{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T13:50:13Z","timestamp":1760709013569},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,2,23]],"date-time":"2016-02-23T00:00:00Z","timestamp":1456185600000},"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":["Math. Program."],"published-print":{"date-parts":[[2016,6]]},"DOI":"10.1007\/s10107-016-0988-4","type":"journal-article","created":{"date-parts":[[2016,2,23]],"date-time":"2016-02-23T11:57:41Z","timestamp":1456228661000},"page":"451-481","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["An approximation scheme for a class of risk-averse stochastic equilibrium problems"],"prefix":"10.1007","volume":"157","author":[{"given":"Juan Pablo","family":"Luna","sequence":"first","affiliation":[]},{"given":"Claudia","family":"Sagastiz\u00e1bal","sequence":"additional","affiliation":[]},{"given":"Mikhail","family":"Solodov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,2,23]]},"reference":[{"issue":"11","key":"988_CR1","doi-asserted-by":"crossref","first-page":"1998","DOI":"10.1109\/JPROC.2005.857484","volume":"93","author":"R Baldick","year":"2005","unstructured":"Baldick, R., Helman, U., Hobbs, B., O\u2019Neill, R.: Design of efficient generation markets. Proc. IEEE 93(11), 1998\u20132012 (2005). doi: 10.1109\/JPROC.2005.857484","journal-title":"Proc. IEEE"},{"key":"988_CR2","doi-asserted-by":"crossref","first-page":"1457","DOI":"10.1137\/120889812","volume":"23","author":"J Burke","year":"2013","unstructured":"Burke, J., Hoheisel, T.: Epi-convergent smoothing with applications to convex composite functions. SIAM J. Optim. 23, 1457\u20131479 (2013)","journal-title":"SIAM J. Optim."},{"key":"988_CR3","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/BF00249052","volume":"5","author":"C Chen","year":"1996","unstructured":"Chen, C., Mangasarian, O.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97\u2013138 (1996)","journal-title":"Comput. Optim. Appl."},{"key":"988_CR4","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10107-012-0569-0","volume":"134","author":"X Chen","year":"2012","unstructured":"Chen, X.: Smoothing methods for nonsmooth, novonvex minimization. Math. Program. 134, 71\u201399 (2012)","journal-title":"Math. Program."},{"issue":"4","key":"988_CR5","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1287\/moor.1050.0160","volume":"30","author":"X Chen","year":"2005","unstructured":"Chen, X., Fukushima, M.: Expected residual minimization method for stochastic linear complementarity problems. Math. Oper. Res. 30(4), 1022\u20131038 (2005). doi: 10.1287\/moor.1050.0160","journal-title":"Math. Oper. Res."},{"issue":"2","key":"988_CR6","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1137\/110825248","volume":"22","author":"X Chen","year":"2012","unstructured":"Chen, X., Wets, R.J.B., Zhang, Y.: Stochastic variational inequalities: residual minimization smoothing sample average approximations. SIAM J. Optim. 22(2), 649\u2013673 (2012)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"988_CR7","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/s10107-007-0163-z","volume":"117","author":"X Chen","year":"2009","unstructured":"Chen, X., Zhang, C., Fukushima, M.: Robust solution of monotone stochastic linear complementarity problems. Math. Program. 117(1\u20132), 51\u201380 (2009). doi: 10.1007\/s10107-007-0163-z","journal-title":"Math. Program."},{"key":"988_CR8","unstructured":"Collado, R.A., Powell, W.B.: Threshold risk measures part 1: finite horizon http:\/\/castlelab.princeton.edu\/Papers\/Collado-Threshold-Measures-FINITE_ONLY-April-22-2013.pdf (2013)"},{"issue":"3","key":"988_CR9","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1109\/TPWRS.2004.831652","volume":"19","author":"A Conejo","year":"2004","unstructured":"Conejo, A., Nogales, F., Arroyo, J., Garcia-Bertrand, R.: Risk-constrained self-scheduling of a thermal power producer. IEEE Trans. Power Syst. 19(3), 1569\u20131574 (2004). doi: 10.1109\/TPWRS.2004.831652","journal-title":"IEEE Trans. Power Syst."