{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T18:48:12Z","timestamp":1776106092268,"version":"3.50.1"},"reference-count":64,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T00:00:00Z","timestamp":1580688000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T00:00:00Z","timestamp":1580688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"NTNU Norwegian University of Science and Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2020,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper studies mathematical programming formulations for solving optimization problems with piecewise polynomial (PWP) constraints. We elaborate on suitable polynomial bases as a means of efficiently representing PWPs in mathematical programs, comparing and drawing connections between the monomial basis, the Bernstein basis, and B-splines. The theory is presented for both continuous and semi-continuous PWPs. Using a disjunctive formulation, we then exploit the characteristic of common polynomial basis functions to significantly reduce the number of nonlinearities, and to suggest a bound-tightening technique for PWP constraints. We derive several extensions using Bernstein cuts, an expanded Bernstein basis, and an expanded monomial basis, which upon a standard big-M reformulation yield a set of new MINLP models. The formulations are compared by globally solving six test sets of MINLPs and a realistic petroleum production optimization problem. The proposed framework shows promising numerical performance and facilitates the solution of PWP-constrained optimization problems using standard MINLP software.<\/jats:p>","DOI":"10.1007\/s10898-020-00881-4","type":"journal-article","created":{"date-parts":[[2020,2,3]],"date-time":"2020-02-03T07:02:33Z","timestamp":1580713353000},"page":"455-486","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Mathematical programming formulations for piecewise polynomial functions"],"prefix":"10.1007","volume":"77","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3197-6968","authenticated-orcid":false,"given":"Bjarne","family":"Grimstad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brage R.","family":"Knudsen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,2,3]]},"reference":[{"issue":"3","key":"881_CR1","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1137\/0606047","volume":"6","author":"E Balas","year":"1985","unstructured":"Balas, E.: Disjunctive programming and a hierarchy of relaxations for discret optimization problems. SIAM J. Algebraic Discrete Methods 6(3), 466\u2013486 (1985)","journal-title":"SIAM J. Algebraic Discrete Methods"},{"issue":"447\u2013454","key":"881_CR2","first-page":"99","volume":"69","author":"EML Beale","year":"1970","unstructured":"Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. OR 69(447\u2013454), 99 (1970)","journal-title":"OR"},{"issue":"3","key":"881_CR3","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/0377-2217(90)90419-C","volume":"48","author":"N Beaumont","year":"1990","unstructured":"Beaumont, N.: An algorithm for disjunctive programs. Eur. J. Oper. Res. 48(3), 362\u2013371 (1990)","journal-title":"Eur. J. Oper. Res."},{"key":"881_CR4","doi-asserted-by":"crossref","unstructured":"Biegler, L.T.: Simultaneous methods for dynamic optimization. In: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, Chap. 10, pp. 287\u2013324. SIAM, New York (2010)","DOI":"10.1137\/1.9780898719383.ch10"},{"issue":"2","key":"881_CR5","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s11081-011-9141-7","volume":"13","author":"C Bragalli","year":"2012","unstructured":"Bragalli, C., D\u2019Ambrosio, C., Lee, J., Lodi, A., Toth, P.: On the optimal design of water distribution networks: a practical MINLP approach. Optim. Eng. 13(2), 219\u2013246 (2012)","journal-title":"Optim. Eng."},{"issue":"3","key":"881_CR6","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1007\/s101070050106","volume":"86","author":"S Ceria","year":"1999","unstructured":"Ceria, S., Soares, J.: Convex programming for disjunctive convex optimization. Math. Program. 