{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T21:44:35Z","timestamp":1772833475307,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,1,10]],"date-time":"2021-01-10T00:00:00Z","timestamp":1610236800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,10]],"date-time":"2021-01-10T00:00:00Z","timestamp":1610236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Southampton Presidential Scholarship"},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/P022553\/1"],"award-info":[{"award-number":["EP\/P022553\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This article studies Gauss\u2013Newton-type methods for over-determined systems to find solutions to bilevel programming problems. To proceed, we use the lower-level value function reformulation of bilevel programs and consider necessary optimality conditions under appropriate assumptions. First, under strict complementarity for upper- and lower-level feasibility constraints, we prove the convergence of a Gauss\u2013Newton-type method in computing points satisfying these optimality conditions under additional tractable qualification conditions. Potential approaches to address the shortcomings of the method are then proposed, leading to alternatives such as the pseudo or smoothing Gauss\u2013Newton-type methods for bilevel optimization. Our numerical experiments conducted on 124 examples from the recently released Bilevel Optimization LIBrary (BOLIB) compare the performance of our method under different scenarios and show that it is a tractable approach to solve bilevel optimization problems with continuous variables.<\/jats:p>","DOI":"10.1007\/s10589-020-00254-3","type":"journal-article","created":{"date-parts":[[2021,1,10]],"date-time":"2021-01-10T09:05:51Z","timestamp":1610269551000},"page":"793-824","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Gauss\u2013Newton-type methods for bilevel optimization"],"prefix":"10.1007","volume":"78","author":[{"given":"J\u00f6rg","family":"Fliege","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrey","family":"Tin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1265-4178","authenticated-orcid":false,"given":"Alain","family":"Zemkoho","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,10]]},"reference":[{"key":"254_CR1","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10107-010-0342-1","volume":"131","author":"GB Allende","year":"2012","unstructured":"Allende, G.B., Still, G.: Solving bi-level programs with the KKT-approach. Math. Program. 131, 37\u201348 (2012)","journal-title":"Math. Program."},{"key":"254_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2836-1","volume-title":"Practical Bilevel Optimization: Algorithms and Applications","author":"JF Bard","year":"1998","unstructured":"Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Cambridge (1998)"},{"issue":"222","key":"254_CR3","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1090\/S0025-5718-98-00932-6","volume":"67","author":"X Chen","year":"1998","unstructured":"Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67(222), 519\u2013540 (1998)","journal-title":"Math. Comput."},{"issue":"2","key":"254_CR4","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1023\/A:1008735010803","volume":"15","author":"S Dempe","year":"2000","unstructured":"Dempe, S.: A bundle algorithm applied to bilevel programming problems with non-unique lower level solutions. Comput. Optim. Appl. 15(2), 145\u2013166 (2000)","journal-title":"Comput. Optim. Appl."},{"key":"254_CR5","volume-title":"Foundations of Bilevel Programming","author":"S Dempe","year":"2002","unstructured":"Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Cambridge (2002)"},{"key":"254_CR6","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s10107-010-0342-1","volume":"131","author":"S Dempe","year":"2010","unstructured":"Dempe, S., Dutta, J.: Is bilevel programming a special case of mathematical programming with equilibrium constraints? Math. Program. 131, 37\u201348 (2010)","journal-title":"Math. Program."},{"issue":"5\u20136","key":"254_CR7","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1080\/02331930701617551","volume":"56","author":"S Dempe","year":"2007","unstructured":"Dempe, S., Dutta, J., Mordukhovich, B.S.: New necessary optimality conditions in optimistic bilevel programming. Optimization 56(5\u20136), 577\u2013604 (2007)","journal-title":"Optimization"},{"issue":"3","key":"254_CR8","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/BF00247793","volume":"6","author":"S Dempe","year":"1996","unstructured":"Dempe, S., Schmidt, H.: On an algorithm solving two-level programming problems with nonunique lower level solutions. Comput. Optim. Appl. 6(3), 227\u2013249 (1996)","journal-title":"Comput. Optim. Appl."},{"key":"254_CR9","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/s10107-011-0508-5","volume":"138","author":"S Dempe","year":"2013","unstructured":"Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualification and optimality conditions. Math. Program. 138, 447\u2013473 (2013)","journal-title":"Math. Program."