},{"key":"988_CR10","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-7421-1","volume-title":"Decision Making Under Uncertainty in Electricity Markets. International Series in Operations Research & Management Science","author":"AJ Conejo","year":"2010","unstructured":"Conejo, A.J., Carri\u00f3n, M., Morales, J.M.: Decision Making Under Uncertainty in Electricity Markets. International Series in Operations Research & Management Science. Springer, New York (2010)"},{"key":"988_CR11","doi-asserted-by":"crossref","unstructured":"David, A., Wen, F.: Strategic bidding in competitive electricity markets: a literature survey. In: Power Engineering Society Summer Meeting, 2000. IEEE, vol. 4, pp. 2168\u20132173 (2000). doi: 10.1109\/PESS.2000.866982","DOI":"10.1109\/PESS.2000.866982"},{"key":"988_CR12","volume-title":"Lectures on Stochastic Programming","author":"D Dentcheva","year":"2009","unstructured":"Dentcheva, D., Ruszczy\u0144ski, A., Shapiro, A.: Lectures on Stochastic Programming. SIAM, Philadelphia (2009)"},{"key":"988_CR13","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/10556789508805606","volume":"5","author":"S Dirkse","year":"1995","unstructured":"Dirkse, S., Ferris, M.: The PATH solver : a nonmonotone stabilization scheme for mixed complementarity problems. Optim. Methods Softw. 5, 123\u2013156 (1995)","journal-title":"Optim. Methods Softw."},{"issue":"2","key":"988_CR14","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/j.ejor.2012.11.024","volume":"226","author":"R Egging","year":"2013","unstructured":"Egging, R.: Benders decomposition for multi-stage stochastic mixed complementarity problems applied to a global natural gas market model. Eur. J. Oper. Res. 226(2), 341\u2013353 (2013). doi: 10.1016\/j.ejor.2012.11.024","journal-title":"Eur. J. Oper. Res."},{"issue":"17","key":"988_CR15","doi-asserted-by":"crossref","first-page":"2762","DOI":"10.1016\/j.enpol.2005.04.018","volume":"34","author":"RG Egging","year":"2006","unstructured":"Egging, R.G., Gabriel, S.A.: Examining market power in the European natural gas market. Energy Policy 34(17), 2762\u20132778 (2006). doi: 10.1016\/j.enpol.2005.04.018","journal-title":"Energy Policy"},{"issue":"2","key":"988_CR16","doi-asserted-by":"crossref","first-page":"274","DOI":"10.1287\/opre.1080.0624","volume":"57","author":"A Ehrenmann","year":"2009","unstructured":"Ehrenmann, A., Neuhoff, K.: A comparison of electricity market designs in networks. Oper. Res. 57(2), 274\u2013286 (2009)","journal-title":"Oper. Res."},{"issue":"6","key":"988_CR17","doi-asserted-by":"crossref","first-page":"1332","DOI":"10.1287\/opre.1110.0992","volume":"59","author":"A Ehrenmann","year":"2011","unstructured":"Ehrenmann, A., Smeers, Y.: Generation capacity expansion in risky environment: a stochastic equilibrium analysis. Oper. Res. 59(6), 1332\u20131346 (2011)","journal-title":"Oper. Res."},{"key":"988_CR18","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.orl.2006.03.004","volume":"35","author":"F Facchinei","year":"2007","unstructured":"Facchinei, F., Fischer, A., Piccialli, V.: On generalized Nash games and variational inequalities. Oper. Res. Lett. 35, 159\u2013164 (2007)","journal-title":"Oper. Res. Lett."},{"issue":"1","key":"988_CR19","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.: Generalized nash equilibrium problems. Ann. OR 175(1), 177\u2013211 (2010)","journal-title":"Ann. OR"},{"key":"988_CR20","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research","author":"F Facchinei","year":"2003","unstructured":"Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. II. Springer, New York (2003)"},{"key":"988_CR21","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1023\/A:1008636318275","volume":"12","author":"MC Ferris","year":"1999","unstructured":"Ferris, M.C., Munson, T.S.: Interfaces to PATH 3.0: design, implementation and usage. Comput. Optim. Appl. 12, 207\u2013227 (1999)","journal-title":"Comput. Optim. Appl."},{"key":"988_CR22","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/s10614-005-2519-x","volume":"25","author":"JD Fuller","year":"2005","unstructured":"Fuller, J.D., Chung, W.: Dantzig\u2013Wolfe decomposition of variational inequalities. Comput. Econ. 25, 303\u2013326 (2005). doi: 10.1007\/s10614-005-2519-x","journal-title":"Comput. Econ."},{"issue":"3","key":"988_CR23","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.ejor.2007.12.046","volume":"197","author":"SA Gabriel","year":"2009","unstructured":"Gabriel, S.A., Zhuang, J., Egging, R.: Solving stochastic complementarity problems in energy market modeling using scenario reduction. Eur. J. Oper. Res. 197(3), 1028\u20131040 (2009). doi: 10.1016\/j.ejor.2007.12.046","journal-title":"Eur. J. Oper. Res."},{"key":"988_CR24","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.B., Lemar\u00e9chal, C.: Convex Analysis and Minimization Algorithms. No. 305\u2013306 in Grund. der math. Wiss. Springer, Berlin (1993). (two volumes)","DOI":"10.1007\/978-3-662-02796-7"},{"issue":"2","key":"988_CR25","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1109\/59.867153","volume":"15","author":"B Hobbs","year":"2000","unstructured":"Hobbs, B., Metzler, C., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15(2), 638\u2013645 (2000). doi: 10.1109\/59.867153","journal-title":"IEEE Trans. Power Syst."