86(3), 595\u2013614 (1999)","journal-title":"Math. Program."},{"issue":"1","key":"881_CR7","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, nonconvex minimization. Math. Program. 134(1), 71\u201399 (2012)","journal-title":"Math. Program."},{"issue":"3","key":"881_CR8","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF01581171","volume":"80","author":"AR Conn","year":"1998","unstructured":"Conn, A.R., Mongeau, M.: Discontinuous piecewise linear optimization. Math. Program. 80(3), 315\u2013380 (1998)","journal-title":"Math. Program."},{"issue":"1","key":"881_CR9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/BF02788653","volume":"17","author":"HB Curry","year":"1966","unstructured":"Curry, H.B., Schoenberg, I.J.: On P\u00f3lya frequency functions IV: the fundamental spline functions and their limits. J. d\u2019Anal. Math. 17(1), 71\u2013107 (1966)","journal-title":"J. d\u2019Anal. Math."},{"issue":"1","key":"881_CR10","doi-asserted-by":"crossref","first-page":"30","DOI":"10.2307\/1905292","volume":"28","author":"GB Dantzig","year":"1960","unstructured":"Dantzig, G.B.: On the significance of solving linear programming problems with some integer variables. Econometrica 28(1), 30\u201344 (1960)","journal-title":"Econometrica"},{"issue":"10","key":"881_CR11","doi-asserted-by":"crossref","first-page":"101004","DOI":"10.1115\/1.3201977","volume":"131","author":"B Demeulenaere","year":"2009","unstructured":"Demeulenaere, B., Pipeleers, G., De Caigny, J., Swevers, J., De Schutter, J., Vandenberghe, L.: Optimal splines for rigid motion systems: a convex programming framework. ASME J. Mech. Des. 131(10), 101004\u2013101004-11 (2009)","journal-title":"ASME J. Mech. Des."},{"issue":"2","key":"881_CR12","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/ss\/1038425655","volume":"11","author":"PH Eilers","year":"1996","unstructured":"Eilers, P.H., Marx, B.D.: Flexible smoothing with B-splines and penalties. Stat. Sci. 11(2), 89\u2013102 (1996)","journal-title":"Stat. Sci."},{"issue":"6","key":"881_CR13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.cagd.2012.03.001","volume":"29","author":"RT Farouki","year":"2012","unstructured":"Farouki, R.T.: The Bernstein polynomial basis: a centennial retrospective. Comput. Aided Geom. Des. 29(6), 379\u2013419 (2012)","journal-title":"Comput. Aided Geom. Des."},{"issue":"4","key":"881_CR14","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s11590-017-1190-1","volume":"12","author":"B Grimstad","year":"2018","unstructured":"Grimstad, B.: A MIQCP formulation for B-spline constraints. Optim. Lett. 12(4), 713\u2013725 (2018)","journal-title":"Optim. Lett."},{"key":"881_CR15","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.compchemeng.2015.08.022","volume":"84","author":"B Grimstad","year":"2016","unstructured":"Grimstad, B., Foss, B., Heddle, R., Woodman, M.: Global optimization of multiphase flow networks using spline surrogate models. Comput. Chem. Eng. 84, 237\u2013254 (2016)","journal-title":"Comput. Chem. Eng."},{"issue":"3","key":"881_CR16","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/s10898-015-0358-4","volume":"65","author":"B Grimstad","year":"2016","unstructured":"Grimstad, B., Sandnes, A.: Global optimization with spline constraints: a new branch-and-bound method based on B-splines. J. Glob. Optim. 65(3), 401\u2013439 (2016)","journal-title":"J. Glob. Optim."},{"key":"881_CR17","unstructured":"Grimstad, B., et al.: SPLINTER: a library for multivariate function approximation with splines (2015). http:\/\/github.com\/bgrimstad\/splinter. Accessed 16 May 2015"},{"issue":"4","key":"881_CR18","doi-asserted-by":"crossref","first-page":"338","DOI":"10.2514\/3.20223","volume":"10","author":"C Hargraves","year":"1987","unstructured":"Hargraves, C., Paris, S.W.: Direct trajectory optimization using nonlinear programming and collocation. J. Guid. Control Dyn. 10(4), 338\u2013342 (1987)","journal-title":"J. Guid. Control Dyn."