{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T00:27:38Z","timestamp":1775003258971,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2013,12,3]],"date-time":"2013-12-03T00:00:00Z","timestamp":1386028800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2014,5]]},"DOI":"10.1007\/s10589-013-9622-z","type":"journal-article","created":{"date-parts":[[2013,12,2]],"date-time":"2013-12-02T20:31:39Z","timestamp":1386016299000},"page":"249-272","source":"Crossref","is-referenced-by-count":7,"title":["Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions"],"prefix":"10.1007","volume":"58","author":[{"given":"Bryan","family":"Gardiner","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khan","family":"Jakee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yves","family":"Lucet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2013,12,3]]},"reference":[{"key":"9622_CR1","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1023\/A:1014299228104","volume":"16","author":"P. Alfeld","year":"2002","unstructured":"Alfeld, P., Schumaker, L.L.: Smooth macro-elements based on Powell-Sabin triangle splits. Adv. Comput. Math. 16, 29\u201346 (2002)","journal-title":"Adv. Comput. Math."},{"key":"9622_CR2","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1137\/070687542","volume":"19","author":"H.H. Bauschke","year":"2008","unstructured":"Bauschke, H.H., Goebel, R., Lucet, Y., Wang, X.: The proximal average: basic theory. SIAM J. Optim. 19, 768\u2013785 (2008)","journal-title":"SIAM J. Optim."},{"key":"9622_CR3","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1137\/060664513","volume":"50","author":"H.H. Bauschke","year":"2008","unstructured":"Bauschke, H.H., Lucet, Y., Trienis, M.: How to transform one convex function continuously into another. SIAM Rev. 50, 115\u2013132 (2008)","journal-title":"SIAM Rev."},{"key":"9622_CR4","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1137\/060675794","volume":"46","author":"H.H. Bauschke","year":"2007","unstructured":"Bauschke, H.H., Lucet, Y., Wang, X.: Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators. SIAM J. Control Optim. 46, 2031\u20132051 (2007)","journal-title":"SIAM J. Control Optim."},{"key":"9622_CR5","series-title":"Springer Optimization and Its Applications","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/978-1-4419-9569-8_2","volume-title":"Fixed-Point Algorithms for Inverse Problems in Science and Engineering","author":"H.H. Bauschke","year":"2011","unstructured":"Bauschke, H.H., Moffat, S.M., Wang, X.: Self-dual smooth approximations of convex functions via the proximal average. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications, vol.\u00a049, pp.\u00a023\u201332. Springer, New York (2011)"},{"key":"9622_CR6","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10107-009-0319-0","volume":"123","author":"H.H. Bauschke","year":"2010","unstructured":"Bauschke, H.H., Wang, X., Yao, L.: Autoconjugate representers for linear monotone operators. Math. Program. 123, 5\u201324 (2010)","journal-title":"Math. Program."},{"key":"9622_CR7","first-page":"587","volume":"308","author":"Y. Brenier","year":"1989","unstructured":"Brenier, Y.: Un algorithme rapide pour le calcul de transform\u00e9es de Legendre\u2013Fenchel discr\u00e8tes. C.\u00a0R. Acad. Sci. Paris, S\u00e9r. I, Math. 308, 587\u2013589 (1989)","journal-title":"C.\u00a0R. Acad. Sci. Paris, S\u00e9r. I, Math."},{"key":"9622_CR8","series-title":"Computers & Graphics-UK","first-page":"341","volume-title":"IEEE International Conference on Shape Modeling and Applications","author":"J. Cao","year":"2009","unstructured":"Cao, J., Li, X., Wang, G., Qin, H.: Surface reconstruction using bivariate simplex splines on Delaunay configurations. In: IEEE International Conference on Shape Modeling and Applications. Computers & Graphics-UK, Tsinghua Univ, Beijing, P.R. China, Jun. 26\u201328, 2009, vol.\u00a033, pp.