},{"issue":"5","key":"988_CR26","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1287\/opre.1070.0431","volume":"55","author":"X Hu","year":"2007","unstructured":"Hu, X., Ralph, D.: Using EPECs to model bilevel games in restructured electricity markets with locational prices. Oper. Res. 55(5), 809\u2013827 (2007)","journal-title":"Oper. Res."},{"issue":"5","key":"988_CR27","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1080\/10556788.2012.676756","volume":"28","author":"A Kannan","year":"2013","unstructured":"Kannan, A., Shanbhag, U.V., Kim, H.M.: Addressing supply-side risk in uncertain power markets: stochastic nash models, scalable algorithms and error analysis. Optim. Methods Softw. 28(5), 1095\u20131138 (2013). doi: 10.1080\/10556788.2012.676756","journal-title":"Optim. Methods Softw."},{"key":"988_CR28","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.automatica.2011.09.042","volume":"48","author":"A Kulkarni","year":"2012","unstructured":"Kulkarni, A., Shanbhag, U.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45\u201355 (2012)","journal-title":"Automatica"},{"key":"988_CR29","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s10957-011-9981-5","volume":"154","author":"AA Kulkarni","year":"2012","unstructured":"Kulkarni, A.A., Shanbhag, U.V.: Revisiting generalized Nash games and variational inequalities. J. Optim. Theory Appl. 154, 175\u2013186 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"1\u20132","key":"988_CR30","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s10107-007-0119-3","volume":"116","author":"GH Lin","year":"2009","unstructured":"Lin, G.H., Chen, X., Fukushima, M.: Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Program. 116(1\u20132), 343\u2013368 (2009)","journal-title":"Math. Program."},{"key":"988_CR31","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s10107-012-0599-7","volume":"143","author":"JP Luna","year":"2014","unstructured":"Luna, J.P., Sagastiz\u00e1bal, C., Solodov, M.: A class of Dantzig\u2013Wolfe type decomposition methods for variational inequality problems. Math. Program. 143, 177\u2013209 (2014). doi: 10.1007\/s10107-012-0599-7","journal-title":"Math. Program."},{"key":"988_CR32","first-page":"237","volume-title":"International Series in Operations Research and Management Science, vol. 199, Chap. 10","author":"JP Luna","year":"2014","unstructured":"Luna, J.P., Sagastiz\u00e1bal, C., Solodov, M.: Complementarity and game-theoretical models for equilibria in energy markets: deterministic and risk-averse formulations. In: Kovacevic, R., Pflug, G., Vespucci, M. (eds.) International Series in Operations Research and Management Science, vol. 199, Chap. 10, pp. 237\u2013264. Springer, Berlin (2014)"},{"issue":"2","key":"988_CR33","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s10589-010-9328-4","volume":"50","author":"F Meng","year":"2011","unstructured":"Meng, F., Sun, J., Goh, M.: A smoothing sample average approximation method for stochastic optimization problems with CVaR risk measure. Comput. Optim. Appl. 50(2), 379\u2013401 (2011)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"988_CR34","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s10107-010-0395-1","volume":"125","author":"JS Pang","year":"2010","unstructured":"Pang, J.S.: Three modeling paradigms in mathematical programming. Math. Program. 125(2), 297\u2013323 (2010). doi: 10.1007\/s10107-010-0395-1","journal-title":"Math. Program."},{"issue":"1","key":"988_CR35","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10287-004-0010-0","volume":"2","author":"JS Pang","year":"2005","unstructured":"Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 2(1), 21\u201356 (2005). doi: 10.1007\/s10287-004-0010-0","journal-title":"Comput. Manag. Sci."},{"issue":"3","key":"988_CR36","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/s10287-009-0093-8","volume":"6","author":"JS Pang","year":"2009","unstructured":"Pang, J.S., Fukushima, M.: Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games. Comput. Manag. Sci. 6(3), 373\u2013375 (2009). doi: 10.1007\/s10287-009-0093-8","journal-title":"Comput. Manag. Sci."},{"issue":"4","key":"988_CR37","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0167-6377(99)00052-8","volume":"25","author":"M Patriksson","year":"1999","unstructured":"Patriksson, M., Wynter, L.: Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25(4), 159\u2013167 (1999). doi: 10.1016\/S0167-6377(99)00052-8","journal-title":"Oper. Res. Lett."},{"key":"988_CR38","doi-asserted-by":"crossref","unstructured":"Ralph, D., Smeers, Y.: EPECs as models for electricity markets. In: Power Systems Conference and Exposition, 2006. PSCE \u201906. 