Optimization Books

Book SynopsisGames and elections are fundamental activities in society with applications in economics, political science, and sociology. These topics offer familiar, current, and lively subjects for a course in mathematics. This classroom-tested textbook, primarily intended for a general education course in game theory at the freshman or sophomore level, provides an elementary treatment of games and elections. Starting with basics such as gambling, zero-sum and combinatorial games, Nash equilibria, social dilemmas, and fairness and impossibility theorems for elections, the text then goes further into the theory with accessible proofs of advanced topics such as the SpragueâGrundy theorem and Arrow's impossibility theorem. â Uses an integrative approach to probability, game, and social choice theory â Provides a gentle introduction to the logic of mathematical proof, thus equipping readers with the necessary tools for further mathematical studies â Contains numerous exercises and examples of varying Trade Review'Sam Smith's book offers an intriguing juxtaposition of chance, strategy, and elections. The mathematical analysis is rigorous without being too formal or forbidding. The applications to topics in economics and political science - including auctions, power, and voting - as well as to parlor games like poker will engage both students and professionals.' Steven Brams, New York University'I like the logical flow and length of the chapters and I like that the layout is simple (no excessively boxed theorems, etc.). There are numerous chapters, with one key concept explained in each. One could envision that each chapter would roughly be covered in a class period.' John Cullinan, Bard College, New York'The author's approach does seem as if it would appeal to a broad range of instructors and students: there are enough chapters that an instructor could choose a collection of topics according to his or her interest. Furthermore, the inclusion of proof-based sections would allow an instructor to use the text for a course targeted at math majors and minors rather than at a general nontechnical audience.' James Parson, Hood College, Maryland'The book is well written and interesting. Students should have little difficulty reading and understanding this book … The book covers the topics with clarity and applies game theory to 'real-world' problems.' Dan Cunningham, State University of New York, Buffalo'While some of Smith's material has origins more than 100 years old, the author engages the reader through modern developments, such as the minimax theorem (1928), the work of John Nash and Kenneth Arrow (1950s) and even more recent developments by Steven Brams, William Zwicker and Alan Taylor (1980s–2000s). The author does an effective job of presenting this material to an audience of non-science majors with no prerequisites. A unique feature of the text is the treatment of combinatorial games such as Nim and Hackenbush alongside traditional two person game theory.' David Vella, Skidmore College, New York'Chance, Strategy, and Choice fits an important niche for general audience textbooks about games, elections, and other introductory material related to social choice theory … One of my favorite features of the book is that it does an excellent job of integrating the topics of games and elections to illustrate the interconnections between the different areas of social choice theory, often through illustrative examples.' Adam Graham-Squire, MAA ReviewsTable of Contents1. Introduction; 2. Games and elections; 3. Chance; 4. Strategy; 5. Choice; 6. Strategy and choice; 7. Choice and chance; 8. Chance and strategy; 9. Nash equilibria; 10. Proofs and counterexamples; 11. Laws of probability; 12. Fairness in elections; 13. Weighted voting; 14. Gambling games; 15. Zero-sum games; 16. Partial conflict games; 17. Take-away games; 18. Fairness and impossibility; 19. Paradoxes and puzzles in probability; 20. Combinatorial games; 21. Borda versus Condorcet; 22. The Sprague–Grundy theorem; 23. Arrow's impossibility theorem.

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Book SynopsisDiscover the subject of optimization in a new light with this modern and unique treatment. Includes a thorough exposition of applications and algorithms in sufficient detail for practical use, while providing you with all the necessary background in a self-contained manner. Features a deeper consideration of optimal control, global optimization, optimization under uncertainty, multiobjective optimization, mixed-integer programming and model predictive control. Presents a complete coverage of formulations and instances in modelling where optimization can be applied for quantitative decision-making. As a thorough grounding to the subject, covering everything from basic to advanced concepts and addressing real-life problems faced by modern industry, this is a perfect tool for advanced undergraduate and graduate courses in chemical and biochemical engineering.Trade Review'This book offers a very clear, uncluttered presentation of key ideas of optimisation in rigorous form and with plenty of examples from a decade of research and educational experience. It offers an exceptional resource for educators and students of optimisation methods, as well as a valuable reference text to practitioners.' Alexei Lapkin, University of Cambridge'This excellent book brings together important and up-to-date elements of the theory and practice of optimisation with application to chemical and biochemical engineering. It's an ideal reference for students on advanced courses or for researchers in the field.' Nilay Shah, Imperial CollegeTable of ContentsPart I. Overview of Optimization: 1. Introduction to optimization; Part II. From General Mathematical Background to General Nonlinear Programming Problems (NLP): 2. General concepts; 3. Convexity; 4. Quadratic functions; 5. Minimization in one dimension; 6. Unconstrained multivariate gradient-based minimization; 7. Constrained nonlinear programming problems (NLP); 8. Penalty and barrier function methods; 9. Interior point methods (IPMs), a detailed analysis; Part III. Formulation and Solution of Linear Programming (LP) Problem Models: 10. Introduction to LP models; 11. Numerical solution of LP problems using the simplex method; 12. A sampler of LP problem formulations; 13. Regression revisited, using LP to fit linear models; 14. Network flow problems; 15, LP and sensitivity analysis, in brief; Part IV. Further Topics in Optimization: 16. Multiobjective optimilzation problem (MOP); 17. Stochastic optimization problem (SOP); 18. Mixed integer programming; 19. Global optimization; 20. Optical control problems (dynamic optimization); 21. System identification and model predictive control.

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Book SynopsisOptimization is an essential technique for solving problems in areas as diverse as accounting, computer science and engineering. Assuming only basic linear algebra and with a clear focus on the fundamental concepts, this textbook is the perfect starting point for first- and second-year undergraduate students from a wide range of backgrounds and with varying levels of ability. Modern, real-world examples motivate the theory throughout. The authors keep the text as concise and focused as possible, with more advanced material treated separately or in starred exercises. Chapters are self-contained so that instructors and students can adapt the material to suit their own needs and a wide selection of over 140 exercises gives readers the opportunity to try out the skills they gain in each section. Solutions are available for instructors. The book also provides suggestions for further reading to help students take the next step to more advanced material.Table of ContentsPreface; 1. Introduction; 2. Solving linear programs; 3. Duality through examples; 4. Duality theory; 5. Applications of duality; 6. Solving integer programs; 7. Nonlinear optimization; Appendix A. Computational complexity; References; Index.

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Book SynopsisInteger linear programming (ILP) is a versatile modeling and optimization technique that is increasingly used in non-traditional ways in biology, with the potential to transform biological computation. However, few biologists know about it. This how-to and why-do text introduces ILP through the lens of computational and systems biology. It uses in-depth examples from genomics, phylogenetics, RNA, protein folding, network analysis, cancer, ecology, co-evolution, DNA sequencing, sequence analysis, pedigree and sibling inference, haplotyping, and more, to establish the power of ILP. This book aims to teach the logic of modeling and solving problems with ILP, and to teach the practical ''work flow'' involved in using ILP in biology. Written for a wide audience, with no biological or computational prerequisites, this book is appropriate for entry-level and advanced courses aimed at biological and computational students, and as a source for specialists. Numerous exercises and accompanying soTrade Review'In his classic accessible teaching style, Gusfield teaches us why integer linear programming (ILP) is the most useful mathematical idea you've probably never heard of. Read this book to learn how what you don't know can hurt you, and why ILP should be your new favorite method.' Trey Ideker, University of California, San Diego'Once again, Dan Gusfield has written an accessible book that shows that algorithmic rigor need not be sacrificed when solving real-world problems. He explains integer linear programming in the context of real-world biology. In doing so, the reader has an enriched understanding of both algorithmic details and the challenges in modern biology.' Russ Altman, Stanford University, CaliforniaTable of ContentsPreface; Part I: 1. A fly-over introduction; 2. Biological networks and graphs; 3. Character compatibility; 4. Near-cliques; 5. Parsimony in phylogenetics; 6. RNA folding; 7. Protein problems; 8. Tanglegrams; 9. TSP in genomics; 10. Molecular sequence analysis; 11. Metabolic networks and engineering; 12. ILP idioms; Part II: 13. Communities and cuts; 14. Corrupted data and extensions in phylogenetics; 15. More tanglegrams and trees; 16. Return to Steiner-trees; 17. Exploiting protein networks; 18. More strings and sequences; 19. Max-likelihood pedigrees; 20. Haplotyping; 21. Extended exercises; 22. What's next?; Epilogue: opinionated comments.

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Book SynopsisIn the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.Trade Review'The field of mathematical programming has two major themes: linear programming and convex programming. The far-reaching impact of the first theory in computer science, game theory and engineering is well known. We are now witnessing the growth of the second theory as it finds its way into diverse fields such as machine learning, mathematical economics and quantum computing. This much-awaited book with its unique approach, steeped in the modern theory of algorithms, will go a long way in making this happen.' Vijay V. Vazirani, Distinguished Professor at University of California, Irvine'I had thought that there is no need for new books about convex optimization but this book proves me wrong. It treats both classic and cutting-edge topics with an unparalleled mix of clarity and rigor, building intuitions about key ideas and algorithms driving the field. A must read for anyone interested in optimization!' Aleksander Madry, Massachusetts Institute of Technology'Vishnoi's book provides an exceptionally good introduction to convex optimization for students and researchers in computer science, operations research, and discrete optimization. The book gives a comprehensive introduction to classical results as well as to some of the most recent developments. Concepts and ideas are introduced from first principles, conveying helpful intuitions. There is significant emphasis on bridging continuous and discrete optimization, in particular, on recent breakthroughs on flow problems using convex optimization methods; the book starts with an enlightening overview of the interplay between these areas.' László Végh, LSE'Recommended.' M. Bona, Choice ConnectTable of Contents1. Bridging continuous and discrete optimization; 2. Preliminaries; 3. Convexity; 4. Convex optimization and efficiency; 5. Duality and optimality; 6. Gradient descent; 7. Mirror descent and multiplicative weights update; 8. Accelerated gradient descent; 9. Newton's method; 10. An interior point method for linear programming; 11. Variants of the interior point method and self-concordance; 12. Ellipsoid method for linear programming; 13. Ellipsoid method for convex optimization.