\u00a0341\u2013350 (2009)"},{"key":"9622_CR9","unstructured":"CGAL: Computational geometry algorithms library. http:\/\/www.cgal.org"},{"key":"9622_CR10","unstructured":"CGLAB a Scilab toolbox for geometry based on CGAL. http:\/\/cglab.gforge.inria.fr\/"},{"key":"9622_CR11","unstructured":"Consortium, S.: Scilab (1994). http:\/\/www.scilab.org"},{"key":"9622_CR12","doi-asserted-by":"crossref","first-page":"1534","DOI":"10.1137\/S0036142993260208","volume":"33","author":"L. Corrias","year":"1996","unstructured":"Corrias, L.: Fast Legendre\u2013Fenchel transform and applications to Hamilton\u2013Jacobi equations and conservation laws. SIAM J. Numer. Anal. 33, 1534\u20131558 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"9622_CR13","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1090\/S0025-5718-1988-0942151-1","volume":"51","author":"M. D\u00e6hlen","year":"1988","unstructured":"D\u00e6hlen, M., Lyche, T.: Bivariate interpolation with quadratic box splines. Math. Comput. 51, 219\u2013230 (1988)","journal-title":"Math. Comput."},{"key":"9622_CR14","series-title":"Algorithms and Applications","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-77974-2","volume-title":"Computational Geometry","author":"M. Berg de","year":"2008","unstructured":"de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry, 3rd edn. Algorithms and Applications. Springer, Berlin (2008)","edition":"3"},{"key":"9622_CR15","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/imanum\/12.2.271","volume":"12","author":"P. Dierckx","year":"1992","unstructured":"Dierckx, P., Van Leemput, S., Vermeire, T.: Algorithms for surface fitting using Powell-Sabin splines. IMA J. Numer. Anal. 12, 271\u2013299 (1992)","journal-title":"IMA J. Numer. Anal."},{"key":"9622_CR16","first-page":"779","volume":"16","author":"B. Gardiner","year":"2009","unstructured":"Gardiner, B., Lucet, Y.: Numerical computation of Fitzpatrick functions. J. Convex Anal. 16, 779\u2013790 (2009)","journal-title":"J. Convex Anal."},{"key":"9622_CR17","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1007\/s11228-010-0157-5","volume":"18","author":"B. Gardiner","year":"2010","unstructured":"Gardiner, B., Lucet, Y.: Convex hull algorithms for piecewise linear-quadratic functions in computational convex analysis. Set-Valued Var. Anal. 18, 467\u2013482 (2010)","journal-title":"Set-Valued Var. Anal."},{"key":"9622_CR18","series-title":"Springer Optimization and Its Applications","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/978-1-4419-9569-8_12","volume-title":"Fixed-Point Algorithms for Inverse Problems in Science and Engineering","author":"B. Gardiner","year":"2011","unstructured":"Gardiner, B., Lucet, Y.: Graph-matrix calculus for computational convex analysis. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications, vol.\u00a049, pp.\u00a0243\u2013259. Springer, New York (2011)"},{"key":"9622_CR19","doi-asserted-by":"crossref","unstructured":"Gardiner, B., Lucet, Y.: Computing the conjugate of convex piecewise linear-quadratic bivariate functions. Math. Program. 1\u201324 (2013)","DOI":"10.1007\/s10589-013-9622-z"},{"key":"9622_CR20","unstructured":"Goebel, R.: Convexity, convergence and feedback in optimal control. PhD thesis, University of Washington, Seattle (2000)"},{"key":"9622_CR21","first-page":"179","volume":"15","author":"R. Goebel","year":"2008","unstructured":"Goebel, R.: Self-dual smoothing of convex and saddle functions. J. Convex Anal. 15, 179\u2013190 (2008)","journal-title":"J. Convex Anal."},{"key":"9622_CR22","doi-asserted-by":"crossref","first-page":"650","DOI":"10.1137\/07070913X","volume":"20","author":"W. Hare","year":"2009","unstructured":"Hare, W.: A proximal average for nonconvex functions: a proximal stability perspective. SIAM J. Optim. 20, 650\u2013666 (2009)","journal-title":"SIAM J. Optim."},{"key":"9622_CR23","series-title":"Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","volume-title":"Convex Analysis and Minimization Algorithms","author":"J.-B. Hiriart-Urruty","year":"1993","unstructured":"Hiriart-Urruty, J.-B., Lemar\u00e9chal, C.: Convex Analysis and Minimization Algorithms. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.