INTOPT_90 swMATH ID: 4705 Software Authors: Kearfott, R.Baker Description: Rigorous global search: continuous problems The monograph gives a survey of interval arithmetic based methods for solving systems of equations and global optimization problems. The connection between these two themes is that zero finding is an almost unavoidable part of optimization. The interval arithmetic philosophy of selfvalidation is followed in the book as indicated by the term “rigorous” in the title. That means that for each of the two problems all solutions are to be found and enclosed in sufficiently small boxes. In addition the solutions ought to be unique in these boxes. – The book contains standard results, new results and ones that need further development. Algorithmic and practical tools are emphasized, theoretical considerations are sporadically present. Many extensive remarks and explanations are connected with INTLIB, INTLIB 90 and INTOPT 90 which are the authors’s software and library packages for interval arithmetic and the solvers for the above mentioned problems in FORTRAN 77 resp. FORTRAN 90. par The book consists of seven chapters and a bibliography of 247 items. The first chapter (69 pages) introduces interval arithmetic, solving linear interval equations, automatic differentiation and code list generation, interval Newton method, and a short glance at the topological degree idea, which seems a bit outside the interval scope of the book. A groad chapter (42 pages) about interval software follows. Described are primarily the packages mentioned before which allow, among others, the generation of code lists and automatic differentiation. The chapter on preconditioning (32 pages) contains some of the authors own results and it provides techniques to transform linear interval equations into ones which might have tighter solution intervals. The chapter about solving nonlinear systems of equations (23 pages) features a rather practical and numerical approach in a branch and bound pattern touching many software details. The chapter on global optimization (40 pages) only admits equality constraints (within a box frame). The solution procedures are based on branch and bound, infeasibility test, interval Newton algorithm applied to the John conditions and computationally executed proofs of the existence of feasible points generalizing Hansen and Walster’s pioneering concept of 1987 (Nonlinear equations and optimization. Preprint.). The chapter about non-differentiable problems (17 pages) studies function expressions containing if-then-else connectives obtaining some new results with respect to zero search. The final chapter about intermediate values (8 pages) remains on the very surface of such a topic and contains only introductory examples of how these values can be used to reduce the overestimation. Homepage: http://interval.louisiana.edu/kearfott.html Programming Languages: Fortran 90 Operating Systems: None Dependencies: None Keywords: interval arithmetic based methods; systems of equations; global optimization; automatic differentiation; interval Newton method; preconditioning Related Software: INTLAB; C-XSC; PROFIL/BIAS; C-XSC 2.0; BARON; Numerica; GlobSol; filib++; Cosy; Global Optimization Toolbox For Maple; minpack; COCONUT; Matlab; AMIGO; Genocop; GAMS; PNM; RealPaver; SeDuMi; CPLEX Cited in: 311 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Rigorous global search: continuous problems. Zbl 0876.90082Kearfott, R. Baker 1996 all top 5 Cited by 349 Authors 19 Kearfott, Baker 17 Kreinovich, Vladik Yakovlevich 13 Neumaier, Arnold 12 Casado, Leocadio G. 12 Csendes, Tibor 12 Gazdag-Tóth, Boglárka 11 García, Inmaculada F. 10 Fernandez Hernandez, Jose 9 Lang, Bruno 9 Messine, Frédéric 9 Schichl, Hermann 7 Nataraj, Paluri S. V. 7 Pelegrín, Blas 7 Sergeev, Yaroslav Dmitrievich 7 Stadtherr, Mark A. 6 Allahviranloo, Tofigh 6 Chen, Xiaojun 6 Frommer, Andreas 6 Kubica, Bartłomiej Jacek 6 Markót, Mihály Csaba 5 Beelitz, Thomas 5 Jansson, Christian 5 Plastria, Frank 4 Bedregal, Benjamín René Callejas 4 Berz, Martin 4 Bhunia, Asoke Kumar 4 Bischof, Christian H. 4 Chabert, Gilles 4 Dian, Jianwei 4 Hansen, Pierre 4 Jaulin, Luc 4 Makino, Kyoko 4 Mönnigmann, Martin 4 Puig, Vicenç 