{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T12:56:27Z","timestamp":1769086587376,"version":"3.49.0"},"reference-count":107,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2015,9,16]],"date-time":"2015-09-16T00:00:00Z","timestamp":1442361600000},"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":["Ann Oper Res"],"published-print":{"date-parts":[[2016,5]]},"DOI":"10.1007\/s10479-015-2013-3","type":"journal-article","created":{"date-parts":[[2015,9,16]],"date-time":"2015-09-16T10:57:37Z","timestamp":1442401057000},"page":"271-299","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Merit functions: a bridge between optimization and equilibria"],"prefix":"10.1007","volume":"240","author":[{"given":"Massimo","family":"Pappalardo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giandomenico","family":"Mastroeni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mauro","family":"Passacantando","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,9,16]]},"reference":[{"issue":"3","key":"2013_CR1","first-page":"667","volume":"2","author":"L Altangerel","year":"2006","unstructured":"Altangerel, L., Bo\u0163, R. I., & Wanka, G. (2006). On gap functions for equilibrium problems via Fenchel duality. Pacific Journal of Optimization, 2(3), 667\u2013678.","journal-title":"Pacific Journal of Optimization"},{"issue":"3","key":"2013_CR2","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1142\/S0217595907001309","volume":"24","author":"L Altangerel","year":"2007","unstructured":"Altangerel, L., Bo\u0163, R. I., & Wanka, G. (2007). On the construction of gap functions for variational inequalities via conjugate duality. Asia-Pacific Journal of Operational Research, 24(3), 353\u2013371.","journal-title":"Asia-Pacific Journal of Operational Research"},{"issue":"1","key":"2013_CR3","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1023\/B:NETS.0000015653.52983.61","volume":"4","author":"E Altman","year":"2004","unstructured":"Altman, E., & Wynter, L. (2004). Equilibrium, games, and pricing in transportation and telecommunication networks. Networks and Spatial Economics, 4(1), 7\u201321.","journal-title":"Networks and Spatial Economics"},{"key":"2013_CR4","doi-asserted-by":"crossref","unstructured":"Ardagna, D., Panicucci, B., & Passacantando, M. (2011). A game theoretic formulation of the service provisioning problem in cloud systems. In Proceedings of the 20th international conference on World Wide Web, WWW \u201911 (pp. 177\u2013186). New York, NY: ACM.","DOI":"10.1145\/1963405.1963433"},{"issue":"4","key":"2013_CR5","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1109\/TSC.2012.14","volume":"6","author":"D Ardagna","year":"2013","unstructured":"Ardagna, D., Panicucci, B., & Passacantando, M. (2013). Generalized nash equilibria for the service provisioning problem in cloud systems. IEEE Transactions on Services Computing, 6(4), 429\u2013442.","journal-title":"IEEE Transactions on Services Computing"},{"issue":"9\u201310","key":"2013_CR6","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1080\/01630568908816335","volume":"10","author":"G Auchmuty","year":"1989","unstructured":"Auchmuty, G. (1989). Variational principles for variational inequalities. Numerical Functional Analysis and Optimization, 10(9\u201310), 863\u2013874.","journal-title":"Numerical Functional Analysis and Optimization"},{"key":"2013_CR7","volume-title":"Optimisation. M\u00e9thodes num\u00e9riques","author":"A Auslender","year":"1976","unstructured":"Auslender, A. (1976). Optimisation. M\u00e9thodes num\u00e9riques. Paris: Masson."},{"issue":"3","key":"2013_CR8","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1007\/s10957-011-9898-z","volume":"151","author":"D Aussel","year":"2011","unstructured":"Aussel, D., Correa, R., & Marechal, M. (2011). Gap functions for quasivariational inequalities and generalized Nash equilibrium problems. Journal of Optimization Theory and Applications, 151(3), 474\u2013488.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2013_CR9","volume-title":"Variational and quasivariational inequalities","author":"C Baiocchi","year":"1984","unstructured":"Baiocchi, C., & Capelo, A. (1984). Variational and quasivariational inequalities. New York: Wiley."},{"key":"2013_CR10","first-page":"A1189","volume":"276","author":"A Bensoussan","year":"1973","unstructured":"Bensoussan, A., & Lions, J. L. (1973). Nouvelle formulation de probl\u00e8mes de contr\u00f4le impulsionnel et applications. C R Academic Science Paris S\u00e9ries A-B, 276, A1189\u2013A1192.","journal-title":"C R Academic Science Paris S\u00e9ries A-B"},{"key":"2013_CR11","first-page":"A1279","volume":"276","author":"A Bensoussan","year":"1973","unstructured":"Bensoussan, A., Goursat, M., & Lions, J. L. (1973). Contr\u00f4le impulsionnel et in\u00e9quations quasi-variationnelles stationnaires. C R Academic Science Paris S\u00e9ries A-B, 276, A1279\u2013A1284.","journal-title":"C R Academic Science Paris S\u00e9ries A-B"},{"issue":"1","key":"2013_CR12","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1080\/10556780903157885","volume":"25","author":"G