\u00a0305\u2013306. Springer, Berlin (1993). Vol I: Fundamentals, Vol II: Advanced theory and bundle methods"},{"key":"9622_CR24","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s10957-010-9747-5","volume":"148","author":"J. Johnstone","year":"2011","unstructured":"Johnstone, J., Koch, V., Lucet, Y.: Convexity of the proximal average. J. Optim. Theory Appl. 148, 107\u2013124 (2011)","journal-title":"J. Optim. Theory Appl."},{"key":"9622_CR25","series-title":"Encyclopedia of Mathematics and Its Applications","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511721588","volume-title":"Spline Functions on Triangulations","author":"M.-J. Lai","year":"2007","unstructured":"Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and Its Applications, vol.\u00a0110. Cambridge University Press, Cambridge (2007)"},{"key":"9622_CR26","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/BF00248008","volume":"6","author":"Y. Lucet","year":"1996","unstructured":"Lucet, Y.: A fast computational algorithm for the Legendre\u2013Fenchel transform. Comput. Optim. Appl. 6, 27\u201357 (1996)","journal-title":"Comput. Optim. Appl."},{"key":"9622_CR27","unstructured":"Lucet, Y.: Computational convex analysis library, 1996\u20132011. https:\/\/people.ok.ubc.ca\/ylucet\/cca.html"},{"key":"9622_CR28","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1023\/A:1019191114493","volume":"16","author":"Y. Lucet","year":"1997","unstructured":"Lucet, Y.: Faster than the fast Legendre transform, the linear-time Legendre transform. Numer. Algorithms 16, 171\u2013185 (1997)","journal-title":"Numer. Algorithms"},{"key":"9622_CR29","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s11075-006-9056-0","volume":"43","author":"Y. Lucet","year":"2006","unstructured":"Lucet, Y.: Fast Moreau envelope computation I: numerical algorithms. Numer. Algorithms 43, 235\u2013249 (2006)","journal-title":"Numer. Algorithms"},{"key":"9622_CR30","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1137\/100788458","volume":"52","author":"Y. Lucet","year":"2010","unstructured":"Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52, 505\u2013542 (2010)","journal-title":"SIAM Rev."},{"key":"9622_CR31","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10589-007-9124-y","volume":"43","author":"Y. Lucet","year":"2009","unstructured":"Lucet, Y., Bauschke, H.H., Trienis, M.: The piecewise linear-quadratic model for computational convex analysis. Comput. Optim. Appl. 43, 95\u2013118 (2009)","journal-title":"Comput. Optim. Appl."},{"key":"9622_CR32","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s10444-006-9025-0","volume":"26","author":"C. Manni","year":"2007","unstructured":"Manni, C., Sablonni\u00e8re, P.: Quadratic spline quasi-interpolants on Powell-Sabin partitions. Adv. Comput. Math. 26, 283\u2013304 (2007)","journal-title":"Adv. Comput. Math."},{"key":"9622_CR33","unstructured":"Moffat, S.M.: On the kernel average of n functions. Master\u2019s thesis, Department of Mathematics, University of British Columbia (Dec 2009)"},{"key":"9622_CR34","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/BF01575032","volume":"9","author":"A. Noullez","year":"1994","unstructured":"Noullez, A., Vergassola, M.: A fast Legendre transform algorithm and applications to the adhesion model. J. Sci. Comput. 9, 259\u2013281 (1994)","journal-title":"J. Sci. Comput."},{"key":"9622_CR35","doi-asserted-by":"crossref","unstructured":"Patrinos, P., Sarimveis, H.: Convex parametric piecewise quadratic optimization: theory, algorithms and control applications. Tech. rep., National Technical University of Athens, Greece (2010)","DOI":"10.1016\/j.automatica.2011.04.003"},{"key":"9622_CR36","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.1016\/j.automatica.2010.06.008","volume":"46","author":"P. Patrinos","year":"2010","unstructured":"Patrinos, P., Sarimveis, H.: A new algorithm for solving convex parametric quadratic programs based on graphical derivatives of solution mappings. Automatica 46, 1405\u20131418 (2010)","journal-title":"Automatica"},{"key":"9622_CR37","doi-asserted-by":"crossref","first-page":"1770","DOI":"10.1016\/j.automatica.2011.04.003","volume":"47","author":"P. Patrinos","year":"2011","unstructured":"Patrinos, P., Sarimveis, H.: Convex parametric piecewise quadratic optimization: theory and algorithms. Automatica 47, 1770\u20131777 (2011)","journal-title":"Automatica"},{"key":"9622_CR38","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1016\/j.na.2004.05.018","volume":"58","author":"J.