{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T07:34:27Z","timestamp":1761896067490},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2015,4,18]],"date-time":"2015-04-18T00:00:00Z","timestamp":1429315200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2016,2]]},"DOI":"10.1007\/s10898-015-0304-5","type":"journal-article","created":{"date-parts":[[2015,4,17]],"date-time":"2015-04-17T12:00:40Z","timestamp":1429272040000},"page":"217-231","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An extension of the $$\\alpha \\hbox {BB}$$ \u03b1 BB -type underestimation to linear parametric Hessian matrices"],"prefix":"10.1007","volume":"64","author":[{"given":"Milan","family":"Hlad\u00edk","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,4,18]]},"reference":[{"issue":"9","key":"304_CR1","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1016\/S0098-1354(98)00218-X","volume":"22","author":"CS Adjiman","year":"1998","unstructured":"Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: A global optimization method, $$\\alpha \\text{ BB }$$ \u03b1 BB , for general twice-differentiabe constrained NLPs - II. Implementation and computational results. Comput. Chem. Eng. 22(9), 1159\u20131179 (1998)","journal-title":"Comput. Chem. Eng."},{"issue":"9","key":"304_CR2","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1016\/S0098-1354(98)00027-1","volume":"22","author":"CS Adjiman","year":"1998","unstructured":"Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, $$\\alpha \\text{ BB }$$ \u03b1 BB , for general twice-differentiable constrained NLPs - I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137\u20131158 (1998)","journal-title":"Comput. Chem. Eng."},{"issue":"3","key":"304_CR3","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1023\/B:JOGO.0000044768.75992.10","volume":"29","author":"IG Akrotirianakis","year":"2004","unstructured":"Akrotirianakis, I.G., Floudas, C.A.: Computational experience with a new class of convex underestimators: box-constrained NLP problems. J. Glob. Optim. 29(3), 249\u2013264 (2004)","journal-title":"J. Glob. Optim."},{"issue":"4","key":"304_CR4","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s10898-004-6455-4","volume":"30","author":"IG Akrotirianakis","year":"2004","unstructured":"Akrotirianakis, I.G., Floudas, C.A.: A new class of improved convex underestimators for twice continuously differentiable constrained NLPs. J. Glob. Optim. 30(4), 367\u2013390 (2004)","journal-title":"J. Glob. Optim."},{"key":"304_CR5","unstructured":"Akrotirianakis, I.G., Meyer, C.A., Floudas, C.A.: The role of the off-diagonal elements of the hessian matrix in the construction of tight convex underestimators for nonconvex functions. In: Discovery Through Product and Process Design. Sixth International Conference on Foundations of Computer-Aided Process Design, pp. 501\u2013504. FOCAPD 2004, Princeton, New Jersey (2004)"},{"issue":"4","key":"304_CR6","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/BF01099647","volume":"7","author":"IP Androulakis","year":"1995","unstructured":"Androulakis, I.P., Maranas, C.D., Floudas, C.A.: $$\\alpha BB$$ \u03b1 B B : a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337\u2013363 (1995)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"304_CR7","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s10107-012-0602-3","volume":"136","author":"KM Anstreicher","year":"2012","unstructured":"Anstreicher, K.M.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136(2), 233\u2013251 (2012)","journal-title":"Math. Program."},{"issue":"1","key":"304_CR8","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1007\/BF01934696","volume":"28","author":"E Baumann","year":"1988","unstructured":"Baumann, E.: Optimal centered forms. BIT 28(1), 80\u201387 (1988)","journal-title":"BIT"},{"issue":"1\u20133","key":"304_CR9","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/S0024-3795(98)10044-7","volume":"281","author":"O Beaumont","year":"1998","unstructured":"Beaumont, O.: Solving interval linear systems with linear programming techniques. Linear Algebra Appl. 281(1\u20133), 293\u2013309 (1998)","journal-title":"Linear Algebra Appl."},{"issue":"3","key":"304_CR10","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s10898-011-9722-1","volume":"53","author":"F Domes","year":"2012","unstructured":"Domes, F., Neumaier, A.: Rigorous filtering using linear relaxations. J. Glob. Optim. 53(3), 441\u2013473 (2012)","journal-title":"J. Glob. Optim."},{"issue":"6","key":"304_CR11","doi-asserted-by":"crossref","first-page":"1185","DOI":"10.1016\/j.compchemeng.2005.02.006","volume":"29","author":"C Floudas","year":"2005","unstructured":"Floudas, C., Akrotirianakis, I., Caratzoulas, S., Meyer, C., Kallrath, J.: Global optimization in the 21st century: advances and challenges. Comput. Chem. Eng. 29(6), 1185\u20131202 (2005)","journal-title":"Comput. Chem. Eng."