},{"issue":"7","key":"881_CR19","doi-asserted-by":"crossref","first-page":"1880","DOI":"10.1002\/aic.12109","volume":"56","author":"MMF Hasan","year":"2010","unstructured":"Hasan, M.M.F., Karimi, I.: Piecewise linear relaxation of bilinear programs using bivariate partitioning. AIChE J. 56(7), 1880\u20131893 (2010)","journal-title":"AIChE J."},{"key":"881_CR20","volume-title":"The Elements of Statistical Learning, Springer Series in Statistics","author":"T Hastie","year":"2009","unstructured":"Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, Springer Series in Statistics, vol. 1, 2nd edn. Springer, New York (2009)","edition":"2"},{"key":"881_CR21","volume-title":"Convex Analysis and Minimization Algorithms $$II$$\u2014Advanced Theory and Bundle Methods","author":"JB Hiriart-Urruty","year":"1993","unstructured":"Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms $$II$$\u2014Advanced Theory and Bundle Methods. Springer, Berlin (1993)"},{"issue":"3","key":"881_CR22","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1214\/009053607000000677","volume":"36","author":"T Hofmann","year":"2008","unstructured":"Hofmann, T., Sch\u00f6lkopf, B., Smola, A.J.: Kernel methods in machine learning. Ann. Stat. 36(3), 1171\u20131220 (2008)","journal-title":"Ann. Stat."},{"key":"881_CR23","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717532","volume-title":"Finite Element Methods with B-Splines","author":"K H\u00f6llig","year":"2003","unstructured":"H\u00f6llig, K.: Finite Element Methods with B-Splines. Society for Industrial and Applied Mathematics, Philadelphia (2003)"},{"issue":"1","key":"881_CR24","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/0377-2217(94)90184-8","volume":"75","author":"K Holmberg","year":"1994","unstructured":"Holmberg, K.: Solving the staircase cost facility location problem with decomposition and piecewise linearization. Eur. J. Oper. Res. 75(1), 41\u201361 (1994)","journal-title":"Eur. J. Oper. Res."},{"key":"881_CR25","doi-asserted-by":"crossref","unstructured":"Jahanshahi, E., Grimstad, B., Foss, B.: Spline fluid models for optimization. In: Proceedings of the IFAC Symposium on DYCOPS, pp. 400\u2013405, Trondheim (2016)","DOI":"10.1016\/j.ifacol.2016.07.374"},{"key":"881_CR26","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/0166-218X(87)90026-6","volume":"17","author":"RG Jeroslow","year":"1987","unstructured":"Jeroslow, R.G.: Representability in mixed integer programming, I: characterization results. Discrete Appl. Math. 17, 223\u2013243 (1987)","journal-title":"Discrete Appl. Math."},{"issue":"2","key":"881_CR27","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/0166-218X(89)90023-1","volume":"23","author":"RG Jeroslow","year":"1989","unstructured":"Jeroslow, R.G.: Representability of functions. Discrete Appl. Math. 23(2), 125\u2013137 (1989)","journal-title":"Discrete Appl. Math."},{"issue":"5","key":"881_CR28","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1287\/opre.1060.0277","volume":"54","author":"AB Keha","year":"2006","unstructured":"Keha, A.B., de Farias Jr, I.R., Nemhauser, G.L.: A branch-and-cut algorithm without binary variables for nonconvex piecewise linear optimization. Oper. Res. 54(5), 847\u2013858 (2006)","journal-title":"Oper. Res."},{"key":"881_CR29","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.compchemeng.2013.05.022","volume":"58","author":"BR Knudsen","year":"2013","unstructured":"Knudsen, B.R., Foss, B.: Shut-in based production optimization of shale-gas systems. Comput. Chem. Eng. 58, 54\u201367 (2013)","journal-title":"Comput. Chem. Eng."},{"issue":"1","key":"881_CR30","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF02592329","volume":"72","author":"W Li","year":"1996","unstructured":"Li, W.: A conjugate gradient method for the unconstrained minimization of strictly convex quadratic splines. Math. Program. 