Bigi","year":"2010","unstructured":"Bigi, G., & Panicucci, B. (2010). A successive linear programming algorithm for nonsmooth monotone variational inequalities. Optimization Methods & Software, 25(1), 29\u201335.","journal-title":"Optimization Methods & Software"},{"issue":"2","key":"2013_CR13","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s10589-012-9481-z","volume":"53","author":"G Bigi","year":"2012","unstructured":"Bigi, G., & Passacantando, M. (2012). Gap functions and penalization for solving equilibrium problems with nonlinear constraints. Computational Optimization and Applications, 53(2), 323\u2013346.","journal-title":"Computational Optimization and Applications"},{"key":"2013_CR14","doi-asserted-by":"crossref","unstructured":"Bigi, G., & Passacantando, M. (2015). D-gap functions and descent techniques for solving equilibrium problems. Journal of Global Optimization, 62(1), 183\u2013203.","DOI":"10.1007\/s10898-014-0223-x"},{"issue":"3","key":"2013_CR15","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1007\/s10957-013-0473-7","volume":"164","author":"G Bigi","year":"2015","unstructured":"Bigi, G., & Passacantando, M. (2015a). Descent and penalization techniques for equilibrium problems with nonlinear constraints. Journal of Optimization Theory and Applications, 164(3), 804\u2013818.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"2","key":"2013_CR16","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/10556788.2014.900552","volume":"30","author":"G Bigi","year":"2015","unstructured":"Bigi, G., & Passacantando, M. (2015b). Twelve monotonicity conditions arising from algorithms for equilibrium problems. Optimization Methods and Software, 30(2), 323\u2013337.","journal-title":"Optimization Methods and Software"},{"issue":"6","key":"2013_CR17","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1080\/10556780902855620","volume":"24","author":"G Bigi","year":"2009","unstructured":"Bigi, G., Castellani, M., & Pappalardo, M. (2009). A new solution method for equilibrium problems. Optimization Methods and Software, 24(6), 895\u2013911.","journal-title":"Optimization Methods and Software"},{"issue":"1","key":"2013_CR18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2012.11.037","volume":"227","author":"G Bigi","year":"2013","unstructured":"Bigi, G., Castellani, M., Pappalardo, M., & Passacantando, M. (2013). Existence and solution methods for equilibria. European Journal of Operational Research, 227(1), 1\u201311.","journal-title":"European Journal of Operational Research"},{"issue":"1\u20134","key":"2013_CR19","first-page":"123","volume":"63","author":"E Blum","year":"1994","unstructured":"Blum, E., & Oettli, W. (1994). From optimization and variational inequalities to equilibrium problems. Mathematics Student-India, 63(1\u20134), 123\u2013145.","journal-title":"Mathematics Student-India"},{"issue":"3","key":"2013_CR20","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1023\/A:1016056327692","volume":"114","author":"E Cavazzuti","year":"2002","unstructured":"Cavazzuti, E., Pappalardo, M., & Passacantando, M. (2002). Nash equilibria, variational inequalities, and dynamical systems. Journal of Optimization Theory and Applications, 114(3), 491\u2013506.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"3\u20134","key":"2013_CR21","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.camwa.2003.05.011","volume":"48","author":"O Chadli","year":"2004","unstructured":"Chadli, O., Konnov, I. V., & Yao, J. C. (2004). Descent methods for equilibrium problems in a Banach space. Computers & Mathematics with Applications, 48(3\u20134), 609\u2013616.","journal-title":"Computers & Mathematics with Applications"},{"issue":"2","key":"2013_CR22","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1287\/moor.7.2.211","volume":"7","author":"D Chan","year":"1982","unstructured":"Chan, D., & Pang, J. S. (1982). The generalized quasivariational inequality problem. Mathematics of Operations Research, 7(2), 211\u2013222.","journal-title":"Mathematics of Operations Research"},{"issue":"2","key":"2013_CR23","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1080\/02331934.2011.583987","volume":"62","author":"C Cherugondi","year":"2013","unstructured":"Cherugondi, C. (2013). A note on D-gap functions for equilibrium problems. Optimization, 62(2), 211\u2013226.","journal-title":"Optimization"},{"issue":"1","key":"2013_CR24","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1287\/trsc.14.1.42","volume":"14","author":"S Dafermos","year":"1980","unstructured":"Dafermos, S. (1980). Traffic equilibrium and variational inequalities. Transportation Science, 14(1), 42\u201354.","journal-title":"Transportation Science"},{"key":"2013_CR25","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1137\/0114053","volume":"14","author":"JM Danskin","year":"1966","unstructured":"Danskin, J. M. (1966). The theory of max\u2013min, with applications. SIAM Journal on Applied Mathematics, 14, 641\u2013664.","journal-title":"SIAM Journal on Applied