4 Ratz, Dietmar 4 Sahinidis, Nikolaos V. 4 Smith, Andrew Paul 3 Amrahov, Sahin Emrah 3 Csallner, András Erik 3 Domes, Ferenc 3 Enszer, Joshua A. 3 Fatullayev, Afet Golayoglu 3 Garloff, Jürgen 3 Gasilov, Nizami A. 3 Georg, Kurt 3 Ghanbari, Mojtaba 3 Grapsa, Theodoula N. 3 Hansen, Eldon Robert 3 Hendrix, Eligius M. T. 3 Karmakar, Samiran 3 Lebbah, Yahia 3 Lin, Youdong 3 Longpré, Luc 3 Martínez García, José Antonio 3 Michel, Claude 3 Montanher, Tiago 3 Neher, Markus 3 Ozdamar, Linet 3 Pintér, János D. 3 Posypkin, Mikhail A. 3 Quevedo, Joseba 3 Rueher, Michel 3 Schnurr, Marco 3 Sharyĭ, Sergeĭ Petrovich 3 Sotiropoulos, Dimitris G. 3 Starks, Scott A. 3 Vinkó, Tamás 3 Walster, G. William 3 Wolfe, Michael A. 3 Xu, Peiliang 2 Alefeld, Götz E. 2 Alolyan, Ibraheem 2 Araya, Ignacio J. 2 Arjmandzadeh, Ziba 2 Armengol, Joaquim 2 Arounassalame, M. 2 Barton, Paul I. 2 Beck, Jan B. 2 Bravo, José Manuel 2 Carrizosa, Emilio 2 Dellnitz, Michael 2 Evtushenko, Yuriĭ Gavrilovich 2 Goldsztejn, Alexandre 2 Granvilliers, Laurent 2 Hladík, Milan 2 Hoefkens, Jens 2 Huyer, Waltraud 2 Jermann, Christophe 2 Kálovics, Ferenc 2 Kolev, Lubomir V. 2 Kotecha, Ketan 2 Lagouanelle, Jean-Louis 2 Lodwick, Weldon Alexander 2 Malinowski, Krzysztof B. 2 Marquardt, Wolfgang 2 Nenov, Ivo P. 2 Nguyen, Hung Trung 2 Ninin, Jordan 2 Pedamallu, Chandra Sekhar 2 Potra, Florian-Alexandru ...and 249 more Authors all top 5 Cited in 90 Serials 46 Journal of Global Optimization 35 Reliable Computing 14 Journal of Computational and Applied Mathematics 11 Numerical Algorithms 8 Applied Mathematics and Computation 8 European Journal of Operational Research 7 Fuzzy Sets and Systems 5 SIAM Journal on Numerical Analysis 5 Mathematical Programming. 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Zeitschrift für Angewandte Mathematik und Mechanik 1 Bulletin of the European Association for Theoretical Computer Science EATCS 1 Sibirskiĭ Zhurnal Vychislitel’noĭ Matematiki 1 Journal of Computational Analysis and Applications 1 Communications in Nonlinear Science and Numerical Simulation 1 Optimization and Engineering 1 RAIRO. Operations Research 1 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 1 Advances in Geometry 1 International Journal of Computational and Numerical Analysis and Applications (IJCNAA) 1 OR Spectrum 1 4OR 1 Networks and Spatial Economics 1 ISNM. International Series of Numerical Mathematics 1 Nonconvex Optimization and Its Applications 1 Applied Mathematical Sciences (Ruse) 1 PAMM. Proceedings in Applied Mathematics and Mechanics 1 SIAM Journal on Imaging Sciences 1 Advances in Operations Research 1 Advances in Fuzzy Systems 1 Springer Optimization and Its Applications 1 Natural Computing Series all top 5 Cited in 39 Fields 185 Numerical analysis (65-XX) 169 Operations research, mathematical programming (90-XX) 27 Computer science (68-XX) 16 Systems theory; control (93-XX) 12 Linear and multilinear algebra; matrix theory (15-XX) 10 Ordinary differential equations (34-XX) 9 Statistics (62-XX) 8 Real functions (26-XX) 8 Biology and other natural sciences (92-XX) 6 Mathematical logic and foundations (03-XX) 6 Functions of a complex variable (30-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 5 Mechanics of particles and systems (70-XX) 5 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 4 Operator theory (47-XX) 4 Convex and discrete geometry (52-XX) 3 Field theory and polynomials (12-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Approximations and expansions (41-XX) 2 General algebraic systems (08-XX) 2 Algebraic geometry (14-XX) 2 Algebraic topology (55-XX) 2 Mechanics of deformable solids (74-XX) 2 Geophysics (86-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Order, lattices, ordered algebraic structures (06-XX) 1 Commutative algebra (13-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Special functions (33-XX) 1 Partial differential equations (35-XX) 1 Difference and functional equations (39-XX) 1 Integral equations (45-XX) 1 General topology (54-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Probability theory and stochastic processes (60-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Information and communication theory, circuits (94-XX) Citations by Year