72(1), 17\u201332 (1996)","journal-title":"Math. Program."},{"key":"881_CR31","volume-title":"Bernstein Polynomials","author":"GG Lorentz","year":"2013","unstructured":"Lorentz, G.G.: Bernstein Polynomials. American Mathematical Soc., New York (2013)"},{"key":"881_CR32","doi-asserted-by":"crossref","unstructured":"Luo, Y.: Simulation-based optimization over discrete sets with noisy constraints. Ph.D. Thesis, University of Miami (2011)","DOI":"10.1109\/WSC.2011.6148091"},{"issue":"3","key":"881_CR33","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/s10898-016-0494-5","volume":"68","author":"N Martinez","year":"2017","unstructured":"Martinez, N., Anahideh, H., Rosenberger, J.M., Martinez, D., Chen, V.C., Wang, B.P.: Global optimization of non-convex piecewise linear regression splines. J. Glob. Optim. 68(3), 563\u2013586 (2017)","journal-title":"J. Glob. Optim."},{"issue":"8","key":"881_CR34","doi-asserted-by":"crossref","first-page":"6098","DOI":"10.1109\/TIE.2018.2874617","volume":"66","author":"T Mercy","year":"2019","unstructured":"Mercy, T., Jacquod, N., Herzog, R., Pipeleers, G.: Spline-based trajectory generation for CNC machines. IEEE Trans. Ind. Electron. 66(8), 6098\u20136107 (2019)","journal-title":"IEEE Trans. Ind. Electron."},{"issue":"1","key":"881_CR35","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10898-012-9874-7","volume":"57","author":"R Misener","year":"2013","unstructured":"Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. 57(1), 3\u201350 (2013)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"881_CR36","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s10898-014-0166-2","volume":"59","author":"R Misener","year":"2014","unstructured":"Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous\/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2), 503\u2013526 (2014)","journal-title":"J. Glob. Optim."},{"issue":"4\u20135","key":"881_CR37","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1080\/10556780802614507","volume":"24","author":"JM Natali","year":"2009","unstructured":"Natali, J.M., Pinto, J.M.: Piecewise polynomial interpolations and approximations of one-dimensional functions through mixed integer linear programming. Optim. Methods Softw. 24(4\u20135), 783\u2013803 (2009)","journal-title":"Optim. Methods Softw."},{"key":"881_CR38","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s10107-004-0552-5","volume":"152","author":"Y Nesterov","year":"2005","unstructured":"Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 152, 127\u2013152 (2005)","journal-title":"Math. Program."},{"issue":"1","key":"881_CR39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0167-6377(00)00028-6","volume":"27","author":"M Padberg","year":"2000","unstructured":"Padberg, M.: Approximating separable nonlinear functions via mixed zero-one programs. Oper. Res. Lett. 27(1), 1\u20135 (2000)","journal-title":"Oper. Res. Lett."},{"issue":"3","key":"881_CR40","doi-asserted-by":"crossref","first-page":"1030","DOI":"10.1109\/59.260897","volume":"8","author":"J Park","year":"1993","unstructured":"Park, J., Kim, Y., Eom, I., Lee, K.: Economic load dispatch for piecewise quadratic cost function using Hopfield neural network. IEEE Trans. Power Syst. 8(3), 1030\u20131038 (1993)","journal-title":"IEEE Trans. Power Syst."},{"issue":"8","key":"881_CR41","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, algorithms and control applications. Automatica 47(8), 1770\u20131777 (2011)","journal-title":"Automatica"},{"key":"881_CR42","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-59223-2","volume-title":"The NURBS Book","author":"LA Piegl","year":"1997","unstructured":"Piegl, L.A., Tiller, W.: The NURBS Book. Springer, Berlin (1997)"},{"key":"881_CR43","doi-asserted-by":"crossref","unstructured":"Posa, M., Kuindersma, S., Tedrake, R.: Optimization and stabilization of trajectories for constrained dynamical systems. In: 2016 IEEE International Conference on Robotics and Automation (ICRA), pp. 1366\u20131373 (2016)","DOI":"10.1109\/ICRA.2016.7487270"},{"issue":"1760","key":"881_CR44","doi-asserted-by":"crossref","first-page":"2573","DOI":"10.1098\/rsta.1999.0449","volume":"357","author":"P Prandoni","year":"1999","unstructured":"Prandoni, P., Vetterli, M.: Approximation and compression of piecewise smooth functions. Philos. Trans. Math. Phys. Eng. Sci. 357(1760), 2573\u20132591 (1999)","journal-title":"Philos. Trans. Math. Phys. Eng. Sci."