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Book SynopsisA unique text integrating numerics, mathematics and applications to provide a hands-on approach to using optimization techniques, this mathematically accessible textbook emphasises conceptual understanding and importance of theorems rather than elaborate proofs. It allows students to develop fundamental optimization methods before delving into MATLAB''s optimization toolbox, and to link MATLAB''s results with the results from their own code. Following a practical approach, the text demonstrates several applications, from error-free analytic examples to truss (size) optimization, and 2D and 3D shape optimization, where numerical errors are inevitable. The principle of minimum potential energy is discussed to highlight the deep relationship between engineering and optimization. MATLAB code in every chapter illustrates key concepts and the text demonstrates the coupling between MATLAB and SOLIDWORKS for design optimization. A wide variety of optimization problems are covered including conTrade Review'Design Optimization using MATLAB and SOLIDWORKS by Dr. Suresh provides an excellent review of various optimization methods, especially for structural problems. Its introduction to MATLAB would help students who have had little experience with this software to become familiar with it quickly and apply it to some of the basic optimization problems.' Hamid Torab, Gannon University'Dr. Suresh's text brings his contributions to shape optimization into the classroom by connecting optimization, MATLAB, SOLIDWORKS, and SOLIDLAB into a single textbook. This text enables the reader to build upon this research accomplishment. I look forward to seeing what my students can achieve with this textbook at their fingertips.' Cameron Turner, Clemson UniversityTable of ContentsPreface; Table of Contents; 1. Introduction; 2. Modeling; 3. Introduction to MATLAB; 4. Unconstrained Optimization: Theory; 5. Unconstrained Optimization: Algorithms; 6. MATLAB Optimization Toolbox; 7. Constrained Optimization; 8. Special Classes of Problems; 9. Truss Analysis; 10. Size Optimization of Trusses; 11. Gradient Computation; 12. Finite Element Analysis in 2D; 13. Shape Optimization in 2D; 14. Finite Element Analysis in 3D; 15. SOLIDLAB: A SOLIDWORKS-MATLAB Interface; 16. Shape Optimization using SOLIDLAB; 17. Appendix; 18. References.

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Book SynopsisUsing a step-by-step approach, this textbook provides a modern treatment of the fundamental concepts, analytical techniques, and software tools used to perform multi-domain modeling, system analysis and simulation, linear control system design and implementation, and advanced control engineering. Chapters follow a progressive structure, which builds from modeling fundamentals to analysis and advanced control while showing the interconnections between topics, and solved problems and examples are included throughout. Students can easily recall key topics and test understanding using Review Note and Concept Quiz boxes, and over 200 end-of-chapter homework exercises with accompanying Concept Keys are included. Focusing on practical understanding, students will gain hands-on experience of many modern MATLAB tools, including Simulink and physical modeling in Simscape. With a solutions manual, MATLAB code, and Simulink/Simscape files available online, this is ideal for senior undergraduates Trade Review'Lucid and easy to read. It methodically explains classic control theory, from modeling of multi-domain systems to digital control. The detailed examples and end-of-chapter problems make it an excellent choice as a textbook for students in different engineering and science disciplines. MATLAB® and Simulink® instructions are a big plus.' Pezhman Hassanpour, California State Polytechnic University'Dynamic Systems and Control Engineering by Jalili and Candelino is one of the most organized and easily understood basic texts in this area. They have taken what is a nebulous subject for many students and made it less daunting through their use of numerous examples across several disciplines. The text is laid out well, logical from the basic systems modeling, to their analyses, and their control. They have taken a progressive approach to build on previous knowledge as the topics become more advanced. Their straightforward mathematical models are reinforced through MATLAB® and Simulink®, with basic user guides for the software. This allows the student to learn about controls by doing controls.' Robert Rabb, Penn State University'I have enjoyed reading this book very much for several reasons. This is the most complete text on dynamic systems and automatic control, with a rich set of examples and deep analytic treatment, along with an application viewpoint. The authors have rich experience in research and teaching in dynamic systems and control engineering in several world-class universities worldwide.' Reza N. Jazar, Royal Melbourne Institute of Technology UniversityTable of ContentsPart I. Modeling of Multi-Domain Dynamic Systems: 1. Introduction to Dynamic Systems; 2. Modeling of Mechanical Systems; 3. Modeling of Electrical Systems; 4. Modeling of Multi-Domain Systems; Part II. Analysis of Multi-Domain Dynamic Systems: 5. Dynamic System Response; 6. System Response Characteristics; 7. System Transfer Function Analysis; Part III. Introduction to Feedback Systems: 8. Analysis of Feedback Control Systems; 9. Root Locus Techniques; 10. Frequency Domain Methods; 11. Implementation of Feedback Control Systems; Part IV. Analysis and Feedback Control of Modern Systems: 12. State-Space Representation and Analysis; 13. State-Space Control System Design; 14. Advanced Topics in Control Engineering.

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Book SynopsisBased on the author''s forty years of teaching experience, this unique textbook covers both basic and advanced concepts of optimization theory and methods for process systems engineers. Topics covered include continuous, discrete and logic optimization (linear, nonlinear, mixed-integer and generalized disjunctive programming), optimization under uncertainty (stochastic programming and flexibility analysis), and decomposition techniques (Lagrangean and Benders decomposition). Assuming only a basic background in calculus and linear algebra, it enables easy understanding of mathematical reasoning, and numerous examples throughout illustrate key concepts and algorithms. End-of-chapter exercises involving theoretical derivations and small numerical problems, as well as in modeling systems like GAMS, enhance understanding and help put knowledge into practice. Accompanied by two appendices containing web links to modeling systems and models related to applications in PSE, this is an essentialTrade Review'Authored by Ignacio Grossmann, the creator and key developer of the field of mixed integer nonlinear programming, this outstanding textbook provides a thorough and comprehensive treatment of fundamental concepts, optimization models and effective solution strategies for discrete and continuous optimization. It is an essential, 'must-have' reference for all students, researchers and practitioners in process systems engineering.' Lorenz Biegler, Carnegie Mellon University'From the globally recognized leading authority in the field of process systems engineering, this long-awaited book will definitely become the standard reference for anyone interested in optimization. It is very well thought and written, with excellent presentation of the material. The theory is described in a very effective, rigorous, and clear way, with appropriate explanations and examples used throughout, covering traditional topics such as linear and nonlinear optimization concepts and mixed-integer linear programming, along with more advanced topics, such as disjunctive programming, global optimization, and stochastic programming. A real gem and a must read!' Stratos Pistikopoulos, Texas A & M University'Excellent coverage of the basic concepts and approaches developed in the area of process systems engineering in the last forty years. A unique book that can be easily adapted to advanced undergraduate and graduate-level classes to provide overall guidance to different tools that can be used to model and optimize complex engineering problems. I am certainly looking forward to using it in my class on mathematical modeling and optimization principles.' Marianthi Ierapetritou, University of DelawareTable of ContentsPreface; 1. Optimization in process systems engineering; 2. Solving nonlinear equations; 3. Basic theoretical concepts in optimization; 4. Nonlinear programming algorithms; 5. Linear programming; 6. Mixed-integer programming models; 7. Systematic modeling of constraints with logic; 8. Mixed-integer linear programming; 9 Mixed-integer nonlinear programming; 10. Generalized disjunctive programming; 11. Constraint programming; 12. Nonconvex optimization; 13. Lagrangean decomposition; 14. Stochastic programming; 15. Flexibility analysis; Appendix A. Modeling systems and optimization software; Appendix B. Optimization models for process systems engineering; References; Index.

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Book SynopsisUsing a pedagogical, unified approach, this book presents both the analytic and combinatorial aspects of convexity and its applications in optimization. On the structural side, this is done via an exposition of classical convex analysis and geometry, along with polyhedral theory and geometry of numbers. On the algorithmic/optimization side, this is done by the first ever exposition of the theory of general mixed-integer convex optimization in a textbook setting. Classical continuous convex optimization and pure integer convex optimization are presented as special cases, without compromising on the depth of either of these areas. For this purpose, several new developments from the past decade are presented for the first time outside technical research articles: discrete Helly numbers, new insights into sublinear functions, and best known bounds on the information and algorithmic complexity of mixed-integer convex optimization. Pedagogical explanations and more than 300 exercises make this book ideal for students and researchers.

