PirateBay Download | sergeyev y numerical infinities infinitesimals opt 2022

sergeyev y numerical infinities infinitesimals opt 2022

Type:: E-books
Files:: 1
Size:: 6.4 MB
Uploaded On:: July 9, 2022, 11:06 a.m.
Added By:: andryold1
Seeders:: 1
Leechers:: 0
Info Hash:: C93AC4226F35BBFBD53B06F5586F6B9C25F2E22E

Textbook in PDF format This book provides a friendly introduction to the paradigm and proposes a broad panorama of killing applications of the Infinity Computer in optimization: radically new numerical algorithms, great theoretical insights, efficient software implementations, and interesting practical case studies. This is the first book presenting to the readers interested in optimization the advantages of a recently introduced supercomputing paradigm that allows to numerically work with different infinities and infinitesimals on the Infinity Computer patented in several countries. One of the editors of the book is the creator of the Infinity Computer, and another editor was the first who has started to use it in optimization. Their results were awarded by numerous scientific prizes. This engaging book opens new horizons for researchers, engineers, professors, and students with interests in supercomputing paradigms, optimization, decision making, game theory, and foundations of mathematics and computer science. “Mathematicians have never been comfortable handling infinities… But an entirely new type of mathematics looks set to by-pass the problem… Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem… ” MIT Technology Review “These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.” A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing. “I am sure that the new approach … will have a very deep impact both on Mathematics and Computer Science.” D. Trigiante, Computational Management Science. “Within the grossone framework, it becomes feasible to deal computationally with infinite quantities, in a way that is both new (in the sense that previously intractable problems become amenable to computation) and natural”. R. Gangle, G. Caterina, F. Tohmé, Soft Computing. “The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort.” P. Amodio, L. Brugnano, F. Iavernaro & F. Mazzia, Soft Computing Preface Contributors Theoretical Background A New Computational Paradigm Using Grossone-Based Numerical Infinities and Infinitesimals Introduction Numeral Systems Used to Express Finite and Infinite Quantities Three Methodological Postulates A New Way of Counting and the Infinite Unit of Measure ① Positional Numeral System with the Infinite Base ① Some Paradoxes of Infinity Related to Divergent Series Conclusion References Nonlinear Optimization: A Brief Overview Introduction Convex Sets and Functions Unconstrained Optimization Necessary and Sufficient Optimality Conditions Algoritms for Unconstrained Optimization The Gradient Method The Newton's Method The Conjugate Gradient Method Quasi-Newton's Methods Constrained Optimization Necessary and Sufficient Optimality Conditions Duality in Constrained Optimization Penalty and Augmented Lagrangian Methods Sequential Quadratic Programming References New Computational Tools in Optimization The Role of grossone in Nonlinear Programming and Exact Penalty Methods Introduction Exact Penalty Methods Equality Constraints Case Equality and Inequality Constraints Case Quadratic Case A General Scheme for the New Exact Penalty Function Conclusions References Krylov-Subspace Methods for Quadratic Hypersurfaces: A Grossone–based Perspective Introduction The CG Method and the Lanczos Process for Matrix Tridiagonalization Basics on the Lanczos Process How the CG and the Lanczos Process Compare: A Path to Degeneracy Coupling the CG with Grossone: A Marriage of Interest The Geometry Behind CG Degeneracy A New Perspective for CG Degeneracy Using Grossone Grossone for the Degenerate Step k of the CG Large Scale (unconstrained) Optimization Problems: The Need of Negative Curvatures A Theoretical Path to the Assessment of Negative Curvature Directions CG① for the Computation of Negative Curvature Directions Conclusions References Multi-objective Lexicographic Mixed-Integer Linear Programming: An Infinity Computer Approach Introduction Lexicographic Multi-objective Linear Programming The Preemptive Scheme The Nonpreemptive Scheme Based on Appropriate Finite Weights Grossone-Based Reformulation of the LMOLP Problem The GrossSimplex Algorithm Lexicographic Multi-objective Mixed-Integer Linear Programming A Grossone-Based Extension of the Branch-and-Bound