},{"key":"304_CR12","volume-title":"Deterministic Global Optimization. Theory, Methods and Applications, Nonconvex Optimization and its Applications","author":"CA Floudas","year":"2000","unstructured":"Floudas, C.A.: Deterministic Global Optimization. Theory, Methods and Applications, Nonconvex Optimization and its Applications, vol. 37. Kluwer, Dordrecht (2000)"},{"issue":"1","key":"304_CR13","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s10898-008-9332-8","volume":"45","author":"CA Floudas","year":"2009","unstructured":"Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Glob. Optim. 45(1), 3\u201338 (2009)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"304_CR14","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s11590-006-0003-8","volume":"1","author":"CA Floudas","year":"2007","unstructured":"Floudas, C.A., Kreinovich, V.: On the functional form of convex underestimators for twice continuously differentiable functions. Optim. Lett. 1(2), 187\u2013192 (2007)","journal-title":"Optim. Lett."},{"key":"304_CR15","doi-asserted-by":"crossref","unstructured":"Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization, 2nd edn. Springer, New York (2009)","DOI":"10.1007\/978-0-387-74759-0"},{"issue":"1","key":"304_CR16","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/s10898-008-9288-8","volume":"42","author":"CE Gounaris","year":"2008","unstructured":"Gounaris, C.E., Floudas, C.A.: Tight convex underestimators for $${\\cal C}^2$$ C 2 -continuous problems. II: Multivariate functions. J. Glob. Optim. 42(1), 69\u201389 (2008)","journal-title":"J. Glob. Optim."},{"key":"304_CR17","volume-title":"Global Optimization Using Interval Analysis","author":"ER Hansen","year":"2004","unstructured":"Hansen, E.R., Walster, G.W.: Global Optimization Using Interval Analysis, 2nd edn. Marcel Dekker, New York (2004)","edition":"2"},{"issue":"2","key":"304_CR18","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/s10898-006-9045-9","volume":"37","author":"P Hansen","year":"2007","unstructured":"Hansen, P., Lagouanelle, J.L., Messine, F.: Comparison between Baumann and admissible simplex forms in interval analysis. J. Glob. Optim. 37(2), 215\u2013228 (2007)","journal-title":"J. Glob. Optim."},{"key":"304_CR19","unstructured":"Hedar, A.R.: Global optimization test problems (2014). Web pages http:\/\/www-optima.amp.i.kyoto-u.ac.jp\/member\/student\/hedar\/Hedar_files\/TestGO.htm"},{"key":"304_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-88670-1","volume-title":"Introduction to Nonlinear and Global Optimization, Optimization and its Applications","author":"EMT Hendrix","year":"2010","unstructured":"Hendrix, E.M.T., Gazdag-T\u00f3th, B.: Introduction to Nonlinear and Global Optimization, Optimization and its Applications, vol. 37. Springer, New York (2010)"},{"issue":"3","key":"304_CR21","doi-asserted-by":"crossref","first-page":"561","DOI":"10.2478\/v10006-012-0043-4","volume":"22","author":"M Hlad\u00edk","year":"2012","unstructured":"Hlad\u00edk, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561\u2013574 (2012)","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"issue":"10","key":"304_CR22","doi-asserted-by":"crossref","first-page":"5584","DOI":"10.1016\/j.amc.2012.11.075","volume":"219","author":"M Hlad\u00edk","year":"2013","unstructured":"Hlad\u00edk, M.: Bounds on eigenvalues of real and complex interval matrices. Appl. Math. Comput. 219(10), 5584\u20135591 (2013)","journal-title":"Appl. Math. Comput."},{"key":"304_CR23","unstructured":"Hlad\u00edk, M.: The effect of Hessian evaluations in the global optimization $$\\alpha {\\rm BB}$$ \u03b1 BB method (2013). Preprint, http:\/\/arxiv.org\/abs\/1307.2791"},{"issue":"2","key":"304_CR24","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1007\/s10898-014-0161-7","volume":"61","author":"M Hlad\u00edk","year":"2015","unstructured":"Hlad\u00edk, M.: On the efficient Gerschgorin inclusion usage in the global optimization $$\\alpha {\\rm BB}$$ \u03b1 BB method. J. Glob. Optim. 61(2), 235\u2013253 (2015)","journal-title":"J. Glob. Optim."},{"issue":"4","key":"304_CR25","doi-asserted-by":"crossref","first-page":"2116","DOI":"10.1137\/090753991","volume":"31","author":"M Hlad\u00edk","year":"2010","unstructured":"Hlad\u00edk, M., Daney, D., Tsigaridas, E.: Bounds on real eigenvalues and singular values of interval matrices. SIAM J. Matrix Anal. Appl. 31(4), 2116\u20132129 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"12","key":"304_CR26","doi-asserted-by":"crossref","first-page":"5236","DOI":"10.1016\/j.amc.2010.09.066","volume":"217","author":"M Hlad\u00edk","year":"2011","unstructured":"Hlad\u00edk, M., Daney, D., Tsigaridas, E.P.: A filtering method for the interval eigenvalue problem. Appl. Math. Comput. 217(12), 5236\u20135242 (2011)","journal-title":"Appl. Math. Comput."},{"key":"304_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2495-0","volume-title":"Rigorous Global Search: Continuous Problems","author":"RB Kearfott","year":"1996","unstructured":"Kearfott, R.B.: Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht (1996)"},{"issue":"2","key":"304_CR28","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1080\/10556781003636851","volume":"26","author":"RB Kearfott","year":"2011","unstructured":"Kearfott, R.B.: Interval computations, rigour and non-rigour in deterministic continuous global optimization. Optim. Methods Softw. 