-P. Penot","year":"2004","unstructured":"Penot, J.-P.: The relevance of convex analysis for the study of monotonicity. Nonlinear Anal. 58, 855\u2013871 (2004)","journal-title":"Nonlinear Anal."},{"key":"9622_CR39","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1145\/355759.355761","volume":"3","author":"M.J.D. Powell","year":"1977","unstructured":"Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316\u2013325 (1977)","journal-title":"ACM Trans. Math. Softw."},{"key":"9622_CR40","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"R.T. Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"9622_CR41","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"R.T. Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"key":"9622_CR42","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1016\/j.apnum.2008.03.040","volume":"59","author":"D. Sbibih","year":"2009","unstructured":"Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell-Sabin partitions. Appl. Numer. Math. 59, 938\u2013958 (2009)","journal-title":"Appl. Numer. Math."},{"key":"9622_CR43","first-page":"97","volume":"3","author":"D. Sbibih","year":"2010","unstructured":"Sbibih, D., Serghini, A., Tijini, A.: Bivariate simplex spline quasi-interpolants. Numer. Math. 3, 97\u2013118 (2010)","journal-title":"Numer. Math."},{"key":"9622_CR44","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/BF02096551","volume":"148","author":"Z.-S. She","year":"1992","unstructured":"She, Z.-S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623\u2013641 (1992)","journal-title":"Commun. Math. Phys."},{"key":"9622_CR45","series-title":"Mod. Methods Math.","first-page":"423","volume-title":"Approximation Theory XI: Gatlinburg 2004","author":"T. Sorokina","year":"2005","unstructured":"Sorokina, T., Zeilfelder, F.: Optimal quasi-interpolation by quadratic C 1-splines on type-2 triangulations. In: Approximation Theory XI: Gatlinburg 2004, Mod. Methods Math., pp.\u00a0423\u2013438. Nashboro Press, Brentwood (2005)"},{"key":"9622_CR46","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.cagd.2008.05.001","volume":"26","author":"H. Speleers","year":"2009","unstructured":"Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell-Sabin B-splines. Comput. Aided Geom. Des. 26, 174\u2013191 (2009)","journal-title":"Comput. Aided Geom. Des."},{"key":"9622_CR47","series-title":"Eurographics Association","first-page":"133","volume-title":"Proceedings of the Symposium on Data Visualisation 2002, VISSYM \u201902, Aire-la-Ville","author":"D.F. Wiley","year":"2002","unstructured":"Wiley, D.F., Childs, H.R., Hamann, B., Joy, K.I., Max, N.L.: Best quadratic spline approximation for hierarchical visualization. In: Proceedings of the Symposium on Data Visualisation 2002, VISSYM \u201902, Aire-la-Ville. Eurographics Association, Switzerland, pp.\u00a0133\u2013140 (2002)"},{"key":"9622_CR48","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0377-0427(94)90082-5","volume":"56","author":"K. Willemans","year":"1994","unstructured":"Willemans, K., Dierckx, P.: Surface fitting using convex Powell-Sabin splines. J. Comput. Appl. Math. 56, 263\u2013282 (1994)","journal-title":"J. Comput. Appl. Math."}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-013-9622-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-013-9622-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-013-9622-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,18]],"date-time":"2022-03-18T12:12:54Z","timestamp":1647605574000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-013-9622-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,12,3]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014,5]]}},"alternative-id":["9622"],"URL":"https:\/\/doi.org\/10.1007\/s10589-013-9622-z","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,12,3]]}}}