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Book SynopsisA UNIQUE ENGINEERING AND STATISTICAL APPROACH TO OPTIMAL RESOURCE ALLOCATION Optimal Resource Allocation: With Practical Statistical Applications and Theory features the application of probabilistic and statistical methods used in reliability engineering during the different phases of life cycles of technical systems. Bridging the gap between reliability engineering and applied mathematics, the book outlines different approaches to optimal resource allocation and various applications of models and algorithms for solving real-world problems. In addition, the fundamental background on optimization theory and various illustrative numerical examples are provided. The book also features: An overview of various approaches to optimal resource allocation, from classical Lagrange methods to modern algorithms based on ideas of evolution in biology Numerous exercises and case studies from a variety of areas, including communications, transportation, eTable of ContentsPreface xi 1 BASIC MATHEMATICAL REDUNDANCY MODELS 1 1.1 Types of Models 2 1.2 Non-repairable Redundant Group with Active Redundant Units 3 1.3 Non-repairable Redundant Group with Standby Redundant Units 7 1.4 Repairable Redundant Group with Active Redundant Units 10 1.5 Repairable Redundant Group with Standby Redundant Units 13 1.6 Multi-level Systems and System Performance Estimation 15 1.7 Brief Review of Other Types of Redundancy 16 1.8 Time Redundancy 24 1.9 Some Additional Optimization Problems 27 Chronological Bibliography of Main Monographs on Reliability Theory (with topics on Optimization) 30 2 FORMULATION OF OPTIMAL REDUNDANCY PROBLEMS 33 2.1 Problem Description 33 2.2 Formulation of the Optimal Redundancy Problem with a Single Restriction 35 2.3 Formulation of Optimal Redundancy Problems with Multiple Constraints 39 2.4 Formulation of Multi-Criteria Optimal Redundancy Problems 43 Chronological Bibliography 45 3 METHOD OF LAGRANGE MULTIPLIERS 48 Chronological Bibliography 55 4 STEEPEST DESCENT METHOD 56 4.1 The Main Idea of SDM 56 4.2 Description of the Algorithm 57 4.3 The Stopping Rule 60 4.5 Approximate Solution 66 Chronological Bibliography 68 5 DYNAMIC PROGRAMMING 69 5.1 Bellman’s Algorithm 69 5.2 Kettelle’s Algorithm 73 Chronological Bibliography 84 6 UNIVERSAL GENERATING FUNCTIONS 85 6.1 Generating Function 85 6.2 Universal GF (U-function) 87 Chronological Bibliography 94 7 GENETIC ALGORITHMS 96 7.1 Introduction 96 7.2 Structure of Steady-State Genetic Algorithms 100 7.3 Related Techniques 102 Chronological Bibliography 104 8 MONTE CARLO SIMULATION 107 8.1 Introductory Remarks 107 8.2 Formulation of Optimal Redundancy Problems in Statistical Terms 108 8.3 Algorithm for Trajectory Generation 108 8.4 Description of the Idea of the Solution 111 8.5 Inverse Optimization Problem 114 8.6 Direct Optimization Problem 124 Chronological Bibliography 129 9 COMMENTS ON CALCULATION METHODS 130 9.1 Comparison of Methods 130 9.2 Sensitivity Analysis of Optimal Redundancy Solutions 135 10 OPTIMAL REDUNDANCY WITH SEVERAL LIMITING FACTORS 142 10.1 Method of “Weighing Costs” 142 10.2 Method of Generalized Generating Functions 146 Chronological Bibliography 149 11 OPTIMAL REDUNDANCY IN MULTISTATE SYSTEMS 150 Chronological Bibliography 170 12 CASE STUDIES 172 12.1 Spare Supply System for Worldwide Telecommunication System Globalstar 172 12.2 Optimal Capacity Distribution of Telecommunication Backbone Network Resources 179 12.3 Optimal Spare Allocation for Mobile Repair Station 183 Chronological Bibliography 190 13 COUNTER-TERRORISM: PROTECTION RESOURCES ALLOCATION 191 13.1 Introduction 191 13.2 Written Description of the Problem 192 13.3 Evaluation of Expected Loss 195 13.4 Algorithm of Resource Allocation 197 13.5 Branching System Protection 201 13.6 Fictional Case Study 210 13.7 Measures of Defense, Their Effectiveness, and Related Expenses 217 13.8 Antiterrorism Resource Allocation under Fuzzy Subjective Estimates 223 13.9 Conclusion 232 Chronological Bibliography 232 About the author 235