26(2), 259\u2013279 (2011)","journal-title":"Optim. Methods Softw."},{"issue":"18","key":"304_CR29","first-page":"2657","volume":"16","author":"V Kreinovich","year":"2010","unstructured":"Kreinovich, V., Kubica, B.J.: From computing sets of optima, Pareto sets, and sets of Nash equilibria to general decision-related set computations. J. Univ. Comput. Sci. 16(18), 2657\u20132685 (2010)","journal-title":"J. Univ. Comput. Sci."},{"issue":"4","key":"304_CR30","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1137\/10078760X","volume":"32","author":"M M\u00f6nnigmann","year":"2011","unstructured":"M\u00f6nnigmann, M.: Fast calculation of spectral bounds for hessian matrices on hyperrectangles. SIAM J. Matrix Anal. Appl. 32(4), 1351\u20131366 (2011)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"304_CR31","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717716","volume-title":"Introduction to Interval Analysis","author":"RE Moore","year":"2009","unstructured":"Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)"},{"key":"304_CR32","volume-title":"Interval Methods for Systems of Equations","author":"A Neumaier","year":"1990","unstructured":"Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)"},{"key":"304_CR33","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1017\/S0962492904000194","volume":"13","author":"A Neumaier","year":"2004","unstructured":"Neumaier, A.: Complete search in continuous global optimization and constraint satisfaction. Acta Numer. 13, 271\u2013369 (2004)","journal-title":"Acta Numer."},{"issue":"4","key":"304_CR34","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1007\/s10898-010-9531-y","volume":"50","author":"J Ninin","year":"2011","unstructured":"Ninin, J., Messine, F.: A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms. J. Glob. Optim. 50(4), 629\u2013644 (2011)","journal-title":"J. Glob. Optim."},{"issue":"4","key":"304_CR35","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1137\/120870359","volume":"33","author":"ED Popova","year":"2012","unstructured":"Popova, E.D.: Explicit description of AE solution sets for parametric linear systems. SIAM J. Matrix Anal. Appl. 33(4), 1172\u20131189 (2012)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"304_CR36","doi-asserted-by":"crossref","unstructured":"Rump, S.M.: INTLAB\u2014INTerval LABoratory. In: T. Csendes (ed.) Developments in Reliable Computing, pp. 77\u2013104. Kluwer Academic Publishers, Dordrecht (1999). http:\/\/www.ti3.tu-harburg.de\/rump\/","DOI":"10.1007\/978-94-017-1247-7_7"},{"issue":"4","key":"304_CR37","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s10898-011-9664-7","volume":"51","author":"JK Scott","year":"2011","unstructured":"Scott, J.K., Stuber, M.D., Barton, P.I.: Generalized McCormick relaxations. J. Glob. Optim. 51(4), 569\u2013606 (2011)","journal-title":"J. Glob. Optim."},{"issue":"3","key":"304_CR38","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s10898-013-0057-y","volume":"58","author":"A Skj\u00e4l","year":"2014","unstructured":"Skj\u00e4l, A., Westerlund, T.: New methods for calculating $$\\alpha BB$$ \u03b1 B B -type underestimators. J. Glob. Optim. 58(3), 411\u2013427 (2014)","journal-title":"J. Glob. Optim."},{"issue":"2","key":"304_CR39","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1007\/s10957-012-0033-6","volume":"154","author":"A Skj\u00e4l","year":"2012","unstructured":"Skj\u00e4l, A., Westerlund, T., Misener, R., Floudas, C.A.: A generalization of the classical $$\\alpha BB$$ \u03b1 B B convex underestimation via diagonal and nondiagonal quadratic terms. J. Optim. Theory Appl. 154(2), 462\u2013490 (2012)","journal-title":"J. Optim. Theory Appl."},{"issue":"2\u20134","key":"304_CR40","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s00607-011-0170-z","volume":"94","author":"M Zimmer","year":"2012","unstructured":"Zimmer, M., Kr\u00e4mer, W., Popova, E.D.: Solvers for the verified solution of parametric linear systems. Computing 94(2\u20134), 109\u2013123 (2012)","journal-title":"Computing"}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-015-0304-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10898-015-0304-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-015-0304-5","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T04:59:09Z","timestamp":1559278749000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10898-015-0304-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,18]]},"references-count":40,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2016,2]]}},"alternative-id":["304"],"URL":"https:\/\/doi.org\/10.1007\/s10898-015-0304-5","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"value":"0925-5001","type":"print"},{"value":"1573-2916","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,4,18]]}}}