{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:44:22Z","timestamp":1757619862496,"version":"3.44.0"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T00:00:00Z","timestamp":1753488000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T00:00:00Z","timestamp":1753488000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider numerical aspects of finding classical J. Nash\u2019s equilibrium in concave <jats:italic>n<\/jats:italic>-persons game, nonlinear equilibrium (NE), as an alternative to primal and dual linear programming (LP) problems, and recently introduced nonlinear production-consumption equilibrium (NPCE). The problems are particular cases of a general nonlinear equilibrium problem, which is equivalent to a variational inequality (VI). The corresponding VIs have simple feasible sets, that the projection on them is a low cost operation. Therefore, we apply two projection methods for finding the equilibrium: pseudo-gradient projection (PGP) and extra pseudo-gradient (EPG). We present and analyze results obtained on random generated sets of these three classes of problems. The obtained results show expected advantages of the EPG over PGP. What is most important: the number of iterations requited by EPG method to find an approximation for the equilibrium with a given accuracy grows linearly with the number of products in case of NE and NPCE, or with the number of active strategies in case of J. Nash\u2019s equilibrium. The number of operations, or solution time grows as a cube of the corresponding parameters. These results corroborate the complexity bounds established in [18\u201320] under reasonable assumptions on the input data.<\/jats:p>","DOI":"10.1007\/s10957-025-02787-1","type":"journal-article","created":{"date-parts":[[2025,7,26]],"date-time":"2025-07-26T14:04:00Z","timestamp":1753538640000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finding Equilibrium in Some Economics and Game Models"],"prefix":"10.1007","volume":"207","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2291-233X","authenticated-orcid":false,"given":"Igor","family":"Griva","sequence":"first","affiliation":[]},{"given":"Roman","family":"Polyak","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,26]]},"reference":[{"key":"2787_CR1","unstructured":"Antipin, A.: The gradient and extragradient approaches in bilinear equilibrium programming, Dorodnizin Computing Center RAS (in Russian) (2002)"},{"issue":"1","key":"2787_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/01630563.2019.1633665","volume":"41","author":"M Balashov","year":"2020","unstructured":"Balashov, M., Polyak, B.T., Tremba, A.: Gradient projection and conditional gradient methods for constrained nonconvex minimization. Numer. Funct. Anal. Optim. 41(1), 1\u201328 (2020)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"2787_CR3","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1080\/02331934.2010.539689","volume":"5","author":"Y Censor","year":"2012","unstructured":"Censor, Y., Gibali, A., Reich, S.: Extensions of Korpelevich\u2019s extragradient method for the variational inequality problem in Euclidean space. Optimization 5, 1119\u20131132 (2012)","journal-title":"Optimization"},{"key":"2787_CR4","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1090\/S0002-9904-1964-11178-2","volume":"70","author":"A Goldstein","year":"1964","unstructured":"Goldstein, A.: Convex programming in Hilbert space. Bull. Am. Math. Soc. 70, 709\u2013710 (1964)","journal-title":"Bull. Am. Math. Soc."},{"issue":"8","key":"2787_CR5","doi-asserted-by":"publisher","first-page":"1259","DOI":"10.1080\/02331931003603133","volume":"59","author":"A Iusem","year":"2010","unstructured":"Iusem, A., Sosa, W.: On the proximal point methods for equilibrium problems in Hilbert spaces. Optimization 59(8), 1259\u20131274 (2010)","journal-title":"Optimization"},{"issue":"4","key":"2787_CR6","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1080\/02331939708844365","volume":"42","author":"A Iusem","year":"1997","unstructured":"Iusem, A., Svaiter, B.: A variant of Korpelevich\u2019s method for variational inequalities with new search strategy. Optimization 42(4), 309\u2013321 (1997)","journal-title":"Optimization"},{"issue":"10","key":"2787_CR7","first-page":"1562","volume":"27","author":"E Khobotov","year":"1987","unstructured":"Khobotov, E.: A modification of the extragradient method for solving variational inequalities and some optimization problems (Russian). USSR Comput. Math. Math. Phys. 27(10), 1562\u20131597 (1987)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"2787_CR8","doi-asserted-by":"crossref","unstructured":"Konnov, I.: Combined relaxation methods for variational inequalities. Lecture Notes in Economics and Mathematical Systems, vol. 495. Springer-Verlag, Berlin (2001)","DOI":"10.1007\/978-3-642-56886-2"},{"issue":"4","key":"2787_CR9","first-page":"747","volume":"12","author":"G Korpelevich","year":"1976","unstructured":"Korpelevich, G.: An extragradient method for finding saddle points and for other problems. Matecon 12(4), 747\u2013756 (1976)","journal-title":"Matecon"},{"key":"2787_CR10","doi-asserted-by":"crossref","unstructured":"Kunh, H.: On theorem of Wald. In: Kunh, H.W., Tucker, A.W. (eds.): Linear Inequalities and Related Systems, Vol. 38, Princeton University Press (1957)","DOI":"10.1515\/9781400881987-017"},{"key":"2787_CR11","volume-title":"Math. Econ","author":"K Lancaster","year":"1968","unstructured":"Lancaster, K.: Math. Econ. The Macmillan Company, New York (1968)"},{"key":"2787_CR12","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1016\/0041-5553(66)90114-5","volume":"6","author":"E Levitin","year":"1966","unstructured":"Levitin, E., Polyak, B.T.: Minimization methods in the presence of constraints. USSR Comput. Math. Math. Phys. 6, 787\u2013823 (1966)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"2787_CR13","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/BF01351927","volume":"152","author":"T Lezanski","year":"1963","unstructured":"Lezanski, T.: Uber das minimumproblem fur funktionale in Banachschen raumen. Math. Ann. 152, 271\u2013274 (1963)","journal-title":"Math. Ann."},{"key":"2787_CR14","unstructured":"Lojasiewicz, S.: Une propriet e topologique des sous-ensembles analytiques reels, Les Equations aux Derivees Partielles, CNRS, 87-89, Paris (1963)"},{"issue":"3\u20134","key":"2787_CR15","doi-asserted-by":"publisher","first-page":"2086","DOI":"10.1016\/j.na.2009.10.009","volume":"72","author":"J Mashreghi","year":"2010","unstructured":"Mashreghi, J., Nasri, M.: Forcing strong convergence of Korpelevich\u2019s method in Banach spaces with its applications in game theory. Nonlinear Anal. Theory Methods Appl. 72(3\u20134), 2086\u20132099 (2010)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"2787_CR16","doi-asserted-by":"crossref","unstructured":"Nash, J.: Non-cooperative games, Ann. of Math. 54 (2), (1951)","DOI":"10.2307\/1969529"},{"issue":"4","key":"2787_CR17","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1016\/0041-5553(63)90382-3","volume":"3","author":"BT Polyak","year":"1963","unstructured":"Polyak, B.T.: Gradient methods for the minimization of functionals. USSR Comput. Math. Math. Phys. 3(4), 643\u2013653 (1963)","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"2787_CR18","unstructured":"Polyak, R.: Nonlinear equilibrium for optimal resources allocation, Contemporary Mathematics 636 AMS, 1-17 (2015)"},{"key":"2787_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-68713-7","volume-title":"Introduction to Continuous Optimization","author":"RA Polyak","year":"2021","unstructured":"Polyak, R.A.: Introduction to Continuous Optimization. Springer, Switzerland (2021)"},{"key":"2787_CR20","unstructured":"Polyak, R.A.: Finding nonlinear production-consumption equilibrium, In: Khan, M.A., Sagara, N., and Zaslavski, A.J. (eds.): David Gale: Mathematical Economist: Essays in Appreciation On His 100th Birthday, Series Monographs in Mathematical Economics, Springer, New York (forthcoming)"},{"key":"2787_CR21","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1137\/0109044","volume":"9","author":"J Rozen","year":"1961","unstructured":"Rozen, J.: The gradient projection method for nonlinear programming, part 2: nonlinear constraints. J. SIAM Appl. Math. 9, 514\u2013553 (1961)","journal-title":"J. SIAM Appl. Math."},{"key":"2787_CR22","doi-asserted-by":"publisher","first-page":"520","DOI":"10.2307\/1911749","volume":"33","author":"J Rozen","year":"1965","unstructured":"Rozen, J.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520\u2013534 (1965)","journal-title":"Econometrica"},{"key":"2787_CR23","unstructured":"Wang, W., Carreira-Perpinan, M.: Projection onto the probability simplex: an efficient algorithm with a simple proof and an application, arXiv:1309.1541 [cs LG] (2013)"}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02787-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02787-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02787-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T01:43:14Z","timestamp":1757295794000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02787-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,26]]},"references-count":23,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["2787"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02787-1","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"type":"print","value":"0022-3239"},{"type":"electronic","value":"1573-2878"}],"subject":[],"published":{"date-parts":[[2025,7,26]]},"assertion":[{"value":"15 September 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"30"}}