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; PROFIL/BIAS; C-XSC; C-XSC 2.0; BARON; Numerica; GlobSol; Cosy; Global Optimization Toolbox For Maple; filib++; minpack; COCONUT; Matlab; AMIGO; Genocop; GAMS; PNM; RealPaver; SeDuMi; CPLEX
Cited in: 304 Publications

Standard Articles

1 Publication describing the Software, including 1 Publication in zbMATH Year
Rigorous global search: continuous problems. Zbl 0876.90082
Kearfott, R. Baker
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Cited by 338 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 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 Kubica, Bartłomiej Jacek
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 Jaulin, Luc
3 Karmakar, Samiran
3 Lebbah, Yahia
3 Lin, Youdong
3 Longpré, Luc
3 Michel, Claude
3 Montanher, Tiago
3 Neher, Markus
3 Ozdamar, Linet
3 Pintér, János D.
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 Kaelo, Pro
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 Posypkin, Mikhail A.
2 Potra, Florian-Alexandru
...and 238 more Authors
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Cited in 88 Serials

45 Journal of Global Optimization
35 Reliable Computing
14 Journal of Computational and Applied Mathematics
10 Numerical Algorithms
8 Applied Mathematics and Computation
8 European Journal of Operational Research
6 Fuzzy Sets and Systems
5 SIAM Journal on Numerical Analysis
5 Mathematical Programming. Series A. Series B
5 Optimization Methods & Software
4 Computing
4 Information Sciences
4 SIAM Journal on Scientific Computing
3 Journal of Optimization Theory and Applications
3 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods
3 Numerische Mathematik
3 Journal of Complexity
3 Computers & Operations Research
3 Computational and Applied Mathematics
3 Soft Computing
2 Artificial Intelligence
2 Automatica
2 Opsearch
2 Japan Journal of Industrial and Applied Mathematics
2 Linear Algebra and its Applications
2 International Journal of Robust and Nonlinear Control
2 SIAM Journal on Optimization
2 Cybernetics and Systems Analysis
2 Constraints
2 International Journal of Applied Mathematics and Computer Science
2 Optimization Letters
2 International Journal of Applied and Computational Mathematics
1 Bulletin of the Australian Mathematical Society
1 Computers & Mathematics with Applications
1 IMA Journal of Numerical Analysis
1 Mathematical Biosciences
1 Mathematics of Computation
1 ACM Transactions on Mathematical Software
1 Journal of Combinatorial Theory. Series A
1 Mathematics and Computers in Simulation
1 Theoretical Computer Science
1 Applied Numerical Mathematics
1 Journal of Symbolic Computation
1 Discrete & Computational Geometry
1 International Journal of Approximate Reasoning
1 Journal of the American Mathematical Society
1 SIAM Journal on Matrix Analysis and Applications
1 Formal Aspects of Computing
1 Neural Networks
1 Annals of Operations Research
1 Computational Statistics
1 Applied Mathematical Modelling
1 International Journal of Computer Mathematics
1 SIAM Review
1 Celestial Mechanics and Dynamical Astronomy
1 Computational Optimization and Applications
1 Mathematical Logic Quarterly (MLQ)
1 Journal of the Egyptian Mathematical Society
1 Turkish Journal of Mathematics
1 ETNA. Electronic Transactions on Numerical Analysis
1 Top
1 Notes on Intuitionistic Fuzzy Sets
1 Advances in Computational Mathematics
1 Geombinatorics
1 Journal of Geodesy
1 Abstract and Applied Analysis
1 ZAMM. 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 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
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Cited in 39 Fields

182 Numerical analysis (65-XX)
166 Operations research, mathematical programming (90-XX)
26 Computer science (68-XX)
16 Systems theory; control (93-XX)
11 Linear and multilinear algebra; matrix theory (15-XX)
9 Ordinary differential equations (34-XX)
9 Statistics (62-XX)
8 Real functions (26-XX)
8 Biology and other natural sciences (92-XX)
6 Functions of a complex variable (30-XX)
6 Calculus of variations and optimal control; optimization (49-XX)
5 Mathematical logic and foundations (03-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)
4 Mechanics of particles and systems (70-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)

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