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Book SynopsisAn Application-Oriented Introduction to Essential Optimization Concepts and Best Practices Optimization is an inherent human tendency that gained new life after the advent of calculus; now, as the world grows increasingly reliant on complex systems, optimization has become both more important and more challenging than ever before. Engineering Optimization provides a practically-focused introduction to modern engineering optimization best practices, covering fundamental analytical and numerical techniques throughout each stage of the optimization process. Although essential algorithms are explained in detail, the focus lies more in the human function: how to create an appropriate objective function, choose decision variables, identify and incorporate constraints, define convergence, and other critical issues that define the success or failure of an optimization project. Examples, exercises, and homework throughout reinforce the author's do, not studTable of ContentsContents Preface xix Acknowledgments xxvii Nomenclature xxix About the Companion Website xxxvii Section 1 Introductory Concepts 1 1 Optimization: Introduction and Concepts 3 1.1 Optimization and Terminology 3 1.2 Optimization Concepts and Definitions 4 1.3 Examples 6 1.4 Terminology Continued 10 1.4.1 Constraint 10 1.4.2 Feasible Solutions 10 1.4.3 Minimize or Maximize 11 1.4.4 Canonical Form of the Optimization Statement 11 1.5 Optimization Procedure 12 1.6 Issues That Shape Optimization Procedures 16 1.7 Opposing Trends 17 1.8 Uncertainty 20 1.9 Over- and Under-specification in Linear Equations 21 1.10 Over- and Under-specification in Optimization 22 1.11 Test Functions 23 1.12 Significant Dates in Optimization 23 1.13 Iterative Procedures 26 1.14 Takeaway 27 1.15 Exercises 27 2 Optimization Application Diversity and Complexity 33 2.1 Optimization 33 2.2 Nonlinearity 33 2.3 Min, Max, Min–Max, Max–Min, … 34 2.4 Integers and Other Discretization 35 2.5 Conditionals and Discontinuities: Cliffs Ridges/Valleys 36 2.6 Procedures, Not Equations 37 2.7 Static and Dynamic Models 38 2.8 Path Integrals 38 2.9 Economic Optimization and Other Nonadditive Cost Functions 38 2.10 Reliability 39 2.11 Regression 40 2.12 Deterministic and Stochastic 42 2.13 Experimental w.r.t. Modeled OF 43 2.14 Single and Multiple Optima 44 2.15 Saddle Points 45 2.16 Inflections 46 2.17 Continuum and Discontinuous DVs 47 2.18 Continuum and Discontinuous Models 47 2.19 Constraints and Penalty Functions 48 2.20 Ranks and Categorization: Discontinuous OFs 50 2.21 Underspecified OFs 51 2.22 Takeaway 51 2.23 Exercises 51 3 Validation: Knowing That the Answer Is Right 53 3.1 Introduction 53 3.2 Validation 53 3.3 Advice on Becoming Proficient 55 3.4 Takeaway 56 3.5 Exercises 57 Section 2 Univariate Search Techniques 59 4 Univariate (Single DV) Search Techniques 61 4.1 Univariate (Single DV) 61 4.2 Analytical Method of Optimization 62 4.2.1 Issues with the Analytical Approach 63 4.3 Numerical Iterative Procedures 64 4.3.1 Newton’s Methods 64 4.3.2 Successive Quadratic (A Surrogate Model or Approximating Model Method) 68 4.4 Direct Search Approaches 70 4.4.1 Bisection Method 70 4.4.2 Golden Section Method 72 4.4.3 Perspective at This Point 74 4.4.4 Heuristic Direct Search 74 4.4.5 Leapfrogging 76 4.4.6 LF for Stochastic Functions 79 4.5 Perspectives on Univariate Search Methods 82 4.6 Evaluating Optimizers 85 4.7 Summary of Techniques 85 4.7.1 Analytical Method 86 4.7.2 Newton’s (and Variants Like Secant) 86 4.7.3 Successive Quadratic 86 4.7.4 Golden Section Method 86 4.7.5 Heuristic Direct 87 4.7.6 Leapfrogging 87 4.8 Takeaway 87 4.9 Exercises 88 5 Path Analysis 93 5.1 Introduction 93 5.2 Path Examples 93 5.3 Perspective About Variables 96 5.4 Path Distance Integral 97 5.5 Accumulation along a Path 99 5.6 Slope along a Path 101 5.7 Parametric Path Notation 103 5.8 Takeaway 104 5.9 Exercises 104 6 Stopping and Convergence Criteria: 1-D Applications 107 6.1 Stopping versus Convergence Criteria 107 6.2 Determining Convergence 107 6.2.1 Threshold on the OF 108 6.2.2 Threshold on the Change in the OF 108 6.2.3 Threshold on the Change in the DV 108 6.2.4 Threshold on the Relative Change in the DV 109 6.2.5 Threshold on the Relative Change in the OF 109 6.2.6 Threshold on the Impact of the DV on the OF 109 6.2.7 Convergence Based on Uncertainty Caused by the Givens 109 6.2.8 Multiplayer Range 110 6.2.9 Steady-State Convergence 110 6.3 Combinations of Convergence Criteria 111 6.4 Choosing Convergence Threshold Values 112 6.5 Precision 112 6.6 Other Convergence Criteria 113 6.7 Stopping Criteria to End a Futile Search 113 6.7.1 N Iteration Threshold 114 6.7.2 Execution Error 114 6.7.3 Constraint Violation 114 6.8 Choices! 114 6.9 Takeaway 114 6.10 Exercises 115 Section 3 Multivariate Search Techniques 117 7 Multidimension Application Introduction and the Gradient 119 7.1 Introduction 119 7.2 Illustration of Surface and Terms 122 7.3 Some Surface Analysis 123 7.4 Parametric Notation 128 7.5 Extension to Higher Dimension 130 7.6 Takeaway 131 7.7 Exercises 131 8 Elementary Gradient-Based Optimizers: CSLS and ISD 135 8.1 Introduction 135 8.2 Cauchy’s Sequential Line Search 135 8.2.1 CSLS with Successive Quadratic 137 8.2.2 CSLS with Newton/Secant 138 8.2.3 CSLS with Golden Section 138 8.2.4 CSLS with Leapfrogging 138 8.2.5 CSLS with Heuristic Direct Search 139 8.2.6 CSLS Commentary 139 8.2.7 CSLS Pseudocode 140 8.2.8 VBA Code for a 2-DV Application 141 8.3 Incremental Steepest Descent 144 8.3.1 Pseudocode for the ISD Method 144 8.3.2 Enhanced ISD 145 8.3.3 ISD Code 148 8.4 Takeaway 149 8.5 Exercises 149 9 Second-Order Model-Based Optimizers: SQ and NR 155 9.1 Introduction 155 9.2 Successive Quadratic 155 9.2.1 Multivariable SQ 156 9.2.2 SQ Pseudocode 159 9.3 Newton–Raphson 159 9.3.1 NR Pseudocode 162 9.3.2 Attenuate NR 163 9.3.3 Quasi-Newton 166 9.4 Perspective on CSLS, ISD, SQ, and NR 168 9.5 Choosing Step Size for Numerical Estimate of Derivatives 169 9.6 Takeaway 170 9.7 Exercises 170 10 Gradient-Based Optimizer Solutions: LM, RLM, CG, BFGS, RG, and GRG 173 10.1 Introduction 173 10.2 Levenberg–Marquardt (LM) 173 10.2.1 LM VBA Code for a 2-DV Case 175 10.2.2 Modified LM (RLM) 176 10.2.3 RLM Pseudocode 177 10.2.4 RLM VBA Code for a 2-DV Case 178 10.3 Scaled Variables 180 10.4 Conjugate Gradient (CG) 182 10.5 Broyden–Fletcher–Goldfarb–Shanno (BFGS) 183 10.6 Generalized Reduced Gradient (GRG) 184 10.7 Takeaway 186 10.8 Exercises 186 11 Direct Search Techniques 187 11.1 Introduction 187 11.2 Cyclic Heuristic Direct (CHD) Search 188 11.2.1 CHD Pseudocode 188 11.2.2 CHD VBA Code 189 11.3 Hooke–Jeeves (HJ) 192 11.3.1 HJ Code in VBA 195 11.4 Compare and Contrast CHD and HJ Features: A Summary 197 11.5 Nelder–Mead (NM) Simplex: Spendley, Hext, and Himsworth 199 11.6 Multiplayer Direct Search Algorithms 200 11.7 Leapfrogging 201 11.7.1 Convergence Criteria 208 11.7.2 Stochastic Surfaces 209 11.7.3 Summary 209 11.8 Particle Swarm Optimization 209 11.8.1 Individual Particle Behavior 210 11.8.2 Particle Swarm 213 11.8.3 PSO Equation Analysis 215 11.9 Complex Method (CM) 216 11.10 A Brief Comparison 217 11.11 Takeaway 218 11.12 Exercises 219 12 Linear Programming 223 12.1 Introduction 223 12.2 Visual Representation and Concepts 225 12.3 Basic LP Procedure 228 12.4 Canonical LP Statement 228 12.5 LP Algorithm 229 12.6 Simplex Tableau 230 12.7 Takeaway 231 12.8 Exercises 231 13 Dynamic Programming 233 13.1 Introduction 233 13.2 Conditions 236 13.3 DP Concept 237 13.4 Some Calculation Tips 240 13.5 Takeaway 241 13.6 Exercises 241 14 Genetic Algorithms and Evolutionary Computation 243 14.1 Introduction 243 14.2 GA Procedures 243 14.3 Fitness of Selection 245 14.4 Takeaway 250 14.5 Exercises 250 15 Intuitive Optimization 253 15.1 Introduction 253 15.2 Levels 254 15.3 Takeaway 254 15.4 Exercises 254 16 Surface Analysis II 257 16.1 Introduction 257 16.2 Maximize Is Equivalent to Minimize the Negative 257 16.3 Scaling by a Positive Number Does Not Change DV∗ 258 16.4 Scaled and Translated OFs Do Not Change DV∗ 258 16.5 Monotonic Function Transformation Does Not Change DV∗ 258 16.6 Impact on Search Path or NOFE 261 16.7 Inequality Constraints 263 16.8 Transforming DVs 263 16.9 Takeaway 263 16.10 Exercises 263 17 Convergence Criteria 2: N-D Applications 265 17.1 Introduction 265 17.2 Defining an Iteration 265 17.3 Criteria for Single TS Deterministic Procedures 266 17.4 Criteria for Multiplayer Deterministic Procedures 267 17.5 Stochastic Applications 268 17.7 Takeaway 269 17.8 Exercises 269 18 Enhancements to Optimizers 271 18.1 Introduction 271 18.2 Criteria for Replicate Trials 271 18.3 Quasi-Newton 274 18.4 Coarse–Fine Sequence 275 18.5 Number of Players 275 18.6 Search Range Adjustment 276 18.7 Adjustment of Optimizer Coefficient Values or Options in Process 276 18.8 Initialization Range 277 18.9 OF and DV Transformations 277 18.10 Takeaway 278 18.11 Exercises 278 Section 4 Developing Your Application Statements 279 19 Scaled Variables and Dimensional Consistency 281 19.1 Introduction 281 19.2 A Scaled Variable Approach 283 19.3 Sampling of Issues with Primitive Variables 283 19.4 Linear Scaling Options 285 19.5 Nonlinear Scaling 286 19.6 Takeaway 287 19.7 Exercises 287 20 Economic Optimization 289 20.1 Introduction 289 20.2 Annual Cash Flow 290 20.3 Including Risk as an Annual Expense 291 20.4 Capital 293 20.5 Combining Capital and Nominal Annual Cash Flow 293 20.6 Combining Time Value and Schedule of Capital and Annual Cash Flow 296 20.7 Present Value 297 20.8 Including Uncertainty 298 20.8.1 Uncertainty Models 301 20.8.2 Methods to Include Uncertainty in an Optimization 303 20.9 Takeaway 304 20.10 Exercises 304 21 Multiple OF and Constraint Applications 305 21.1 Introduction 305 21.2 Solution 1: Additive Combinations of the Functions 306 21.2.1 Solution 1a: Classic Weighting Factors 307 21.2.2 Solution 1b: Equal Concern Weighting 307 21.2.3 Solution 1c: Nonlinear Weighting 309 21.3 Solution 2: Nonadditive OF Combinations 311 21.4 Solution 3: Pareto Optimal 311 21.5 Takeaway 316 21.6 Exercises 316 22 Constraints 319 22.1 Introduction 319 22.2 Equality Constraints 320 22.2.1 Explicit Equality Constraints 320 22.2.2 Implicit Equality Constraints 321 22.3 Inequality Constraints 321 22.3.1 Penalty Function: Discontinuous 323 22.3.2 Penalty Function: Soft Constraint 323 22.3.3 Inequality Constraints: Slack and Surplus Variables 325 22.4 Constraints: Pass/Fail Categories 329 22.5 Hard Constraints Can Block Progress 330 22.6 Advice 331 22.7 Constraint-Equivalent Features 332 22.8 Takeaway 332 22.9 Exercises 332 23 Multiple Optima 335 23.1 Introduction 335 23.2 Solution: Multiple Starts 337 23.2.1 A Priori Method 340 23.2.2 A Posteriori Method 342 23.2.3 Snyman and Fatti Criterion A Posteriori Method 345 23.3 Other Options 348 23.4 Takeaway 349 23.5 Exercises 350 24 Stochastic Objective Functions 353 24.1 Introduction 353 24.2 Method Summary for Optimizing Stochastic Functions 356 24.2.1 Step 1: Replicate the Apparent Best Player 356 24.2.2 Step 2: Steady-State Detection 357 24.3 What Value to Report? 358 24.4 Application Examples 359 24.4.1 GMC Control of Hot and Cold Mixing 359 24.4.2 MBC of Hot and Cold Mixing 359 24.4.3 Batch Reaction Management 359 24.4.4 Reservoir and Stochastic Boot Print 361 24.4.5 Optimization Results 362 24.5 Takeaway 365 24.6 Exercises 365 25 Effects of Uncertainty 367 25.1 Introduction 367 25.2 Sources of Error and Uncertainty 368 25.3 Significant Digits 370 25.4 Estimating Uncertainty on Values 371 25.5 Propagating Uncertainty on DV Values 372 25.5.1 Analytical Method 373 25.5.2 Numerical Method 375 25.6 Implicit Relations 378 25.7 Estimating Uncertainty in DV∗ and OF∗ 378 25.8 Takeaway 379 25.9 Exercises 379 26 Optimization of Probable Outcomes and Distribution Characteristics 381 26.1 Introduction 381 26.2 The Concept of Modeling Uncertainty 385 26.3 Stochastic Approach 387 26.4 Takeaway 389 26.5 Exercises 389 27 Discrete and Integer Variables 391 27.1 Introduction 391 27.2 Optimization Solutions 394 27.2.1 Exhaustive Search 394 27.2.2 Branch and Bound 394 27.2.3 Cyclic Heuristic 394 27.2.4 Leapfrogging or Other Multiplayer Search 395 27.3 Convergence 395 27.4 Takeaway 395 27.5 Exercises 395 28 Class Variables 397 28.1 Introduction 397 28.2 The Random Keys Method: Sequence 398 28.3 The Random Keys Method: Dichotomous Variables 400 28.4 Comments 401 28.5 Takeaway 401 28.6 Exercises 401 29 Regression 403 29.1 Introduction 403 29.2 Perspective 404 29.3 Least Squares Regression: Traditional View on Linear Model Parameters 404 29.4 Models Nonlinear in DV 405 29.4.1 Models with a Delay 407 29.5 Maximum Likelihood 408 29.5.1 Akaho’s Method 411 29.6 Convergence Criterion 416 29.7 Model Order or Complexity 421 29.8 Bootstrapping to Reveal Model Uncertainty 425 29.8.1 Interpretation of Bootstrapping Analysis 428 29.8.2 Appropriating Bootstrapping 430 29.9 Perspective 431 29.10 Takeaway 431 29.11 Exercises 432 Section 5 Perspective on Many Topics 441 30 Perspective 443 30.1 Introduction 443 30.2 Classifications 443 30.3 Elements Associated with Optimization 445 30.4 Root Finding Is Not Optimization 446 30.5 Desired Engineering Attributes 446 30.6 Overview of Optimizers and Attributes 447 30.6.1 Gradient Based: Cauchy Sequential Line Search, Incremental Steepest Descent, GRG, Etc. 447 30.6.2 Local Surface Characterization Based: Newton–Raphson, Levenberg–Marquardt, Successive Quadratic, RLM, Quasi-Newton, Etc. 448 30.6.3 Direct Search with Single Trial Solution: Cyclic Heuristic, Hooke–Jeeves, and Nelder–Mead 448 30.6.4 Multiplayer Direct Search Optimizers: Leapfrogging, Particle Swarm, and Genetic Algorithms 448 30.7 Choices 448 30.8 Variable Classifications 449 30.8.1 Nominal 449 30.8.2 Ordinal 450 30.8.3 Cardinal 450 30.9 Constraints 451 30.10 Takeaway 453 30.11 Exercises 453 31 Response Surface Aberrations 459 31.1 Introduction 459 31.2 Cliffs (Vertical Walls) 459 31.3 Sharp Valleys (or Ridges) 459 31.4 Striations 463 31.5 Level Spots (Functions 1, 27, 73, 84) 463 31.6 Hard-to-Find Optimum 466 31.7 Infeasible Calculations 468 31.8 Uniform Minimum 468 31.9 Noise: Stochastic Response 469 31.10 Multiple Optima 471 31.11 Takeaway 473 31.12 Exercises 473 32 Identifying the Models, OF, DV, Convergence Criteria, and Constraints 475 32.1 Introduction 475 32.2 Evaluate the Results 476 32.3 Takeaway 482 32.4 Exercises 482 33 Evaluating Optimizers 489 33.1 Introduction 489 33.2 Challenges to Optimizers 490 33.3 Stakeholders 490 33.4 Metrics of Optimizer Performance 490 33.5 Designing an Experimental Test 492 33.6 Takeaway 495 33.7 Exercises 496 34 Troubleshooting Optimizers 499 34.1 Introduction 499 34.2 DV Values Do Not Change 499 34.3 Multiple DV∗ Values for the Same OF∗ Value 499 34.4 EXE Error 500 34.5 Extreme Values 500 34.6 DV∗ Is Dependent on Convergence Threshold 500 34.7 OF∗ Is Irreproducible 501 34.8 Concern over Results 501 34.9 CDF Features 501 34.10 Parameter Correlation 502 34.11 Multiple Equivalent Solutions 504 34.12 Takeaway 504 34.13 Exercises 504 Section 6 Analysis of Leapfrogging Optimization 505 35 Analysis of Leapfrogging 507 35.1 Introduction 507 35.2 Balance in an Optimizer 508 35.3 Number of Initializations to be Confident That the Best Will Draw All Others to the Global Optimum 510 35.3.1 Methodology 511 35.3.2 Experimental 512 35.3.3 Results 513 35.4 Leap-To Window Amplification Analysis 515 35.5 Analysis of α and M to Prevent Convergence on the Side of a Hill 519 35.6 Analysis of α and M to Minimize NOFE 521 35.7 Probability Distribution of Leap-Overs 522 35.7.1 Data 526 35.8 Takeaway 527 35.9 Exercises 528 Section 7 Case Studies 529 36 Case Study 1: Economic Optimization of a Pipe System 531 36.1 Process and Analysis 531 36.1.1 Deterministic Continuum Model 531 36.1.2 Deterministic Discontinuous Model 534 36.1.3 Stochastic Discontinuous Model 535 36.2 Exercises 536 37 Case Study 2: Queuing Study 539 37.1 The Process and Analysis 539 37.2 Exercises 541 38 Case Study 3: Retirement Study 543 38.1 The Process and Analysis 543 38.2 Exercises 550 39 Case Study 4: A Goddard Rocket Study 551 39.1 The Process and Analysis 551 39.2 Pre-Assignment Note 554 39.3 Exercises 555 40 Case Study 5: Reservoir 557 40.1 The Process and Analysis 557 40.2 Exercises 559 41 Case Study 6: Area Coverage 561 41.1 Description and Analysis 561 41.2 Exercises 562 42 Case Study 7: Approximating Series Solution to an ODE 565 42.1 Concepts and Analysis 565 42.2 Exercises 568 43 Case Study 8: Horizontal Tank Vapor–Liquid Separator 571 43.1 Description and Analysis 571 43.2 Exercises 576 44 Case Study 9: In Vitro Fertilization 579 44.1 Description and Analysis 579 44.2 Exercises 583 45 Case Study 10: Data Reconciliation 585 45.1 Description and Analysis 585 45.2 Exercises 588 Section 8 Appendices 591 Appendix A Mathematical Concepts and Procedures 593 Appendix B Root Finding 605 Appendix C Gaussian Elimination 611 Appendix D Steady-State Identification in Noisy Signals 621 Appendix E Optimization Challenge Problems (2-D and Single OF) 635 Appendix F Brief on VBA Programming: Excel in Office 2013 709 Section 9 References and Index 717 References and Additional Resources 719 Index 723

