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.

References in zbMATH (referenced in 306 articles , 1 standard article )

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  1. Carrizosa, Emilio; Messine, Frédéric: An interval branch and bound method for global robust optimization (2021)
  2. Reyes, Victor; Araya, Ignacio: \textscAbsTaylor: upper bounding with inner regions in nonlinear continuous global optimization problems (2021)
  3. Marendet, Antoine; Goldsztejn, Alexandre; Chabert, Gilles; Jermann, Christophe: A standard branch-and-bound approach for nonlinear semi-infinite problems (2020)
  4. Popova, Evgenija D.: New parameterized solution with application to bounding secondary variables in FE models of structures (2020)
  5. Akram, Muhammad; Muhammad, Ghulam; Allahviranloo, Tofigh: Bipolar fuzzy linear system of equations (2019)
  6. Montanher, Tiago; Neumaier, Arnold; Markót, Mihály Csaba; Domes, Ferenc; Schichl, Hermann: Rigorous packing of unit squares into a circle (2019)
  7. Puig, Vicenç; de la Fuente, María Jesús; Armengol, Joaquim: FDI approach (2019)
  8. Semenov, V. Yu.; Semenova, E. V.: Method for localizing the zeros of analytic functions based on the Krawczyk operator (2019)
  9. Bhunia, Asoke Kumar; Biswas, Amiya; Shaikh, Ali Akbar: Extended nondominated sorting genetic algorithm (ENSGA-II) for multi-objective optimization problem in interval environment (2018)
  10. Everitt, Richard G.: Efficient importance sampling in low dimensions using affine arithmetic (2018)
  11. Galántai, A.: Always convergent methods for nonlinear equations of several variables (2018)
  12. Geiping, Jonas; Moeller, Michael: Composite optimization by nonconvex majorization-minimization (2018)
  13. Arjmandzadeh, Ziba; Safi, Mohammadreza: A goal programming approach for solving the random interval linear programming problem (2017)
  14. Arjmandzadeh, Ziba; Safi, Mohammadreza; Nazemi, Alireza: A new neural network model for solving random interval linear programming problems (2017)
  15. Fendl, Hannes; Neumaier, Arnold; Schichl, Hermann: Certificates of infeasibility via nonsmooth optimization (2017)
  16. Fernández, José; Tóth, Boglárka G.; Redondo, Juana L.; Ortigosa, Pilar M.; Arrondo, Aránzazu Gila: A planar single-facility competitive location and design problem under the multi-deterministic choice rule (2017)
  17. Imbach, Rémi; Moroz, Guillaume; Pouget, Marc: A certified numerical algorithm for the topology of resultant and discriminant curves (2017)
  18. Martin, Benjamin; Correia, Marco; Cruz, Jorge: A certified branch & bound approach for reliability-based optimization problems (2017)
  19. Martin, Benjamin; Goldsztejn, Alexandre; Granvilliers, Laurent; Jermann, Christophe: Constraint propagation using dominance in interval branch & bound for nonlinear biobjective optimization (2017)
  20. Montanher, Tiago; Domes, Ferenc; Schichl, Hermann; Neumaier, Arnold: Using interval unions to solve linear systems of equations with uncertainties (2017)

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