},{"issue":"1","key":"254_CR10","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1007\/s10957-010-9744-8","volume":"148","author":"S Dempe","year":"2011","unstructured":"Dempe, S., Zemkoho, A.B.: The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148(1), 46\u201368 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"254_CR11","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971200","volume-title":"Numerical Methods for Unconstrained Optimization and Nonlinear Equations","author":"JE Dennis","year":"1996","unstructured":"Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM Classics in Applied Mathematics, New York (1996)"},{"issue":"3","key":"254_CR12","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1080\/02331939208843795","volume":"24","author":"A Fischer","year":"1992","unstructured":"Fischer, A.: A special Newton-type optimization method. Optimization 24(3), 269\u2013284 (1992)","journal-title":"Optimization"},{"key":"254_CR13","unstructured":"Fischer, A., Zemkoho, A.B., Zhou, S.: Semismooth Newton-type method for bilevel optimization: global convergence and extensive numerical experiments (2019). arXiv:1912.07079"},{"issue":"4","key":"254_CR14","doi-asserted-by":"publisher","first-page":"392","DOI":"10.1093\/comjnl\/10.4.392","volume":"10","author":"R Fletcher","year":"1968","unstructured":"Fletcher, R.: Generalized inverse methods for the best least squares solution of systems of non-linear equations. Comput. J. 10(4), 392\u2013399 (1968)","journal-title":"Comput. J."},{"key":"254_CR15","volume-title":"Practical Methods of Optimization","author":"R Fletcher","year":"1987","unstructured":"Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Hoboken (1987)","edition":"2"},{"key":"254_CR16","doi-asserted-by":"crossref","unstructured":"Fliege, J., Tin, A., Zemkoho, A.B.: Supplementary material for \u201cGauss-Newton-type methods for bilevel optimization\u201d. University of Southampton, UK, School of Mathematical Sciences (2020)","DOI":"10.1007\/s10589-020-00254-3"},{"key":"254_CR17","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1016\/j.cam.2014.04.024","volume":"272","author":"SY Gatilov","year":"2014","unstructured":"Gatilov, S.Y.: Using low-rank approximation of the Jacobian matrix in the Newton\u2013Raphson method to solve certain singular equations. J. Comput. Appl. Math. 272, 8\u201324 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"254_CR18","volume-title":"Matrix Computations","author":"GH Golub","year":"1996","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations. The John Hopkins University Press, Baltimore (1996)"},{"key":"254_CR19","doi-asserted-by":"crossref","first-page":"4332","DOI":"10.1016\/j.amc.2012.10.010","volume":"219","author":"Y Jiang","year":"2013","unstructured":"Jiang, Y., Li, X., Huang, C., Wu, X.: Application of particle swarm optimization based on CHKS smoothing function for solving nonlinear bi-level programming problem. Appl. Math. Comput. 219, 4332\u20134339 (2013)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"254_CR20","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1137\/S0895479894273134","volume":"17","author":"C Kanzow","year":"1996","unstructured":"Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17(4), 851\u2013868 (1996)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"254_CR21","doi-asserted-by":"publisher","first-page":"342","DOI":"10.1137\/S1052623497328781","volume":"9","author":"C Kanzow","year":"1999","unstructured":"Kanzow, C., Pieper, H.: Jacobian smoothing methods for general nonlinear complementarity problems. SIAM J. Optim. 9, 342\u2013372 (1999)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"254_CR22","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/s10898-013-0121-7","volume":"60","author":"P Kleniati","year":"2014","unstructured":"Kleniati, P., Adjiman, C.S.: Branch-and-sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: theoretical development. J. Glob. Optim. 60(3), 425\u2013458 (2014)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"254_CR23","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10898-013-0120-8","volume":"60","author":"P Kleniati","year":"2014","unstructured":"Kleniati, P., Adjiman, C.S.: Branch-and-sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: convergence analysis and numerical results. J. Glob. Optim. 60(3), 459\u2013481 (2014)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"254_CR24","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10589-020-00178-y","volume":"76","author":"L Lampariello","year":"2020","unstructured":"Lampariello, L., Sagratella, S.: Numerically tractable optimistic bilevel problems. Comput. Optim. Appl. 76(2), 277\u2013303 (2020)","journal-title":"Comput. Optim. Appl."},{"issue":"1\u20132","key":"254_CR25","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s10107-013-0633-4","volume":"144","author":"G-H Lin","year":"2014","unstructured":"Lin, G.-H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144(1\u20132), 277\u2013305 (2014)","journal-title":"Math. Program."},{"key":"254_CR26","unstructured":"Mehlitz, P., Minchenko, L.I., Zemkoho, A.B.: A note on partial calmness for bilevel optimization problems with linear structures at the lower level (2020). arXiv preprint arXiv:2003.06138"},{"issue":"4","key":"254_CR27","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1007\/s10898-007-9260-z","volume":"42","author":"A Mitsos","year":"2008","unstructured":"Mitsos, A., Lemonidis, P., Barton, P.I.: Global solution of bilevel programs with a nonconvex inner program. J. Glob. Optim. 