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Book SynopsisAims to provides applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).Trade ReviewFrom the reviews: "This book presents a wealth of results on Young measures on topological spaces in a very general framework. It is very likely that it will become the reference and starting point for any further developments in the field." (Georg K. Dolzmann, Mathematical Reviews, 2005k)Table of ContentsPreface. Generalities, Preliminary results. Young Measures, the four Stable Topologies: S, M, N, W. Convergence in Probability of Young Measures (with some applications to stable convergence). Compactness. Strong Tightness. Young Measures on Banach Spaces. Application. Applications in Control Theory. Semicontinuity of Integral Functionals using Young Measures. Stable Convergence in Limit Theorems of Probability Theory.

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Book SynopsisThis book gives a comprehensive account of financial optimization models used to support decision-making for financial engineers. It starts with the classical static mean-variance analysis and portfolio immunization, moves on to scenario-based models, and builds towards multi-period dynamic portfolio optimization.Trade Review“This volume is both a comprehensive guide to optimization techniques useful in financial decision making and a well-illustrated essay on the relationship between theory and practice. While the real problem may always be more complex than any model of it we build, that does not necessarily imply that the largest, most complex model will serve us best. Zenios supplies the reader with a spectrum of optimization models, from simple to complex, and sage advice on how to use them.” From the Foreword by Harry M. Markowitz, Nobel Laureate in Economics “Most books on portfolio optimization focus on continuous time stochastic control models. By contrast, Zenios’s decision to focus on mathematical programming models in financial engineering is an auspicious one. The book is well organized and clearly written, and uses a minimum of technical prerequisites (both mathematical and financial). It should therefore be accessible and of interest to a broad audience: industry practitioners interested in the potential application of optimization to the problems they face, students curious about how optimization is applied in finance, and professional researchers who would like a comprehensive overview of the uses of mathematical programming in financial engineering.” David Saunders, University of WaterlooTable of ContentsForeword: Harry M. Markowitz. Preface. Acknowledgments. Notation. List of Models. I. Introduction. 1. An Optimization View of Financial Engineering. 2. Basics of Risk Management. II. Portfolio Optimization Models. 3. Mean-Variance Analysis. 4. Portfolio Models for Fixed Income. 5. Scenario Optimization. 6. Dynamic Portfolio Optimization with Stochastic Programming. 7. Index Funds. 8. Designing Financial Products. 9. Scenario Generation. III. Applications. 10. International Asset Allocation. 11. Corporate Bond Portfolios. 12. Insurance Policies with Guarantees. 13. Personal Financial Planning. IV. Library of Financial Optimization Models. 14. FINLIB: A Library of Financial Optimization Models. Bibliography. Index

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Book SynopsisThis book gives a comprehensive account of financial optimization models used to support decision-making for financial engineers. It starts with the classical static mean-variance analysis and portfolio immunization, moves on to scenario-based models, and builds towards multi-period dynamic portfolio optimization.Trade Review"This volume is both a comprehensive guide to optimization techniques useful in financial decision making and a well-illustrated essay on the relationship between theory and practice. While the real problem may always be more complex than any model of it we build, that does not necessarily imply that the largest, most complex model will serve us best. Zenios supplies the reader with a spectrum of optimization models, from simple to complex, and sage advice on how to use them."From the Foreword by Harry M. Markowitz, Nobel Laureate in Economics "Most books on portfolio optimization focus on continuous time stochastic control models. By contrast, Zenios's decision to focus on mathematical programming models in financial engineering is an auspicious one. The book is well organized and clearly written, and uses a minimum of technical prerequisites (both mathematical and financial). It should therefore be accessible and of interest to a broad audience: industry practitioners interested in the potential application of optimization to the problems they face, students curious about how optimization is applied in finance, and professional researchers who would like a comprehensive overview of the uses of mathematical programming in financial engineering."David Saunders, University of WaterlooTable of ContentsForeword. Preface. Acknowledgements. List of Models. Notation. I. Introduction. 1. An Optimization View of Financial Engineering. 2. Basics of Risk Management. II. Portfolio Optimization Models. 3. Mean-Variance Analysis. 4. Portfolio Models for Fixed Income. 5. Scenario Optimization. 6. Dynamic Portfolio Optimization with Stochastic Programming. 7. Index Funds. 8. Designing Financial Products. 9. Scenario Generation. III. Applications. 10. Application I: International Asset Allocation. 11. Application II: Corporate Bond Portfolios. 12. Application III: Insurance Policies with Guarantees. 13. Application IV: Personal Financial Planning. IV. Library of Financial Optimization Models. 14. FINLIB: A Library of Financial Optimization Models A. Basics of Optimization. B. Basics of Probability Theory. C. Stochastic Processes. Bibliography. Index.