},{"key":"881_CR45","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s10107-017-1165-0","volume":"170","author":"JO Royset","year":"2018","unstructured":"Royset, J.O.: Approximations and solution estimates in optimization. Math. Program. 170, 479\u2013506 (2018)","journal-title":"Math. Program."},{"issue":"2","key":"881_CR46","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/BF00138693","volume":"8","author":"NV Sahinidis","year":"1996","unstructured":"Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8(2), 201\u2013205 (1996)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"881_CR47","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1287\/opre.1030.0102","volume":"52","author":"S Scholtes","year":"2004","unstructured":"Scholtes, S.: Nonconvex structures in nonlinear programming. Oper. Res. 52(3), 368\u2013383 (2004)","journal-title":"Oper. Res."},{"issue":"1","key":"881_CR48","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1137\/0802008","volume":"2","author":"H Schramm","year":"1992","unstructured":"Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121\u2013152 (1992)","journal-title":"SIAM J. Optim."},{"key":"881_CR49","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511618994","volume-title":"Spline Functions: Basic Theory","author":"LL Schumaker","year":"2007","unstructured":"Schumaker, L.L.: Spline Functions: Basic Theory, 3rd edn. Cambridge University Press, Cambridge (2007)","edition":"3"},{"issue":"4","key":"881_CR50","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S0167-6377(01)00063-3","volume":"28","author":"HD Sherali","year":"2001","unstructured":"Sherali, H.D.: On mixed-integer zero-one representations for separable lower-semicontinuous piecewise-linear functions. Oper. Res. Lett. 28(4), 155\u2013160 (2001)","journal-title":"Oper. Res. Lett."},{"issue":"3","key":"881_CR51","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1109\/TIP.2004.840710","volume":"14","author":"R Shukla","year":"2005","unstructured":"Shukla, R., Dragotti, P.L., Do, M.N., Vetterli, M.: Rate-distortion optimized tree structured compression algorithms for piecewise smooth images. IEEE Trans. Image Process. 14(3), 343\u2013359 (2005)","journal-title":"IEEE Trans. Image Process."},{"key":"881_CR52","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1007\/s101070050103","volume":"86","author":"RA Stubbs","year":"1999","unstructured":"Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0\u20131 mixed convex programming. Math. Program. 86, 515\u2013532 (1999)","journal-title":"Math. Program."},{"issue":"3","key":"881_CR53","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/S0098-1354(02)00220-X","volume":"27","author":"A Vecchietti","year":"2003","unstructured":"Vecchietti, A., Lee, S., Grossmann, I.E.: Modeling of discrete\/continuous optimization problems: characterization and formulation of disjunctions and their relaxations. Comput. Chem. Eng. 27(3), 433\u2013448 (2003)","journal-title":"Comput. Chem. Eng."},{"issue":"1","key":"881_CR54","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/130915303","volume":"57","author":"JP Vielma","year":"2015","unstructured":"Vielma, J.P.: Mixed integer linear programming formulation techniques. SIAM Rev. 57(1), 3\u201357 (2015)","journal-title":"SIAM Rev."