Mathematics"},{"issue":"5","key":"2013_CR26","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1080\/10556788.2013.796376","volume":"29","author":"D Lorenzo Di","year":"2014","unstructured":"Di Lorenzo, D., Passacantando, M., & Sciandrone, M. (2014). A convergent inexact solution method for equilibrium problems. Optimization Methods and Software, 29(5), 979\u2013991.","journal-title":"Optimization Methods and Software"},{"issue":"1","key":"2013_CR27","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1006\/jmaa.1999.6405","volume":"235","author":"H Dietrich","year":"1999","unstructured":"Dietrich, H. (1999). A smooth dual gap function solution to a class of quasivariational inequalities. Journal of Mathematical Analysis and Applications, 235(1), 380\u2013393.","journal-title":"Journal of Mathematical Analysis and Applications"},{"issue":"1\u20132","key":"2013_CR28","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1080\/02331930108844521","volume":"49","author":"H Dietrich","year":"2001","unstructured":"Dietrich, H. (2001). Optimal control problems for certain quasivariational inequalities. Optimization, 49(1\u20132), 67\u201393.","journal-title":"Optimization"},{"issue":"1\u20132","key":"2013_CR29","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/s10287-007-0043-2","volume":"5","author":"L Drouet","year":"2008","unstructured":"Drouet, L., Haurie, A., Moresino, F., Vial, J. P., Vielle, M., & Viguier, L. (2008). An oracle based method to compute a coupled equilibrium in a model of international climate policy. Computational Management Science, 5(1\u20132), 119\u2013140.","journal-title":"Computational Management Science"},{"issue":"1","key":"2013_CR30","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1007\/BF01584073","volume":"1","author":"BC Eaves","year":"1971","unstructured":"Eaves, B. C. (1971). On the basic theorem of complementarity. Math Programming, 1(1), 68\u201375.","journal-title":"Math Programming"},{"key":"2013_CR31","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s10479-009-0653-x","volume":"175","author":"F Facchinei","year":"2010","unstructured":"Facchinei, F., & Kanzow, C. (2010). Generalized Nash equilibrium problems. Annals of Operations Research, 175, 177\u2013211.","journal-title":"Annals of Operations Research"},{"key":"2013_CR32","volume-title":"Finite-dimensional variational inequalities and complementarity problems","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., & Pang, J. S. (2003). Finite-dimensional variational inequalities and complementarity problems. New York: Springer Series in Operations Research, Springer."},{"issue":"2","key":"2013_CR33","first-page":"225","volume":"9","author":"F Facchinei","year":"2013","unstructured":"Facchinei, F., Kanzow, C., & Sagratella, S. (2013). QVILIB: A library of quasi-variational inequality test problems. Pacific Journal of Optimization, 9(2), 225\u2013250.","journal-title":"Pacific Journal of Optimization"},{"issue":"1\u20132, Ser. A","key":"2013_CR34","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10107-013-0637-0","volume":"144","author":"F Facchinei","year":"2014","unstructured":"Facchinei, F., Kanzow, C., & Sagratella, S. (2014). Solving quasi-variational inequalities via their KKT conditions. Mathematical Programming, 144(1\u20132, Ser. A), 369\u2013412.","journal-title":"Mathematical Programming"},{"issue":"4","key":"2013_CR35","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1137\/S0036144595285963","volume":"39","author":"MC Ferris","year":"1997","unstructured":"Ferris, M. C., & Pang, J. S. (1997). Engineering and economic applications of complementarity problems. Siam Review, 39(4), 669\u2013713.","journal-title":"Siam Review"},{"issue":"2\u20133","key":"2013_CR36","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1023\/A:1026598214921","volume":"17","author":"A Fischer","year":"2000","unstructured":"Fischer, A., & Jiang, H. (2000). Merit functions for complementarity and related problems: A survey. Computational Optimization and Applications, 17(2\u20133), 159\u2013182.","journal-title":"Computational Optimization and Applications"},{"issue":"1","key":"2013_CR37","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.ejor.2003.06.025","volume":"160","author":"F Forg\u00f3","year":"2005","unstructured":"Forg\u00f3, F., F\u00fcl\u00f6p, J., & Prill, M. (2005). Game theoretic models for climate change negotiations. European Journal of Operational Research, 160(1), 252\u2013267.","journal-title":"European Journal of Operational Research"},{"issue":"1, Ser. A","key":"2013_CR38","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01585696","volume":"53","author":"M Fukushima","year":"1992","unstructured":"Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming, 53(1, Ser. A), 99\u2013110.