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Book Synopsis0 Review of Optimization in ?d.- Problems.- One Basic Theory.- 1 Standard Optimization Problems.- 2 Linear Spaces and Gâteaux Variations.- 3 Minimization of Convex Functions.- 4 The Lemmas of Lagrange and Du Bois-Reymond.- 5 Local Extrema in Normed Linear Spaces.- 6 The Euler-Lagrange Equations.- Two Advanced Topics.- 7 Piecewise C1 Extremal Functions.- 8 Variational Principles in Mechanics.- 9 Sufficient Conditions for a Minimum.- Three Optimal Control.- 10 Control Problems and Sufficiency Considerations.- 11 Necessary Conditions for Optimality.- A.1. The Intermediate and Mean Value Theorems.- A.2. The Fundamental Theorem of Calculus.- A.3. Partial Integrals: Leibniz' Formula.- A.4. An Open Mapping Theorem.- A.5. Families of Solutions to a System of Differential Equations.- A.6. The Rayleigh Ratio.- Historical References.- Answers to Selected Problems.Table of Contents0 Review of Optimization in ?d.- Problems.- One Basic Theory.- 1 Standard Optimization Problems.- 1.1. Geodesic Problems.- (a) Geodesics in ?d.- (b) Geodesics on a Sphere.- (c) Other Geodesic Problems.- 1.2. Time-of-Transit Problems.- (a) The Brachistochrone.- (b) Steering and Control Problems.- 1.3. Isoperimetric Problems.- 1.4. Surface Area Problems.- (a) Minimal Surface of Revolution.- (b) Minimal Area Problem.- (c) Plateau’s Problem.- 1.5. Summary: Plan of the Text.- Notation: Uses and Abuses.- Problems.- 2 Linear Spaces and Gâteaux Variations.- 2.1. Real Linear Spaces.- 2.2. Functions from Linear Spaces.- 2.3. Fundamentals of Optimization.- Constraints.- Rotating Fluid Column.- 2.4. The Gâteaux Variations.- Problems.- 3 Minimization of Convex Functions.- 3.1. Convex Functions.- 3.2. Convex Integral Functions.- Free End-Point Problems.- 3.3. [Strongly] Convex Functions.- 3.4. Applications.- (a) Geodesics on a Cylinder.- (b) A Brachistochrone.- (c) A Profile of Minimum Drag.- (d) An Economics Problem.- (e) Minimal Area Problem.- 3.5. Minimization with Convex Constraints.- The Hanging Cable.- Optimal Performance.- 3.6. Summary: Minimizing Procedures.- Problems.- 4 The Lemmas of Lagrange and Du Bois-Reymond.- Problems.- 5 Local Extrema in Normed Linear Spaces.- 5.1. Norms for Linear Spaces.- 5.2. Normed Linear Spaces: Convergence and Compactness.- 5.3. Continuity.- 5.4. (Local) Extremal Points.- 5.5. Necessary Conditions: Admissible Directions.- 5.6*. Affine Approximation: The Fréchet Derivative.- Tangency.- 5.7. Extrema with Constraints: Lagrangian Multipliers.- Problems.- 6 The Euler-Lagrange Equations.- 6.1. The First Equation: Stationary Functions.- 6.2. Special Cases of the First Equation.- (a) When f = f(z).- (b) When f = f(x,z).- (c) When f = f(y,z).- 6.3. The Second Equation.- 6.4. Variable End Point Problems: Natural Boundary Conditions.- Jakob Bernoulli’s Brachistochrone.- Transversal Conditions*.- 6.5. Integral Constraints: Lagrangian Multipliers.- 6.6. Integrals Involving Higher Derivatives.- Buckling of a Column under Compressive Load.- 6.7. Vector Valued Stationary Functions.- The Isoperimetric Problem.- Lagrangian Constraints*.- Geodesics on a Surface.- 6.8*. Invariance of Stationarity.- 6.9. Multidimensional Integrals.- Minimal Area Problem.- Natural Boundary Conditions.- Problems.- Two Advanced Topics.- 7 Piecewise C1 Extremal Functions.- 7.1. Piecewise C1 Functions.- (a) Smoothing.- (b) Norms for ?1.- 7.2. Integral Functions on ?1.- 7.3. Extremals in ?1 [a, b]: The Weierstrass-Erdmann Corner Conditions.- A Sturm-Liouville Problem.- 7.4. Minimization Through Convexity.- Internal Constraints.- 7.5. Piecewise C1 Vector-Valued Extremals.- Minimal Surface of Revolution.- Hilbert’s Differentiability Criterion*.- 7.6*. Conditions Necessary for a Local Minimum.- (a) The Weierstrass Condition.- (b) The Legendre Condition.- Bolza’s Problem.- Problems.- 8 Variational Principles in Mechanics.- 8.1. The Action Integral.- 8.2. Hamilton’s Principle: Generalized Coordinates.- Bernoulli’s Principle of Static Equilibrium.- 8.3. The Total Energy.- Spring-Mass-Pendulum System.- 8.4. The Canonical Equations.- 8.5. Integrals of Motion in Special Cases.- Jacobi’s Principle of Least Action.- Symmetry and Invariance.- 8.6. Parametric Equations of Motion.7*. The Hamilton-Jacobi Equation.- 8.8. Saddle Functions and Convexity; Complementary Inequalities.- The Cycloid Is the Brachistochrone.- Dido’s Problem.- 8.9. Continuous Media.- (a) Taut String.- The Nonuniform String.- (b) Stretched Membrane.- Static Equilibrium of (Nonplanar) Membrane.- Problems.- 9 Sufficient Conditions for a Minimum.- 9.1. The Weierstrass Method.- 9.2. [Strict] Convexity of f(x,Y, Z).- 9.3. Fields.- Exact Fields and the Hamilton-Jacobi Equation*.- 9.4. Hilbert’s Invariant Integral.- The Brachistochrone*.- Variable End-Point Problems.- 9.5. Minimization with Constraints.- The Wirtinger Inequality.- 9.6*. Central Fields.- Smooth Minimal Surface of Revolution.- 9.7. Construction of Central Fields with Given Trajectory: The Jacobi Condition.- 9.8. Sufficient Conditions for a Local Minimum.- (a) Pointwise Results.- Hamilton’s Principle.- (b) Trajectory Results.- 9.9*. Necessity of the Jacobi Condition.- 9.10. Concluding Remarks.- Problems.- Three Optimal Control.- 10 Control Problems and Sufficiency Considerations.- 10.1. Mathematical Formulation and Terminology.- 10.2. Sample Problems.- (a) Some Easy Problems.- (b) A Bolza Problem.- (c) Optimal Time of Transit.- (d) A Rocket Propulsion Problem.- (e) A Resource Allocation Problem.- (f) Excitation of an Oscillator.- (g) Time-Optimal Solution by Steepest Descent.- 10.3. Sufficient Conditions Through Convexity.- Linear State-Quadratic Performance Problem.- 10.4. Separate Convexity and the Minimum Principle.- Problems.- 11 Necessary Conditions for Optimality.- 11.1. Necessity of the Minimum Principle.- (a) Effects of Control Variations.- (b) Autonomous Fixed Interval Problems.- Oscillator Energy Problem.- (c) General Control Problems.- 11.2. Linear Time-Optimal Problems.- Problem Statement.- A Free Space Docking Problem.- 11.3. General Lagrangian Constraints.- (a) Control Sets Described by Lagrangian Inequalities.- (b)* Variational Problems with Lagrangian Constraints.- (c) Extensions.- Problems.- A.1. The Intermediate and Mean Value Theorems.- A.2. The Fundamental Theorem of Calculus.- A.3. Partial Integrals: Leibniz’ Formula.- A.4. An Open Mapping Theorem.- A.5. Families of Solutions to a System of Differential Equations.- A.6. The Rayleigh Ratio.- Historical References.- Answers to Selected Problems.

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Book Synopsis1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4Table of Contents1 Euclidean spaces.- 1.1 The real number system.- 1.2 Euclidean En.- 1.3 Elementary geometry of En.- 1.4 Basic topological notions in En.- *1.5 Convex sets.- 2 Elementary topology of En.- 2.1 Functions.- 2.2 Limits and continuity of transformations.- 2.3 Sequences in En.- 2.4 Bolzano-Weierstrass theorem.- 2.5 Relative neighborhoods, continuous transformations.- 2.6 Topological spaces.- 2.7 Connectedness.- 2.8 Compactness.- 2.9 Metric spaces.- 2.10 Spaces of continuous functions.- *2.11 Noneuclidean norms on En.- 3 Differentiation of real-valued functions.- 3.1 Directional and partial derivatives.- 3.2 Linear functions.- **3.3 Difierentiable functions.- 3.4 Functions of class C(q).- 3.5 Relative extrema.- *3.6 Convex and concave functions.- 4 Vector-valued functions of several variables.- 4.1 Linear transformations.- 4.2 Affine transformations.- 4.3 Differentiable transformations.- 4.4 Composition.- 4.5 The inverse function theorem.- 4.6 The implicit function theorem.- 4.7 Manifolds.- 4.8 The multiplier rule.- 5 Integration.- 5.1 Intervals.- 5.2 Measure.- 5.3 Integrals over En.- 5.4 Integrals over bounded sets.- 5.5 Iterated integrals.- 5.6 Integrals of continuous functions.- 5.7 Change of measure under affine transformations.- 5.8 Transformation of integrals.- 5.9 Coordinate systems in En.- 5.10 Measurable sets and functions; further properties.- 5.11 Integrals: general definition, convergence theorems.- 5.12 Differentiation under the integral sign.- 5.13 Lp-spaces.- 6 Curves and line integrals.- 6.1 Derivatives.- 6.2 Curves in En.- 6.3 Differential 1-forms.- 6.4 Line integrals.- *6.5 Gradient method.- *6.6 Integrating factors; thermal systems.- 7 Exterior algebra and differential calculus.- 7.1 Covectors and differential forms of degree 2.- 7.2 Alternating multilinear functions.- 7.3 Multicovectors.- 7.4 Differential forms.- 7.5 Multivectors.- 7.6 Induced linear transformations.- 7.7 Transformation law for differential forms.- 7.8 The adjoint and codifferential.- *7.9 Special results for n = 3.- *7.10 Integrating factors (continued).- 8 Integration on manifolds.- 8.1 Regular transformations.- 8.2 Coordinate systems on manifolds.- 8.3 Measure and integration on manifolds.- 8.4 The divergence theorem.- *8.5 Fluid flow.- 8.6 Orientations.- 8.7 Integrals of r-forms.- 8.8 Stokes’s formula.- 8.9 Regular transformations on submanifolds.- 8.10 Closed and exact differential forms.- 8.11 Motion of a particle.- 8.12 Motion of several particles.- Axioms for a vector space.- Mean value theorem; Taylor’s theorem.- Review of Riemann integration.- Monotone functions.- References.- Answers to problems.

