{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T08:44:46Z","timestamp":1774601086986,"version":"3.50.1"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T00:00:00Z","timestamp":1712534400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T00:00:00Z","timestamp":1712534400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001801","name":"University of Western Australia","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001801","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2024,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Riemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.<\/jats:p>","DOI":"10.1007\/s10957-024-02427-0","type":"journal-article","created":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T19:01:26Z","timestamp":1712602886000},"page":"955-975","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multi-objective Variational Curves"],"prefix":"10.1007","volume":"201","author":[{"given":"C. Yal\u00e7\u0131n","family":"Kaya","sequence":"first","affiliation":[]},{"given":"Lyle","family":"Noakes","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4005-5431","authenticated-orcid":false,"given":"Erchuan","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,4,8]]},"reference":[{"key":"2427_CR1","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1080\/02331934.2011.568619","volume":"62","author":"W Alt","year":"2013","unstructured":"Alt, W., Baier, R., Lempio, F., Gerdts, M.: Approximations of linear control problems with bang-bang solutions. Optimization 62, 9\u201332 (2013)","journal-title":"Optimization"},{"key":"2427_CR2","doi-asserted-by":"crossref","unstructured":"Betts, J.T.: Practical Methods for Optimal Control Using Nonlinear Programming, Third Edition. Advances in Design and Control, SIAM Publications, Philadelphia (2020)","DOI":"10.1137\/1.9781611976199"},{"key":"2427_CR3","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1007\/s10957-010-9709-y","volume":"147","author":"H Bonnel","year":"2010","unstructured":"Bonnel, H., Kaya, C.Y.: Optimization over the efficient set of multi-objective convex optimal control problems. J. Optim. Theory Appl. 147, 93\u201311 (2010)","journal-title":"J. Optim. Theory Appl."},{"key":"2427_CR4","doi-asserted-by":"crossref","unstructured":"Byrd, R.H., Nocedal, J., Waltz, R.A.: KNITRO: An integrated package for nonlinear optimization. In: G. di Pillo and M. Roma, editors, Large-Scale Nonlinear Optimization, 35\u201359, Springer (2006)","DOI":"10.1007\/0-387-30065-1_4"},{"key":"2427_CR5","unstructured":"Do Carmo, M.P.: Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications (2016)"},{"key":"2427_CR6","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1090\/S0025-5718-00-01184-4","volume":"70","author":"AL Dontchev","year":"2001","unstructured":"Dontchev, A.L., Hager, W.W.: The Euler approximation in state constrained optimal control problems. Math. Comput. 70, 173\u2013203 (2001)","journal-title":"Math. Comput."},{"issue":"6","key":"2427_CR7","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1080\/01630560008816979","volume":"21","author":"AL Dontchev","year":"2000","unstructured":"Dontchev, A.L., Hager, W.W., Malanowski, K.: Error bound for Euler approximation of a state and control constrained optimal control problem. Numer. Funct. Anal. Optim. 21(6), 653\u2013682 (2000)","journal-title":"Numer. Funct. Anal. Optim."},{"issue":"1","key":"2427_CR8","doi-asserted-by":"publisher","first-page":"202","DOI":"10.1137\/S0036142999351765","volume":"38","author":"AL Dontchev","year":"2000","unstructured":"Dontchev, A.L., Hager, W.W., Veliov, V.M.: Second-order Runge-Kutta approximations in control constrained optimal control. SIAM J. Num. Anal. 38(1), 202\u2013226 (2000)","journal-title":"SIAM J. Num. Anal."},{"key":"2427_CR9","doi-asserted-by":"publisher","first-page":"1091","DOI":"10.1080\/02331934.2011.587006","volume":"60","author":"J Dutta","year":"2011","unstructured":"Dutta, J., Kaya, C.Y.: A new scalarization and numerical method for constructing weak Pareto front of multi-objective optimization problems. Optimization 60, 1091\u20131104 (2011)","journal-title":"Optimization"},{"key":"2427_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-79159-1","volume-title":"Adaptive Scalarization Methods in Multiobjective Optimization","author":"G Eichfelder","year":"2008","unstructured":"Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization. Springer, Berlin, Heidelberg (2008)"},{"key":"2427_CR11","unstructured":"Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, Second Edition. Brooks\/Cole Publishing Company\/Cengage Learning (2003)"},{"key":"2427_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-17005-8","volume-title":"Vector Optimization: Theory, Applications, and Extensions","author":"J Jahn","year":"2011","unstructured":"Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer-Verlag, Berlin, Heidelberg (2011)"},{"issue":"3","key":"2427_CR13","doi-asserted-by":"publisher","first-page":"685","DOI":"10.1007\/s10589-013-9603-2","volume":"57","author":"CY Kaya","year":"2014","unstructured":"Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57(3), 685\u2013702 (2014)","journal-title":"Comput. Optim. Appl."},{"issue":"3","key":"2427_CR14","doi-asserted-by":"publisher","first-page":"1247","DOI":"10.1007\/s10589-023-00535-7","volume":"86","author":"CY Kaya","year":"2023","unstructured":"Kaya, C.Y., Maurer, H.: Optimization over the Pareto front of nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 86(3), 1247\u20131274 (2023)","journal-title":"Comput. Optim. Appl."},{"key":"2427_CR15","doi-asserted-by":"publisher","first-page":"442","DOI":"10.1137\/12087880X","volume":"51","author":"CY Kaya","year":"2013","unstructured":"Kaya, C.Y., Noakes, J.L.: Finding interpolating curves minimizing $$L^\\infty $$ acceleration in the Euclidean space via optimal control theory. SIAM J. Control Optim. 51, 442\u2013464 (2013)","journal-title":"SIAM J. Control Optim."},{"key":"2427_CR16","unstructured":"Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. John Wiley & Sons (1963)"},{"issue":"3","key":"2427_CR17","doi-asserted-by":"publisher","first-page":"649","DOI":"10.1007\/BF02192163","volume":"86","author":"H Maurer","year":"1995","unstructured":"Maurer, H., Pickenhain, S.: Second-order sufficient conditions for control problems with mixed control-state constraints. J. Optim. Theory Appl. 86(3), 649\u2013667 (1995)","journal-title":"J. Optim. Theory Appl."},{"key":"2427_CR18","volume-title":"Nonlinear Multiobjective Optimization","author":"K Miettinen","year":"1999","unstructured":"Miettinen, K.: Nonlinear Multiobjective Optimization. Kluwer Academic, Boston (1999)"},{"key":"2427_CR19","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1093\/imamci\/dni069","volume":"23","author":"L Noakes","year":"2006","unstructured":"Noakes, L., Popiel, T.: Quadratures and cubics in $$SO(3)$$ and $$SO(1,2)$$. IMA J. Math. Contr. Info. 23, 463\u2013473 (2006)","journal-title":"IMA J. Math. Contr. Info."},{"key":"2427_CR20","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1093\/imamci\/dni040","volume":"22","author":"L Noakes","year":"2005","unstructured":"Noakes, L., Popiel, T.: Null Riemannian cubics in tension in $$SO(3)$$. IMA J. Math. Contr. Info. 22, 477\u2013488 (2005)","journal-title":"IMA J. Math. Contr. Info."},{"key":"2427_CR21","volume-title":"Riemannian Geometry","author":"P Petersen","year":"2006","unstructured":"Petersen, P.: Riemannian Geometry. Springer, New York (2006)"},{"issue":"1","key":"2427_CR22","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1137\/16M1079142","volume":"56","author":"A Pietrus","year":"2018","unstructured":"Pietrus, A., Scarinci, T., Veliov, V.: High order discrete approximations to Mayer\u2019s problems for linear systems. SIAM J. Control Optim. 56(1), 102\u2013119 (2018)","journal-title":"SIAM J. Control Optim."},{"key":"2427_CR23","doi-asserted-by":"publisher","first-page":"1673","DOI":"10.1007\/s10444-018-9601-0","volume":"44","author":"E Zhang","year":"2018","unstructured":"Zhang, E., Noakes, L.: Left Lie reduction for curves in homogeneous spaces. Adv. Comput. Math. 44, 1673\u20131686 (2018)","journal-title":"Adv. Comput. Math."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02427-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-024-02427-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-024-02427-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T15:14:54Z","timestamp":1715267694000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-024-02427-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,8]]},"references-count":23,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["2427"],"URL":"https:\/\/doi.org\/10.1007\/s10957-024-02427-0","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,8]]},"assertion":[{"value":"27 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing, or conflict of, interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}