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR39","doi-asserted-by":"crossref","first-page":"165","DOI":"10.3934\/jimo.2007.3.165","volume":"3","author":"M Fukushima","year":"2007","unstructured":"Fukushima, M. (2007). A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 3(2), 165\u2013171.","journal-title":"Journal of Industrial and Management Optimization"},{"key":"2013_CR40","doi-asserted-by":"crossref","unstructured":"Giannessi, F. (1995). Separation of sets and gap functions for quasi-variational inequalities. In F. Giannessi & A Maugeri (Eds.), Variational inequalities and network equilibrium problems (Erice, 1994), pp. 101\u2013121. New York: Plenum.","DOI":"10.1007\/978-1-4899-1358-6"},{"key":"2013_CR41","doi-asserted-by":"crossref","unstructured":"Giannessi, F. (1998). On Minty variational principle. In F. Giannessi, S. Koml\u00f3si & T. Rapcs\u00e1k (Eds.), New trends in mathematical programming, vol.\u00a013 (pp. 93\u201399). Boston, MA: Kluwer.","DOI":"10.1007\/978-1-4757-2878-1_8"},{"issue":"4","key":"2013_CR42","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1007\/s10898-011-9733-y","volume":"53","author":"R Gupta","year":"2012","unstructured":"Gupta, R., & Mehra, A. (2012). Gap functions and error bounds for quasi variational inequalities. Journal of Global Optimization, 53(4), 737\u2013748.","journal-title":"Journal of Global Optimization"},{"issue":"2, (Ser. B)","key":"2013_CR43","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF01582255","volume":"48","author":"PT Harker","year":"1990","unstructured":"Harker, P. T., & Pang, J. S. (1990). Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming, 48(2, (Ser. B)), 161\u2013220.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR44","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s10957-014-0536-4","volume":"163","author":"N Harms","year":"2014","unstructured":"Harms, N., Hoheisel, T., & Kanzow, C. (2014a). On a smooth dual gap function for a class of quasi-variational inequalities. Journal of Optimization Theory and Applications, 163(2), 413\u2013438.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"4","key":"2013_CR45","doi-asserted-by":"crossref","first-page":"720","DOI":"10.1080\/10556788.2013.841694","volume":"29","author":"N Harms","year":"2014","unstructured":"Harms, N., Kanzow, C., & Stein, O. (2014b). Smoothness properties of a regularized gap function for quasi-variational inequalities. Optimization Methods and Software, 29(4), 720\u2013750.","journal-title":"Optimization Methods and Software"},{"key":"2013_CR46","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/BF02392210","volume":"115","author":"P Hartman","year":"1966","unstructured":"Hartman, P., & Stampacchia, G. (1966). On some non-linear elliptic differential-functional equations. Acta Mathematica, 115, 271\u2013310.","journal-title":"Acta Mathematica"},{"issue":"2","key":"2013_CR47","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s10957-004-1839-7","volume":"125","author":"LR Huang","year":"2005","unstructured":"Huang, L. R., & Ng, K. F. (2005). Equivalent optimization formulations and error bounds for variational inequality problems. Journal of Optimization Theory and Applications, 125(2), 299\u2013314.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"1\u20132","key":"2013_CR48","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/S0167-6377(98)00023-6","volume":"23","author":"C Kanzow","year":"1998","unstructured":"Kanzow, C., & Fukushima, M. (1998a). Solving box constrained variational inequalities by using the natural residual with D-gap function globalization. Operations Research Letters, 23(1\u20132), 45\u201351.","journal-title":"Operations Research Letters"},{"issue":"1, Ser. A","key":"2013_CR49","first-page":"55","volume":"83","author":"C Kanzow","year":"1998","unstructured":"Kanzow, C., & Fukushima, M. (1998b). Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities. Mathematical Programming, 83(1, Ser. A), 55\u201387.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR50","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s11590-006-0004-7","volume":"1","author":"IV Konnov","year":"2007","unstructured":"Konnov, I. V. (2007). On variational inequalities for auction market problems. Optimization Letters, 1(2), 155\u2013162.","journal-title":"Optimization Letters"},{"issue":"1","key":"2013_CR51","doi-asserted-by":"crossref","first-page":"30","DOI":"10.3103\/S1066369X08010052","volume":"52","author":"IV Konnov","year":"2008","unstructured":"Konnov, I. V. (2008a). Spatial equilibrium problems for auction-type systems. Russian Mathematics, 52(1), 30\u201344.","journal-title":"Russian Mathematics"},{"key":"2013_CR52","doi-asserted-by":"crossref","first-page":"29","DOI":"10.2174\/1874243200802010029","volume":"2","author":"IV Konnov","year":"2008","unstructured":"Konnov, I. V. (2008b). Variational inequalities for modeling auction markets with price mappings. Open Operational Research Journal, 2, 29\u201337.","journal-title":"Open Operational Research Journal"},{"issue":"2","key":"2013_CR53","doi-asserted-by":"crossref","first-page":"274","DOI":"10.2478\/cmam-2003-0018","volume":"3","author":"IV Konnov","year":"2003","unstructured":"Konnov, I. V., & Pinyagina, O. V. (2003a). D-gap functions for a class of equilibrium problems in Banach spaces. Computational Methods in Applied Mathematics, 3(2), 274\u2013286.","journal-title":"Computational Methods in Applied Mathematics"},{"issue":"12","key":"2013_CR54","first-page":"67","volume":"47","author":"IV Konnov","year":"2003","unstructured":"Konnov, I. V., & Pinyagina, O. V. (2003b). Descent method with respect to the gap function for nonsmooth equilibrium problems. Russian Mathematics, 47(12), 67\u201373.","journal-title":"Russian Mathematics"},{"issue":"1, Ser. A","key":"2013_CR55","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/BF01582565","volume":"64","author":"T Larsson","year":"1994","unstructured":"Larsson, T., & Patriksson, M. (1994). A class of gap functions for variational inequalities. Mathematical Programming, 64(1, Ser. A), 53\u201379.