2006 IEEE PES, pp. 74\u201380 (2006). doi: 10.1109\/PSCE.2006.296252","DOI":"10.1109\/PSCE.2006.296252"},{"key":"988_CR39","doi-asserted-by":"crossref","unstructured":"Ralph, D., Smeers, Y.: Pricing risk under risk measures: an introduction to stochastic-endogenous equilibria. Social Sci. Res. Netw. http:\/\/ssrn.com\/abstract=1903897 (2011)","DOI":"10.2139\/ssrn.1903897"},{"issue":"3","key":"988_CR40","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s10957-004-1180-1","volume":"124","author":"D Ralph","year":"2005","unstructured":"Ralph, D., Xu, H.: Implicit smoothing and its application to optimization with piecewise smooth equality constraints1. J. Optim. Theory Appl. 124(3), 673\u2013699 (2005)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"988_CR41","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/100792644","volume":"21","author":"U Ravat","year":"2011","unstructured":"Ravat, U., Shanbhag, U.V.: On the characterization of solution sets of smooth and nonsmooth convex stochastic Nash games. SIAM J. Optim. 21(3), 1168\u20131199 (2011)","journal-title":"SIAM J. Optim."},{"issue":"7","key":"988_CR42","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1016\/S0378-4266(02)00271-6","volume":"26","author":"R Rockafellar","year":"2002","unstructured":"Rockafellar, R., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443\u20131471 (2002)","journal-title":"J. Bank. Finance"},{"key":"988_CR43","volume-title":"Variational Analysis","author":"R Rockafellar","year":"1997","unstructured":"Rockafellar, R., Wets, J.B.: Variational Analysis. Springer, New York (1997)"},{"issue":"1","key":"988_CR44","doi-asserted-by":"crossref","first-page":"223243","DOI":"10.1007\/s10957-005-7566-x","volume":"128","author":"A Shapiro","year":"2006","unstructured":"Shapiro, A., Xu, H.: Stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 128(1), 223243 (2006). doi: 10.1007\/s10957-005-7566-x","journal-title":"J. Optim. Theory Appl."},{"key":"988_CR45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-1-4684-9256-9_1","volume-title":"Decision Making Under Uncertainty, The IMA Volumes in Mathematics and Its Applications","author":"RJB Wets","year":"2002","unstructured":"Wets, R.J.B.: Stochastic programming models: wait-and-see versus here-and-now. In: Greengard, C., Ruszczynski, A. (eds.) Decision Making Under Uncertainty, The IMA Volumes in Mathematics and Its Applications, vol. 128, pp. 1\u201315. Springer, New York (2002). doi: 10.1007\/978-1-4684-9256-9_1"},{"issue":"2","key":"988_CR46","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s10107-008-0214-0","volume":"119","author":"H Xu","year":"2009","unstructured":"Xu, H., Zhang, D.: Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications. Math. Program. 119(2), 371\u2013401 (2009)","journal-title":"Math. Program."},{"issue":"3","key":"988_CR47","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1007\/s10589-013-9538-7","volume":"55","author":"H Xu","year":"2013","unstructured":"Xu, H., Zhang, D.: Stochastic Nash equilibrium problems: sample average approximation and applications. Comput. Optim. Appl. 55(3), 597\u2013645 (2013)","journal-title":"Comput. Optim. Appl."},{"issue":"7","key":"988_CR48","doi-asserted-by":"crossref","first-page":"1702","DOI":"10.1109\/TAC.2011.2137590","volume":"56","author":"H Yin","year":"2011","unstructured":"Yin, H., Shanbhag, U., Mehta, P.: Nash equilibrium problems with scaled congestion costs and shared constraints. IEEE Trans. Autom. Control 56(7), 1702\u20131708 (2011). doi: 10.1109\/TAC.2011.2137590","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"988_CR49","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1287\/opre.1090.0771","volume":"58","author":"J Zhao","year":"2010","unstructured":"Zhao, J., Hobbs, B.F., Pang, J.S.: Long-run equilibrium modeling of emissions allowance allocation systems in electric power markets. Oper. Res. 58(3), 529\u2013548 (2010)","journal-title":"Oper. Res."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-016-0988-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10107-016-0988-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-016-0988-4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,4]],"date-time":"2019-09-04T22:19:56Z","timestamp":1567635596000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10107-016-0988-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,23]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2016,6]]}},"alternative-id":["988"],"URL":"https:\/\/doi.org\/10.1007\/s10107-016-0988-4","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2,23]]}}}