Algorithm Pruning Rules for the GrossBB Terminating Conditions, Branching Rules and the GrossBB Algorithm Experimental Results Test Problem 1: The ``Kite'' in 2D Test Problem 2: The Unrotated ``House'' in 3D Test Problem 3: the Rotated ``House'' in 5D Conclusions References The Use of Infinities and Infinitesimals for Sparse Classification Problems Introduction The l0 Pseudo-norm in Optimization Problems Some Approximations of the l0 Pseudo-norm Using ① Applications in Regularization and Classification Problems Elastic Net Regularization Sparse Support Vector Machines Conclusions References The Grossone-Based Diagonal Bundle Method Introduction Grossone-Based Matrix Updates in a Diagonal Bundle Algorithm Computational Experience References On the Use of Grossone Methodology for Handling Priorities in Multi-objective Evolutionary Optimization Introduction Multi-objective Optimization and Evolutionary Algorithms Metrics to Assess the Efficacy of EMO Algorithms Mixed Pareto-Lexicographic Optimization The Priority Chains Model Need of a New Approach for PC-MPL-MOPs Grossone for PC-MPL-MOPs PC-NSGA-II and PC-MOEA/D Test Cases for PC-MPL MOPs Test Problem: PC-1 Test Problem: PC-3 The Priority Levels Model A First Example of a Real-World PL-MPL-MO Problem Need of a New Approach for PL-MPL-MOPs Enhancing PL-MPL Problems by Means of Grossone Handling Precedence in PL-MPL-MOPs A New Definition of Dominance: PL-Dominance PL-Dominance in PL-Crash Problem Algorithm for PL: PL-NSGA-II PL Fast Non-dominated Sort and PL Crowding Distance PL-NSGA-II Test Cases for PL-MPL-MOPs Test Problem: PL-1 PL-GAA: A Real-World Problem Test Problem: PL-Crash Conclusions References Applications and Implementations Exact Numerical Differentiation on the Infinity Computer and Applications in Global Optimization Introduction Numerical Differentiation on the Infinity Computer Application in Lipschitz Global Optimization Local Tuning Techniques General scheme and convergence conditions Numerical experiments References Comparing Linear and Spherical Separation Using Grossone-Based Numerical Infinities in Classification Problems Introduction Linear and Spherical Separability for Supervised Classification Linear Separation Spherical Separation Comparing Linear and Spherical Separation in the Grossone Framework Linear and Spherical Separability for Multiple Instance Learning The SVM Type Model for MIL A Grossone MIL Spherical Model Some Numerical Results Conclusions References Computing Optimal Decision Strategies Using the Infinity Computer: The Case of Non-Archimedean Zero-Sum Games Introduction Zero-Sum Games Non-Archimedean Zero-Sum Games and the Gross-Matrix-Simplex Algorithm Numerical Illustrations Experiment 1: Infinitesimally Perturbed Rock-paper-scissors Experiment 2: A Purely Finite 4-by-3 Game Experiment 3: Infinitesimally Perturbed 4-by-3 Game Experiment 4: High Dimensional Games A Brief Overview of Applications Conclusions References Modeling Infinite Games on Finite Graphs Using Numerical Infinities Introduction The Infinite Unit Axiom and Grossone Infinite Games Strategies Examples and Results An Application: Update Games and Networks Conclusion References Adopting the Infinity Computing in Simulink for Scientific Computing Introduction Background The Infinity Computing and Representation of Numbers The MATLAB/Simulink Environment The Simulink-Based Solution for the Infinity Computer (SSIC) Representation of Grossnumbers in SSIC Architecture Arithmetic Blocks Module Elementary Blocks Module Utility Blocks Module Differentiation Blocks Module Assessment and Evaluation Differentiation of a Univariate Function Higher Order Differentiation of a Univariate Function Computation of the Lie Derivatives for ODEs Conclusion References Addressing Ill-Conditioning in Global Optimization Using a Software Implementation of the Infinity Computer Introduction Problem Statement and Ill-Conditioning Induced by Scaling Lipschitz Global Optimization and Ill-Conditioning Produced by Scaling Functions with Infinite and Infinitesimal Lipschitz Constants A General Scheme Describing Geometric and Information Algorithms Strong Homogeneity of Algorithms Belonging to General Scheme with Finite, Infinite, and Infinitesimal Scaling and Shifting Constants Numerical Illustrations Concluding Remarks References

Get This Torrent

Sergeyev Y. Numerical Infinities and Infinitesimals in Opt. 2022.pdf 6.4 MB

Details for: Sergeyev Y. Numerical Infinities and Infinitesimals in Opt. 2022

sergeyev y numerical infinities infinitesimals opt 2022