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR56","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1137\/070696283","volume":"20","author":"G Li","year":"2009","unstructured":"Li, G., & Ng, K. F. (2009). Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM Journal on Optimization, 20(2), 667\u2013690.","journal-title":"SIAM Journal on Optimization"},{"issue":"11","key":"2013_CR57","doi-asserted-by":"crossref","first-page":"2795","DOI":"10.1016\/j.cam.2009.11.025","volume":"233","author":"G Li","year":"2010","unstructured":"Li, G., Tang, C., & Wei, Z. (2010). Error bound results for generalized D-gap functions of nonsmooth variational inequality problems. Journal of Computational and Applied Mathematics, 233(11), 2795\u20132806.","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"3","key":"2013_CR58","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s10287-006-0010-3","volume":"4","author":"Z Liu","year":"2007","unstructured":"Liu, Z., & Nagurney, A. (2007). Financial networks with intermediation and transportation network equilibria: a supernetwork equivalence and reinterpretation of the equilibrium conditions with computations. Computational Management Science, 4(3), 243\u2013281.","journal-title":"Computational Management Science"},{"issue":"2, Ser. B","key":"2013_CR59","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/BF01585171","volume":"62","author":"OL Mangasarian","year":"1993","unstructured":"Mangasarian, O. L., & Solodov, M. V. (1993). Nonlinear complementarity as unconstrained and constrained minimization. Mathematical Programming, 62(2, Ser. B), 277\u2013297.","journal-title":"Mathematical Programming"},{"issue":"3","key":"2013_CR60","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/BF01584381","volume":"33","author":"P Marcotte","year":"1985","unstructured":"Marcotte, P. (1985). A new algorithm for solving variational inequalities with application to the traffic assignment problem. Mathematical Programming, 33(3), 339\u2013351.","journal-title":"Mathematical Programming"},{"issue":"11","key":"2013_CR61","doi-asserted-by":"crossref","first-page":"1051","DOI":"10.1057\/jors.1987.175","volume":"38","author":"P Marcotte","year":"1987","unstructured":"Marcotte, P. (1987). Algorithms for the network oligopoly problem. Journal of the Operational Research Society, 38(11), 1051\u20131065.","journal-title":"Journal of the Operational Research Society"},{"issue":"1","key":"2013_CR62","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/0167-6377(87)90007-1","volume":"6","author":"P Marcotte","year":"1987","unstructured":"Marcotte, P., & Dussault, J. P. (1987). A note on a globally convergent Newton method for solving monotone variational inequalities. Operations Research Letters, 6(1), 35\u201342.","journal-title":"Operations Research Letters"},{"issue":"6","key":"2013_CR63","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1137\/0327064","volume":"27","author":"P Marcotte","year":"1989","unstructured":"Marcotte, P., & Dussault, J. P. (1989). A sequential linear programming algorithm for solving monotone variational inequalities. SIAM Journal on Control and Optimization, 27(6), 1260\u20131278.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"1","key":"2013_CR64","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1137\/S1052623496309867","volume":"9","author":"P Marcotte","year":"1999","unstructured":"Marcotte, P., & Zhu, D. (1999). Weak sharp solutions of variational inequalities. SIAM Journal on Optimization, 9(1), 179\u2013189.","journal-title":"SIAM Journal on Optimization"},{"key":"2013_CR65","unstructured":"Mastroeni, G. (1999). Minimax and extremum problems associated to a variational inequality. In Proceedings of the workshop \u201cNumerical Methods in Optimization\u201d (Cortona, 1997), vol. 58, pp. 185\u2013196."},{"issue":"4","key":"2013_CR66","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1023\/A:1026050425030","volume":"27","author":"G Mastroeni","year":"2003","unstructured":"Mastroeni, G. (2003). Gap functions for equilibrium problems. Journal of Global Optimization, 27(4), 411\u2013426.","journal-title":"Journal of Global Optimization"},{"key":"2013_CR67","doi-asserted-by":"crossref","unstructured":"Mastroeni, G. (2005). Gap functions and descent methods for Minty variational inequality. In L. Qi, K. Teo & X. Yang (Eds.), Optimization and control with applications, Appl. Optim., vol.