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Book Synopsis Solve SEO problems using data science. This hands-on book is packed with Python code and data science techniques to help you generate data-driven recommendations and automate the SEO workload. This book is a practical, modern introduction to data science in the SEO context using Python. With social media, mobile, changing search engine algorithms, and ever-increasing expectations of users for super web experiences, too much data is generated for an SEO professional to make sense of in spreadsheets. For any modern-day SEO professional to succeed, it is relevant to find an alternate solution, and data science equips SEOs to grasp the issue at hand and solve it. From machine learning to Natural Language Processing (NLP) techniques, Data-Driven SEO with Python provides tried and tested techniques with full explanations for solving both everyday and complex SEO problems. This book is ideal for SEO professionals who want to take their industry skiTable of ContentsData Driven SEO with PythonChapter 1: Meeting the Challenges of SEO with Data1.1 Agents of change in SEO1.2 The Pillars of SEO Strategy1.3 Installing Python1.4 Using Python for SEOChapter 2: Keyword Research2.1 Data Sources2.2 Google Search Console2.4 Google Trends2.5 Google Suggest2.6 Competitor Analytics2.7 SERPsChapter 3: Technical3.1 Improving CTRs3.2 Allocate keywords to pages based on the copy3.3 Allocating parent nodes to the orphaned URLs3.4 Improve interlinking based on copy3.5 Automate Technical AuditsChapter 4: Content & UX4.1 Content that best satisfies the user query4.2 Splitting and merging URLs4.3 Content Strategy: Planning landing page content Chapter 5: Authority5.1 A little SEO history5.1 The source of authority5.2 Finding good linksChapter 6: Competitors6.1 Defining the problem6.2 Data Strategy6.3 Data Sources6.4 Selecting Your Competitors6.5 Get Features6.6 Explore, Clean and Transform6.7 Modelling The SERPS6.8 Evaluating your Model6.9 ActivationChapter 7: Experiments7.1 How experiments fit into the SEO process7.2 Generating Hypotheses7.3 Experiment Design7.4 Running your experiment7.5 Experiment EvaluationChapter 8: Dashboards8.1 Use a Data Layer8.2 Extract, Transform and Load (ETL)8.3 Transform8.4 Querying the Data Warehouse (DW)8.5 Visualization8.6 Making Future ForecastsChapter 9: Site Migrations and Relaunches9.1 Data sources9.2 Establishing the Impact9.3 Segmenting the URLs9.4 Legacy Site URLs9.5 Priority9.6 RoadmapChapter 10: Google Updates10.1 Data sources10.2 Winners and Losers10.3 Quantifying the Impact10.4 Search Intent10.5 Unique URLs10.6 RecommendationsChapter 11: The Future of SEO11.1 Automation11.2 Your journey to SEO science11.3 Suggest resourcesAppendix: CodeGlossaryIndex

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Book SynopsisThis monograph presents smooth, unified, and generalized fractional programming problems, particularly advanced duality models for discrete min-max fractional programming. In the current, interdisciplinary, computer-oriented research environment, these programs are among the most rapidly expanding research areas in terms of their multi-faceted applications including problems ranging from robotics to money market portfolio management. The other more significant aspect of this monograph is in its consideration of minimax fractional integral type problems using higher order sonvexity and sounivexity notions. This is significant for the development of different types of duality models in terms of weak, strong, and strictly converse duality theorems, which can be handled by transforming them into generalized fractional programming problems. Fractional integral type programming is one of the fastest expanding areas of optimization, which feature several types of real-world problems. It can be applied to different branches of engineering (including multi-time multi-objective mechanical engineering problems) as well as to economics, to minimize a ratio of functions between given periods of time. Furthermore, it can be utilized as a resource in order to measure the efficiency or productivity of a system. In these types of problems, the objective function is given as a ratio of functions. For example, we consider a problem that deals with minimizing a maximum of several time-dependent ratios involving integral expressions.Table of ContentsFor more information, please visit our website at:https://novapublishers.com/shop/new-trends-in-fractional-programming/

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Book SynopsisOptimisation is the process of obtaining the most appropriate solution by providing certain constraints for the given purpose or purposes. Mathematically, optimisation can be briefly defined as minimising or maximising a function. In short, optimisation is to look for the best. The best found is called "optimum. Optimisation is used to accelerate decision-making processes and to solve real-life problems in an effective, accurate and real-time manner. In addition to the economic benefits, optimisation is also used as an effective method to include the preferences and constraints of customers, employers and employees in the decision process and to improve the quality of the resources in the system. The purpose of optimisation is to achieve the best result, the best goal. Improvements can be made to the current situation or situations to achieve the best result. One of the major shortcomings in optimisation and robotic is the transformation of theoretical knowledge into practice. The purpose of the book is to introduce students, teachers, researchers, and practitioners to new advances in this area. The book content includes theoretical and practical studies prepared with the academic contributions of scientists working in different fields. It was decided to publish each chapter in the book after being examined by the scientific board. As an editor, my duty is to ensure breadth, while the chapter authors treat the delegated chapters with depth. The book is designed for practitioners or researchers of all levels of expertise from novice to expert. Each of the book's individual topics could be considered as a compact, self-contained mini-book right under its title. The approach is to provide a framework and a set of techniques for evaluating and improving optimisation and robotic. It presents a specific set of solutions, mostly obtained from real world projects and experimental studies, for routine applications. It further highlights promising emerging techniques for research and exploration opportunities. The development team of this book wants to thank their colleagues who made contributions to this book by providing continuous encouragements and thorough reviews of the chapters of the book.Table of ContentsPreface; A Source Seeking Algorithm with Application to a Quadcopter Model; Activation Functions for Deep Learning in Smart Manufacturing; A New Approach to Kaiser and Gaussian Window Based Cosine Modulated Filter Bank Design; A Preliminary Study of Region Of Interest Based Functional Connectivity Analysis for Classification of MDD and Healthy Subjects Using Graph Metrics; Investigation of the Effect of Graphene Oxide and Aluminum Oxide Particles on the Mechanical Properties of Glass / Kevlar /Carbon Fiber Reinforced Epoxy Composites; Conversion from Conventional Power Distribution Networks to Smart Power Distribution Networks; A Parametric Study for Artificial Bee Colony Algorithm Used in Vehicle Routing Problem with Simultaneous Delivery and Pickup; Lightweighting Airborne Vehicles Structural Analysis of Carbon Fiber-Reinforced Aluminum and High Altitude Long Endurance New Drone: Octocopter Robot Unmanned Aerial Vehicles (UAVs) Design by Computational Fluid Dynamics and Finite Element Method; Index.

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Book SynopsisThe quality loss function, introduced by Japanese engineer, statistician and scientist Dr. Genichi Taguchi in the 1980s, is still one of the most interesting topics in applied industrial statistics and quality engineering and management, which presented a paradigm shift in quality loss and product, process and/or system quality conception. Taguchi emphasized a proactive approach toward quality in terms of embedding quality requirements into the design of product, process and/or system, which highly influenced today's quality approaches such as the 'quality-by-design' concept strongly demanded in the era of the fourth industrial revolution that we are currently facing. This book contributes to a further development, extension and application of the Taguchi's quality loss concept, aiming to overcome limitations of the traditional quadratic quality loss function and to address complex demands and circumstances in a dynamic and globalized contemporary industrial sector. It presents essential issues and heterogeneous complementary aspects of the quality loss function, including the theoretical background and advances as well as different application studies. The opening chapter is dedicated to the quality loss functions used in quality engineering, presenting an in-depth theoretical background of the traditional loss function, the bounded loss function concept, i.e. the reflected normal loss function, and the family of inverted loss functions, and proposing the recently developed loss function types. The second chapter is focused on the Taguchi's and inverted quality loss functions, univariate and multivariate types, and their advances and implications in tackling real, heterogeneous industrial problems in statistical quality and process control. The third chapter considers an application of the quality loss and quality cost concepts at a system level, by introducing the quality policy model of an organization, developed and implemented in a middle-sized manufacturing company in the automotive industry. The fourth chapter deals with the comparison and alignment of the Taguchi's orthogonal arrays and the traditional full factorial approach for experimental design, including also the method for analysis of experimental results, depicted by two use cases from different industrial sectors. The last chapter proposes an advanced quality loss-based method for discrete process parameter optimization that tackles processes characterized by multiple correlated responses. The benefits of its implementation are illustrated on heterogeneous process optimization problems, and comparison with several frequently used optimization methods clearly demonstrates its superiority, effectiveness and applicability in real industrial conditions. Therefore, this book offers a unique combination of two aspects relevant for scientists and statisticians, and engineers and managers, respectively: (i) strong scientific background on the quality loss function, its modifications and extensions, and novel, advanced developments; (ii) hands on approach for application of the quality loss function-based methods designed for product, process and/or system quality improvement in different stages, from the experimental design, via analysis of experimental results and process parameter optimization, toward an organizational quality policy implementation.