42(4), 475\u2013513 (2008)","journal-title":"J. Glob. Optim."},{"key":"254_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)"},{"issue":"5","key":"254_CR29","doi-asserted-by":"publisher","first-page":"1109","DOI":"10.1137\/0912058","volume":"12","author":"V Pan","year":"1991","unstructured":"Pan, V., Schreiber, R.: An improved newton iteration for the generalized inverse of a matrix, with applications. SIAM J. Sci. Stat. Comput. 12(5), 1109\u20131130 (1991)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"254_CR30","doi-asserted-by":"publisher","first-page":"106609","DOI":"10.1016\/j.compchemeng.2019.106609","volume":"132","author":"R Paulavicius","year":"2020","unstructured":"Paulavicius, R., Gao, J., Kleniati, P., Adjiman, C.S.: BASBL: branch-and-sandwich bilevel solver. Implementation and computational study with the BASBLib test sets. Comput. Chem. Eng. 132, 106609 (2020)","journal-title":"Comput. Chem. Eng."},{"key":"254_CR31","doi-asserted-by":"crossref","unstructured":"Qi, L., Sun, D.: A survey of some nonsmooth equations and smoothing Newton methods. In: Progress in Optimization Vol. 30 (pp. 121-146), Springer (1999)","DOI":"10.1007\/978-1-4613-3285-5_7"},{"issue":"2","key":"254_CR32","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s10898-007-9228-z","volume":"41","author":"O Stein","year":"2008","unstructured":"Stein, O., Tezel, A.: The semismooth approach for semi-infinite programming under the Reduction Ansatz. J. Glob. Optim. 41(2), 245\u2013266 (2008)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"254_CR33","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1137\/080719765","volume":"20","author":"O Stein","year":"2009","unstructured":"Stein, O., Tezel, A.: The semismooth approach for semi-infinite programming without strict complementarity. SIAM J. Optim. 20(2), 1052\u20131072 (2009)","journal-title":"SIAM J. Optim."},{"issue":"1\u20133","key":"254_CR34","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1023\/A:1008669226453","volume":"13","author":"D Sun","year":"1999","unstructured":"Sun, D., Qi, L.: On NCP Functions. Comput. Optim. Appl. 13(1\u20133), 201\u2013220 (1999)","journal-title":"Comput. Optim. Appl."},{"key":"254_CR35","volume-title":"Optimization Theory and Methods","author":"W Sun","year":"2006","unstructured":"Sun, W., Yuan, Y.-X.: Optimization Theory and Methods. Springer, Berlin (2006)"},{"issue":"1","key":"254_CR36","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1137\/120864015","volume":"23","author":"W Wiesemann","year":"2013","unstructured":"Wiesemann, W., Tsoukalas, A., Kleniati, P., Rustem, B.: Pessimistic bilevel optimization. SIAM J. Optim. 23(1), 353\u2013380 (2013)","journal-title":"SIAM J. Optim."},{"issue":"1\u20132","key":"254_CR37","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/s10589-013-9627-7","volume":"59","author":"M Xu","year":"2014","unstructured":"Xu, M., Ye, J.J.: A smoothing augmented lagrangian method for solving simple bilevel programs. Comput. Optim. Appl. 59(1\u20132), 353\u2013377 (2014)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"254_CR38","doi-asserted-by":"publisher","first-page":"1388","DOI":"10.1137\/140971580","volume":"25","author":"M Xu","year":"2015","unstructured":"Xu, M., Ye, J.J., Zhang, L.: Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs. SIAM J. Optim. 25(3), 1388\u20131410 (2015)","journal-title":"SIAM J. Optim."},{"key":"254_CR39","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1080\/02331939508844060","volume":"33","author":"JJ Ye","year":"1995","unstructured":"Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9\u201327 (1995)","journal-title":"Optimization"},{"key":"254_CR40","unstructured":"Zemkoho, A.B.: Bilevel programming: reformulations, regularity and stationarity. Ph.D. thesis, Department of Mathematics and Computer Science, TU Bergakademie Freiberg, Freiberg, Germany (2012)"},{"key":"254_CR41","volume-title":"Bilevel Optimization: Advances and Next Challenges","author":"S Zhou","year":"2020","unstructured":"Zhou, S., Zemkoho, A.B., Tin, A.: BOLIB: bilevel optimization LIBrary of test problems. In: Dempe, S., Zemkoho, A.B. (eds.) Bilevel Optimization: Advances and Next Challenges. Springer, Berlin (2020)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00254-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-020-00254-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-020-00254-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,19]],"date-time":"2021-04-19T02:38:03Z","timestamp":1618799883000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-020-00254-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,10]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["254"],"URL":"https:\/\/doi.org\/10.1007\/s10589-020-00254-3","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,10]]},"assertion":[{"value":"6 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 December 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 January 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}