\u00a096 (pp. 529\u2013547). New York: Springer.","DOI":"10.1007\/0-387-24255-4_27"},{"issue":"1","key":"2013_CR68","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.ejor.2007.06.052","volume":"191","author":"N Miller","year":"2008","unstructured":"Miller, N., & Ruszczy\u0144ski, A. (2008). Risk-adjusted probability measures in portfolio optimization with coherent measures of risk. European Journal of Operational Research, 191(1), 193\u2013206.","journal-title":"European Journal of Operational Research"},{"key":"2013_CR69","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1090\/S0002-9904-1967-11732-4","volume":"73","author":"GJ Minty","year":"1967","unstructured":"Minty, G. J. (1967). On the generalization of a direct method of the calculus of variations. Bulletin of the American Mathematical Society, 73, 315\u2013321.","journal-title":"Bulletin of the American Mathematical Society"},{"issue":"4","key":"2013_CR70","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1080\/02331930701421079","volume":"56","author":"BS Mordukhovich","year":"2007","unstructured":"Mordukhovich, B. S., Outrata, J. V., & \u010cervinka, M. (2007). Equilibrium problems with complementarity constraints: Case study with applications to oligopolistic markets. Optimization, 56(4), 479\u2013494.","journal-title":"Optimization"},{"issue":"1","key":"2013_CR71","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1007\/BF01585096","volume":"24","author":"FH Murphy","year":"1982","unstructured":"Murphy, F. H., Sherali, H. D., & Soyster, A. L. (1982). A mathematical programming approach for determining oligopolistic market equilibrium. Mathematical Programming, 24(1), 92\u2013106.","journal-title":"Mathematical Programming"},{"key":"2013_CR72","doi-asserted-by":"crossref","unstructured":"Nagurney, A. (1993). Network economics: A variational inequality approach. In Advances in computational economics, vol. 1. Dordrecht: Kluwer.","DOI":"10.1007\/978-94-011-2178-1"},{"issue":"4","key":"2013_CR73","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s10287-009-0095-6","volume":"7","author":"A Nagurney","year":"2010","unstructured":"Nagurney, A. (2010). Formulation and analysis of horizontal mergers among oligopolistic firms with insights into the merger paradox: A supply chain network perspective. Computational Management Science, 7(4), 377\u2013406.","journal-title":"Computational Management Science"},{"issue":"1","key":"2013_CR74","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10957-007-9193-1","volume":"133","author":"KF Ng","year":"2007","unstructured":"Ng, K. F., & Tan, L. L. (2007a). D-gap functions for nonsmooth variational inequality problems. Journal of Optimization Theory and Applications, 133(1), 77\u201397.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"2, Ser. A","key":"2013_CR75","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1007\/s10107-006-0007-2","volume":"110","author":"KF Ng","year":"2007","unstructured":"Ng, K. F., & Tan, L. L. (2007b). Error bounds of regularized gap functions for nonsmooth variational inequality problems. Mathematical Programming, 110(2, Ser. A), 405\u2013429.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR76","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1287\/trsc.18.2.185","volume":"18","author":"S Nguyen","year":"1984","unstructured":"Nguyen, S., & Dupuis, C. (1984). Efficient method for computing traffic equilibria in networks with asymmetric transportation costs. Transportation Science, 18(2), 185\u2013202.","journal-title":"Transportation Science"},{"key":"2013_CR77","doi-asserted-by":"crossref","first-page":"807","DOI":"10.2140\/pjm.1955.5.807","volume":"5","author":"H Nikaid\u00f4","year":"1955","unstructured":"Nikaid\u00f4, H., & Isoda, K. (1955). Note on non-cooperative convex games. Pacific Journal of Mathematics, 5, 807\u2013815.","journal-title":"Pacific Journal of Mathematics"},{"issue":"6","key":"2013_CR78","doi-asserted-by":"crossref","first-page":"3251","DOI":"10.1109\/TSP.2010.2043138","volume":"58","author":"JS Pang","year":"2010","unstructured":"Pang, J. S., Scutari, G., Palomar, D., & Facchinei, F. (2010). Design of cognitive radio systems under temperature-interference constraints: A variational inequality approach. IEEE Transactions on Signal Processing, 58(6), 3251\u20133271.","journal-title":"IEEE Transactions on Signal Processing"},{"issue":"2","key":"2013_CR79","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s10589-007-9132-y","volume":"43","author":"B Panicucci","year":"2009","unstructured":"Panicucci, B., Pappalardo, M., & Passacantando, M. (2009). A globally convergent descent method for nonsmooth variational inequalities. Computational Optimization and Applications, 43(2), 197\u2013211.","journal-title":"Computational Optimization and Applications"},{"issue":"3","key":"2013_CR80","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1023\/A:1015312921888","volume":"113","author":"M Pappalardo","year":"2002","unstructured":"Pappalardo, M., & Passacantando, M. (2002). Stability for equilibrium problems: From variational inequalities to dynamical systems. Journal of Optimization Theory and Applications, 113(3), 567\u2013582.