},{"issue":"2","key":"881_CR55","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1287\/opre.1090.0721","volume":"58","author":"JP Vielma","year":"2010","unstructured":"Vielma, J.P., Ahmed, S., Nemhauser, G.: Mixed-integer models for nonseparable piecewise-linear optimization: unifying framework and extensions. Oper. Res. 58(2), 303\u2013315 (2010)","journal-title":"Oper. Res."},{"issue":"1\u20132","key":"881_CR56","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/s10107-009-0295-4","volume":"128","author":"JP Vielma","year":"2011","unstructured":"Vielma, J.P., Nemhauser, G.L.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Program. 128(1\u20132), 49\u201372 (2011)","journal-title":"Math. Program."},{"key":"881_CR57","first-page":"1","volume":"33","author":"S Vigerske","year":"2017","unstructured":"Vigerske, S., Gleixner, A.: SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optim. Methods Softw. 33, 1\u201331 (2017)","journal-title":"Optim. Methods Softw."},{"issue":"3","key":"881_CR58","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1111\/itor.12292","volume":"24","author":"KK Vu","year":"2017","unstructured":"Vu, K.K., D\u2019Ambrosio, C., Hamadi, Y., Liberti, L.: Surrogate-based methods for black-box optimization. Int. Trans. Oper. Res. 24(3), 393\u2013424 (2017)","journal-title":"Int. Trans. Oper. Res."},{"issue":"2","key":"881_CR59","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1145\/1138450.1138453","volume":"25","author":"W Wang","year":"2006","unstructured":"Wang, W., Pottmann, H., Liu, Y.: Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Trans. Graph. 25(2), 214\u2013238 (2006)","journal-title":"ACM Trans. Graph."},{"issue":"1","key":"881_CR60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10898-013-0060-3","volume":"58","author":"A Wechsung","year":"2014","unstructured":"Wechsung, A., Barton, P.I.: Global optimization of bounded factorable functions with discontinuities. J. Glob. Optim. 58(1), 1\u201330 (2014)","journal-title":"J. Glob. Optim."},{"issue":"382","key":"881_CR61","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1080\/01621459.1983.10477977","volume":"78","author":"EJ Wegman","year":"1983","unstructured":"Wegman, E.J., Wright, I.W.: Splines in statistics. J. Am. Stat. Assoc. 78(382), 351\u2013365 (1983)","journal-title":"J. Am. Stat. Assoc."},{"issue":"3","key":"881_CR62","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1007\/BF00940574","volume":"48","author":"RS Womersley","year":"1986","unstructured":"Womersley, R.S., Fletcher, R.: An algorithm for composite nonsmooth optimization problems. J. Optim. Theory Appl. 48(3), 493\u2013523 (1986)","journal-title":"J. Optim. Theory Appl."},{"issue":"3","key":"881_CR63","doi-asserted-by":"crossref","first-page":"529","DOI":"10.3934\/jimo.2007.3.529","volume":"3","author":"Y Yuan","year":"2007","unstructured":"Yuan, Y., Fan, W., Pu, D.: Spline function smooth support vector machine for classification. J. Ind. Manag. Optim. 3(3), 529\u2013542 (2007)","journal-title":"J. Ind. Manag. Optim."},{"issue":"1","key":"881_CR64","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1287\/moor.6.1.140","volume":"6","author":"I Zang","year":"1981","unstructured":"Zang, I.: Discontinuous optimization by smoothing. Math. Oper. Res. 6(1), 140\u2013152 (1981)","journal-title":"Math. Oper. Res."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-020-00881-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-020-00881-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-020-00881-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,14]],"date-time":"2022-10-14T02:41:27Z","timestamp":1665715287000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-020-00881-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,3]]},"references-count":64,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,7]]}},"alternative-id":["881"],"URL":"https:\/\/doi.org\/10.1007\/s10898-020-00881-4","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,3]]},"assertion":[{"value":"1 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 January 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}