","volume-title":"Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models","author":"F Giannessi","year":"2001","unstructured":"Giannessi, F., Maugeri, A., Pardalos, P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic Press, Holland (2001)"},{"key":"499_CR19","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/s10957-010-9757-3","volume":"148","author":"A Gibali","year":"2011","unstructured":"Gibali, A., Censor, Y., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318\u2013335 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"499_CR20","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1007\/s11784-015-0256-x","volume":"17","author":"A Gibali","year":"2015","unstructured":"Gibali, A., Reich, S., Zalas, R.: Iterative methods for solving variational inequalities in Euclidean space. J. Fixed Point Theory Appl. 17, 775\u2013811 (2015)","journal-title":"J Fixed Point Theory Appl."},{"key":"499_CR21","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1080\/02331934.2016.1271800","volume":"66","author":"A Gibali","year":"2017","unstructured":"Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417\u2013437 (2017)","journal-title":"Optimization"},{"key":"499_CR22","unstructured":"Glowinski, R., Lions, J.L., Tremoliers, R.: Numerical Analysis of Variational Inequalities. North-Holland (1981)"},{"key":"499_CR23","unstructured":"Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings. Dekker (1984)"},{"key":"499_CR24","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF01582255","volume":"48","author":"PT Harker","year":"1990","unstructured":"Harker, P.T., Pang, J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161\u2013220 (1990)","journal-title":"Math. Program."},{"key":"499_CR25","volume-title":"A Class of Implicit Methods for Monotone Variational Inequalities. Reports of the Institute of Mathematics, vol. 95\u20131","author":"BS He","year":"1995","unstructured":"He, B.S.: A Class of Implicit Methods for Monotone Variational Inequalities. Reports of the Institute of Mathematics, vol. 95\u20131. Nanjing University, PR China (1995)"},{"key":"499_CR26","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1023\/A:1013096613105","volume":"112","author":"BS He","year":"2002","unstructured":"He, B.S., Liao, L.Z.: Improvement of some projection methods for monotone variational inequalities. J. Optim. Theory Appl. 112, 111\u2013128 (2002)","journal-title":"J. Optim. Theory Appl."},{"key":"499_CR27","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1186\/s13660-017-1366-3","volume":"2017","author":"SN He","year":"2017","unstructured":"He, S.N., Wu, T.: A modified subgradient extragradient method for solving monotone and variational inequalities. J. Inequal. Appl. 2017, 89 (2017)","journal-title":"J. Inequal. Appl."},{"key":"499_CR28","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1016\/j.jmaa.2004.04.068","volume":"300","author":"BS He","year":"2004","unstructured":"He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362\u2013374 (2004)","journal-title":"J. Math. Anal. Appl."},{"key":"499_CR29","first-page":"245","volume":"10","author":"SN He","year":"2009","unstructured":"He, S.N., Xu, H.K.: Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators. Fixed Point Theory. 10, 245\u2013258 (2009)","journal-title":"Fixed Point Theory."},{"key":"499_CR30","doi-asserted-by":"crossref","unstructured":"He, S.N., Yang, C.P.: Solving the variational inequality problem defined on intersection of finite level sets. Abstr. Appl. Anal. Article ID 942315, 8 p (2013)","DOI":"10.1155\/2013\/942315"},{"key":"499_CR31","doi-asserted-by":"publisher","first-page":"1020","DOI":"10.1016\/j.jmaa.2005.12.064","volume":"324","author":"SA Hirstoaga","year":"2006","unstructured":"Hirstoaga, S.A.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020\u20131035 (2006)","journal-title":"JMath Anal Appl."},{"key":"499_CR32","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719451","volume-title":"An Introduction to Variational Inequalities and Their Applications","author":"D Kinderlehrer","year":"2000","unstructured":"Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. SIAM, Philadelphia (2000)"},{"key":"499_CR33","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1002\/cpa.3160200302","volume":"20","author":"JL Lions","year":"1967","unstructured":"Lions, J.L., Stampacchia, G.: Variational inequalities. Comm. Pure Apl. Math. 20, 493\u2013512 (1967)","journal-title":"Comm. Pure Apl. Math."