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"3\u20134","key":"2013_CR81","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1023\/B:JOGO.0000026455.72523.ed","volume":"28","author":"M Pappalardo","year":"2004","unstructured":"Pappalardo, M., & Passacantando, M. (2004). Gap functions and Lyapunov functions. Journal of Global Optimization, 28(3\u20134), 379\u2013385.","journal-title":"Journal of Global Optimization"},{"issue":"1","key":"2013_CR82","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10288-014-0256-5","volume":"12","author":"M Pappalardo","year":"2014","unstructured":"Pappalardo, M., Mastroeni, G., & Passacantando, M. (2014). Merit functions: A bridge between optimization and equilibria. 4OR, 12(1), 1\u201333.","journal-title":"4OR"},{"key":"2013_CR83","unstructured":"Patriksson, M. (1994). The traffic assignment problem. VSP BV."},{"issue":"3, Ser. A","key":"2013_CR84","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF02614360","volume":"78","author":"JM Peng","year":"1997","unstructured":"Peng, J. M. (1997). Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming, 78(3, Ser. A), 347\u2013355.","journal-title":"Mathematical Programming"},{"issue":"2, Ser. A","key":"2013_CR85","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s101070050094","volume":"86","author":"JM Peng","year":"1999","unstructured":"Peng, J. M., & Fukushima, M. (1999). A hybrid Newton method for solving the variational inequality problem via the D-gap function. Mathematical Programming, 86(2, Ser. A), 367\u2013386.","journal-title":"Mathematical Programming"},{"issue":"3","key":"2013_CR86","first-page":"253","volume":"15","author":"JM Peng","year":"1997","unstructured":"Peng, J. M., & Yuan, Y. (1997). Unconstrained methods for generalized complementarity problems. Journal of Computational Mathematics, 15(3), 253\u2013264.","journal-title":"Journal of Computational Mathematics"},{"issue":"5","key":"2013_CR87","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1080\/10556789908805734","volume":"10","author":"JM Peng","year":"1999","unstructured":"Peng, J. M., Kanzow, C., & Fukushima, M. (1999). A hybrid Josephy\u2013Newton method for solving box constrained variational inequality problems via the D-gap function. Optimization Methods and Software, 10(5), 687\u2013710.","journal-title":"Optimization Methods and Software"},{"issue":"3","key":"2013_CR88","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1023\/B:JOTA.0000006688.13248.04","volume":"119","author":"B Qu","year":"2003","unstructured":"Qu, B., Wang, C. Y., & Zhang, J. Z. (2003). Convergence and error bound of a method for solving variational inequality problems via the generalized D-gap function. Journal of Optimization Theory and Applications, 119(3), 535\u2013552.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"2","key":"2013_CR89","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1007\/s10589-010-9360-4","volume":"51","author":"TD Quoc","year":"2012","unstructured":"Quoc, T. D., & Muu, L. D. (2012). Iterative methods for solving monotone equilibrium problems via dual gap functions. Computational Optimization and Applications, 51(2), 709\u2013728.","journal-title":"Computational Optimization and Applications"},{"key":"2013_CR90","doi-asserted-by":"crossref","unstructured":"Rockafellar, R. T. (1970). Convex analysis. In Princeton mathematical series, no. 28. Princeton, NJ: Princeton University Press.","DOI":"10.1515\/9781400873173"},{"key":"2013_CR91","unstructured":"Scutari, G., Palomar, D. P., & Barbarossa, S. (2010). Competitive optimization of cognitive radio MIMO systems via game theory. In Convex optimization in signal processing and communications (pp. 387\u2013442). Cambridge: Cambridge University Press."},{"key":"2013_CR92","unstructured":"Solodov, M. V., & Tseng, P. (2000). Some methods based on the D-gap function for solving monotone variational inequalities. Comput Optim Appl 17(2\u20133):255\u2013277, nonsmooth and smoothing methods (Hong Kong, 1998)."},{"key":"2013_CR93","unstructured":"Sun, D., Fukushima, M., & Qi, L. (1997). A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems. In M. C. Ferris & J. S. Pang (Eds.), Complementarity and variational problems (Baltimore, MD, 1995) (pp. 452\u2013473). Philadelphia, PA: SIAM."},{"key":"2013_CR94","doi-asserted-by":"crossref","unstructured":"Taji, K. (2008). On gap functions for quasi-variational inequalities. Abstr Appl Anal pp Art. ID 531,361, 7.","DOI":"10.1155\/2008\/531361"},{"issue":"3","key":"2013_CR95","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1137\/S1052623494271199","volume":"6","author":"K Taji","year":"1996","unstructured":"Taji, K., & Fukushima, M. (1996). A new merit function and a successive quadratic programming algorithm for variational inequality problems. SIAM Journal on Optimization, 6(3), 704\u2013713.","journal-title":"SIAM Journal on Optimization"},{"issue":"3, Ser. A","key":"2013_CR96","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF01581276","volume":"58","author":"K Taji","year":"1993","unstructured":"Taji, K., Fukushima, M., & Ibaraki, T. (1993). A globally convergent Newton method for solving strong monotone variational inequalities. Mathematical Programming, 58(3, Ser. A), 369\u2013383.","journal-title":"Mathematical Programming"},{"issue":"2","key":"2013_CR97","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1016\/j.jmaa.2007.01.025","volume":"334","author":"LL Tan","year":"2007","unstructured":"Tan, L. L. (2007). Regularized gap functions for nonsmooth variational inequality problems. Journal of Mathematical Analysis and Applications, 334(2), 1022\u20131038.","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"2013_CR98","unstructured":"Wardrop, J. G. (1952). Road paper. Some theoretical aspects of road traffic research. In ICE Proceedings: Engineering divisions, Thomas Telford, vol.\u00a01, pp. 325\u2013362."},{"issue":"3, Ser. A","key":"2013_CR99","first-page":"281","volume":"61","author":"JH Wu","year":"1993","unstructured":"Wu, J. H., Florian, M., & Marcotte, P. (1993). A general descent framework for the monotone variational inequality problem. Mathematical Programming, 61(3, Ser. A), 281\u2013300.","journal-title":"Mathematical Programming"},{"issue":"1","key":"2013_CR100","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1137\/S0363012994277645","volume":"35","author":"N Yamashita","year":"1997","unstructured":"Yamashita, N., & Fukushima, M. (1997). Equivalent unconstrained minimization and global error bounds for variational inequality problems. SIAM Journal on Control and Optimization, 35(1), 273\u2013284.","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"3","key":"2013_CR101","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1023\/A:1022660704427","volume":"92","author":"N Yamashita","year":"1997","unstructured":"Yamashita, N., Taji, K., & Fukushima, M. (1997). Unconstrained optimization reformulations of variational inequality problems. Journal of Optimization Theory and Applications, 92(3), 439\u2013456.","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"2","key":"2013_CR102","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s00245-003-0771-9","volume":"48","author":"J Zhang","year":"2003","unstructured":"Zhang, J., Wan, C., & Xiu, N. (2003). The dual gap function for variational inequalities. Applied Mathematics and Optimization, 48(2), 129\u2013148.","journal-title":"Applied Mathematics and Optimization"},{"issue":"1","key":"2013_CR103","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.cam.2009.03.006","volume":"231","author":"L Zhang","year":"2009","unstructured":"Zhang, L., & Wu, S. Y. (2009). An algorithm based on the generalized D-gap function for equilibrium problems. Journal of Computational and Applied Mathematics, 231(1), 403\u2013411.","journal-title":"Journal of Computational and Applied Mathematics"},{"issue":"3","key":"2013_CR104","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s10114-008-7096-1","volume":"25","author":"LP Zhang","year":"2009","unstructured":"Zhang, L. P., & Han, J. Y. (2009). Unconstrained optimization reformulations of equilibrium problems. Acta Mathematica Sinica, English Series, 25(3), 343\u2013354.","journal-title":"Acta Mathematica Sinica, English Series"},{"issue":"2","key":"2013_CR105","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.ejor.2007.03.038","volume":"187","author":"L Zhao","year":"2008","unstructured":"Zhao, L., & Nagurney, A. (2008). A network equilibrium framework for internet advertising: Models, qualitative analysis, and algorithms. European Journal of Operational Research, 187(2), 456\u2013472.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"2013_CR106","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0167-6377(93)90103-N","volume":"14","author":"DL Zhu","year":"1993","unstructured":"Zhu, D. L., & Marcotte, P. (1993). Modified descent methods for solving the monotone variational inequality problem. Operations Research Letters, 14(2), 111\u2013120.","journal-title":"Operations Research Letters"},{"issue":"2","key":"2013_CR107","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/BF02192941","volume":"80","author":"DL Zhu","year":"1994","unstructured":"Zhu, D. L., & Marcotte, P. (1994). An extended descent framework for variational inequalities. Journal of Optimization Theory and Applications, 80(2), 349\u2013366.","journal-title":"Journal of Optimization Theory and Applications"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-015-2013-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10479-015-2013-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-015-2013-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,11]],"date-time":"2024-06-11T05:03:17Z","timestamp":1718082197000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10479-015-2013-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,9,16]]},"references-count":107,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016,5]]}},"alternative-id":["2013"],"URL":"https:\/\/doi.org\/10.1007\/s10479-015-2013-3","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,9,16]]}}}