},{"key":"499_CR34","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1006\/jmaa.1999.6615","volume":"241","author":"A Moudafi","year":"2000","unstructured":"Moudafi, A.: Viscosity approximation methods for fixed point problems. J. Math. Anal. Appl. 241, 46\u201355 (2000)","journal-title":"J. Math. Anal. Appl."},{"key":"499_CR35","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1090\/S0002-9904-1967-11761-0","volume":"73","author":"Z Opial","year":"1967","unstructured":"Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Amer. Math. Soc. 73, 595\u2013597 (1967)","journal-title":"Bull. Amer. Math. Soc."},{"key":"499_CR36","first-page":"4413","volume":"258","author":"G Stampacchia","year":"1964","unstructured":"Stampacchia, G.: Formes bilineaires coercivites sur les ensembles convexes. C. R. Acad. Sci. 258, 4413\u20134416 (1964)","journal-title":"C. R. Acad. Sci."},{"key":"499_CR37","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.jat.2008.03.001","volume":"153","author":"N Takahashi","year":"2008","unstructured":"Takahashi, N., Yamada, I.: Parallel algorithms for variational inequalities over the Cartesian product of the intersections of the fixed point sets of nonexpansive mappings. J. Approx Theory. 153, 139\u2013160 (2008)","journal-title":"J. Approx Theory."},{"key":"499_CR38","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1023\/B:JOTA.0000005048.79379.b6","volume":"119","author":"HK Xu","year":"2003","unstructured":"Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185\u2013201 (2003)","journal-title":"J. Optim. Theory Appl."},{"key":"499_CR39","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1016\/j.jmaa.2004.04.059","volume":"298","author":"HK Xu","year":"2004","unstructured":"Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279\u2013291 (2004)","journal-title":"J. Math. Anal. Appl."},{"key":"499_CR40","first-page":"830","volume":"13","author":"CP Yang","year":"2014","unstructured":"Yang, C.P., He, S.N.: Convergence of explicit iterative algorithms for solving a class of variational inequalities. J. WSEAS Trans. 13, 830\u2013839 (2014)","journal-title":"J. WSEAS Trans."},{"key":"499_CR41","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1016\/S0191-2615(96)00030-6","volume":"31","author":"HM Yang","year":"1997","unstructured":"Yang, H.M., Bell, G.H.: Traffic restraint, road pricing and network equilibrium. Trans. Res. B. 31, 303\u2013314 (1997)","journal-title":"Trans. Res. B."},{"key":"499_CR42","doi-asserted-by":"crossref","unstructured":"Yamada, I.: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp 473\u2013504. Amsterdam, North-Holland (2001)","DOI":"10.1016\/S1570-579X(01)80028-8"},{"key":"499_CR43","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1081\/NFA-200045815","volume":"25","author":"I Yamada","year":"2004","unstructured":"Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619\u2013655 (2004)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"499_CR44","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s10957-006-9068-x","volume":"132","author":"LC Zeng","year":"2007","unstructured":"Zeng, L.C., Wong, N.C., Yao, J.C.: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51\u201369 (2007)","journal-title":"J. Optim. Theory Appl."},{"key":"499_CR45","doi-asserted-by":"publisher","first-page":"716","DOI":"10.1007\/s10957-013-0470-x","volume":"161","author":"HY Zhou","year":"2014","unstructured":"Zhou, H.Y., Wang, P.Y.: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 161, 716\u2013727 (2014)","journal-title":"J. Optim. Theory Appl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-018-0499-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0499-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-018-0499-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,11]],"date-time":"2019-10-11T06:16:52Z","timestamp":1570774612000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-018-0499-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,20]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["499"],"URL":"https:\/\/doi.org\/10.1007\/s11075-018-0499-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,20]]},"assertion":[{"value":"16 July 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with Ethical Standards"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}