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Book SynopsisTensors, or hypermatrices, are multi-arrays with more than two indices. In the last decade or so, many concepts and results in matrix theory – some of which are nontrivial – have been extended to tensors and have a wide range of applications (for example, spectral hypergraph theory, higher order Markov chains, polynomial optimization, magnetic resonance imaging, automatic control, and quantum entanglement problems). The authors provide a comprehensive discussion of this new theory of tensors.Tensor Analysis is unique in that it is the first book on the spectral theory of tensors; the theory of special tensors, including nonnegative tensors, positive semidefinite tensors, completely positive tensors, and copositive tensors; and the spectral hypergraph theory via tensors, which is covered in a chapter.Table of Contents List of Figures. List of Algorithms. Preface. Chapter 1: Introduction. Chapter 2: Eigenvalues of Tensors. Chapter 3: Nonnegative Tensors. Chapter 4: Spectral Hypergraph Theory via Tensors. Chapter 5: Positive Semidefinite Tensors. Chapter 6: Completely Positive Tensors and Copositive Tensors. Bibliography. Index.

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Book SynopsisIn the intervening years since this book was published in 1981, the field of optimization has been exceptionally lively. This fertility has involved not only progress in theory, but also faster numerical algorithms and extensions into unexpected or previously unknown areas such as semidefinite programming. Despite these changes, many of the important principles and much of the intuition can be found in this Classics version of Practical Optimization.This book provides model algorithms and pseudocode, useful tools for users who prefer to write their own code as well as for those who want to understand externally provided code. It presents algorithms in a step-by-step format, revealing the overall structure of the underlying procedures and thereby allowing a high-level perspective on the fundamental differences. And it contains a wealth of techniques and strategies that are well suited for optimization in the twenty-first century, and particularly in the now-flourishing fields of data science, “big data,” and machine learning. Practical Optimization is appropriate for advanced undergraduates, graduate students, and researchers interested in methods for solving optimization problems.

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Book SynopsisA compendium of the authors’ recently published results, this book discusses sliding mode control of uncertain nonlinear systems, with a particular emphasis on advanced and optimization based algorithms. The authors survey classical sliding mode control theory and introduce four new methods of advanced sliding mode control. They analyze classical theory and advanced algorithms, with numerical results complementing the theoretical treatment. Case studies examine applications of the algorithms to complex robotics and power grid problems. Advanced and Optimization Based Sliding Mode Control: Theory and Applications is the first book to systematize the theory of optimization based higher order sliding mode control and illustrate advanced algorithms and their applications to real problems. It presents systematic treatment of event-triggered and model based event-triggered sliding mode control schemes, including schemes in combination with model predictive control, and presents adaptive algorithms as well as algorithms capable of dealing with state and input constraints. Additionally, the book includes simulations and experimental results obtained by applying the presented control strategies to real complex systems.

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Book SynopsisEngineering systems operate through actuators, most of which will exhibit phenomena such as saturation or zones of no operation, commonly known as dead zones. These are examples of piecewise-affine characteristics, and they can have a considerable impact on the stability and performance of engineering systems. This book targets controller design for piecewise affine systems, fulfilling both stability and performance requirements.The authors present a unified computational methodology for the analysis and synthesis of piecewise affine controllers, taking an approach that is capable of handling sliding modes, sampled-data, and networked systems. They introduce algorithms that will be applicable to nonlinear systems approximated by piecewise affine systems, and they feature several examples from areas such as switching electronic circuits, autonomous vehicles, neural networks, and aerospace applications.Piecewise Affine Control: Continuous-Time, Sampled-Data, and Networked Systems is intended for graduate students, advanced senior undergraduate students, and researchers in academia and industry. It is also appropriate for engineers working on applications where switched linear and affine models are important.Trade ReviewPiecewise affine systems are widely used as modeling and design tools across a number of applications, ranging from robotics to systems biology. These systems require a delicate touch as they can exhibit complex and sometimes surprising features. This impressive book navigates the world of such systems with clarity, technical depth, and elegance.”- Professor Magnus Egerstedt, Georgia Institute of Technology

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Book SynopsisThe mathematical theory for many application areas depends on a deep understanding of the theory of moments. These areas include medical imaging, signal processing, computer visualization, and data science. The problem of moments has also found novel applications to areas such as control theory, image analysis, signal processing, polynomial optimization, and statistical big data. The Classical Moment Problem and Some Related Questions in Analysis presents: a unified treatment of the development of the classical moment problem from the late 19th century to the middle of the 20th century, important connections between the moment problem and many branches of analysis, a unified exposition of important classical results, which are difficult to read in the original journals, and a strong foundation for many areas in modern applied mathematics.

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Book SynopsisOptimization is presented in most multivariable calculus courses as an application of the gradient, and while this treatment makes sense for a calculus course, there is much more to the theory of optimization. Optimization problems are generated constantly, and the theory of optimization has grown and developed in response to the challenges presented by these problems. This textbook aims to show readers how optimization is done in practice and help them to develop an appreciation for the richness of the theory behind the practice.Exercises, problems (including modeling and computational problems), and implementations are incorporated throughout the text to help students learn by doing. Python notes are inserted strategically to help readers complete computational problems and implementations.The Basics of Practical Optimization, Second Edition is intended for undergraduates who have completed multivariable calculus, as well as anyone interested in optimization. The book is appropriate for a course that complements or replaces a standard linear programming course.

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Book SynopsisEconomics is a science which studies human behaviours as a relationship between ends and scarce means which have alternative uses. Since economic resources are scarce, optimisation forms an integral part in the study of economics. In addition, in the presence of imperfect market structure, externalities, imperfect information or public goods, the market fails to provide an efficient allocation mechanism. Optimisation of economic activities provides an effective remedial measure for market failures. This contributed volume collects advances in the studies of economics and optimisation. Contributions cover areas on analysis of optimal allocation of economic resources, economic optimisation techniques, the interface economics and optimisation, optimisation under market mechanism and history of development of optimisation techniques. The studies assembled in this volume are dedicated to the memory of a pioneering researcher and Nobel Laureate in the field of economic optimisation -- Leonid Vitalyevich Kantorovich. In his 100th birthday tribute in 2012, the International Conference Mathematics, Economic, Management: Kantorovich-100 in St-Petersburg was held in his memory. Selected papers from the conference are included in this Volume. In addition, contributed papers from authors who had worked closely with Kantorovich are also contained.

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Book SynopsisThis book pursues optimal design from the perspective of mechanical properties and resistance to failure caused by cracks and fatigue. The book abandons the scale separation hypothesis and takes up phase-field modeling, which is at the cutting edge of research and is of high industrial and practical relevance. Part 1 starts by testing the limits of the homogenization-based approach when the size of the representative volume element is non-negligible compared to the structure. The book then introduces a non-local homogenization scheme to take into account the strain gradient effects. Using a phase field method, Part 2 offers three significant contributions concerning optimal placement of the inclusion phases. Respectively, these contributions take into account fractures in quasi-brittle materials, interface cracks and periodic composites. The topology optimization proposed has significantly increased the fracture resistance of the composites studied.Table of ContentsIntroduction ix Part 1. Multiscale Topology Optimization in the Context of Non-separated Scales 1 Chapter 1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization 3 1.1. The classical homogenization method 4 1.1.1. Localization problem 4 1.1.2. Definition and computation of the effective material properties 7 1.1.3. Numerical implementation for the local problem with PER 9 1.2. Topology optimization model and procedure 10 1.2.1. Optimization model and sensitivity number 10 1.2.2. Finite element meshes and relocalization scheme 12 1.2.3. Optimization procedure 14 1.3. Numerical examples 16 1.3.1. Doubly clamped elastic domain 17 1.3.2. L-shaped structure 19 1.3.3. MBB beam 24 1.4. Concluding remarks 25 Chapter 2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient 29 2.1. Non-local filter-based homogenization for non-separated scales 30 2.1.1. Definition of local and mesoscopic fields through the filter 30 2.1.2. Microscopic unit cell calculations 33 2.1.3. Mesoscopic structure calculations 39 2.2. Topology optimization procedure 41 2.2.1. Model definition and sensitivity numbers 41 2.2.2. Overall optimization procedure 42 2.3. Validation of the non-local homogenization approach 43 2.4. Numerical examples 45 2.4.1. Cantilever beam with a concentrated load 46 2.4.2. Four-point bending lattice structure 52 2.5. Concluding remarks 55 Chapter 3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures 57 3.1. Optimization model and procedure 58 3.2. Numerical examples 61 3.2.1. A double-clamped beam 61 3.2.2. A cantilever beam 64 3.3. Concluding remarks 66 Part 2. Topology Optimization for Maximizing the Fracture Resistance 67 Chapter 4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites 69 4.1. Phase field modeling of crack propagation 71 4.1.1. Phase field approximation of cracks 71 4.1.2. Thermodynamics of the phase field crack evolution 72 4.1.3. Weak forms of displacement and phase field problems 75 4.1.4. Finite element discretization 76 4.2. Topology optimization model for fracture resistance 78 4.2.1. Model definitions 78 4.2.2. Sensitivity analysis 80 4.2.3. Extended BESO method 85 4.3. Numerical examples 87 4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch 88 4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches 93 4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks 96 4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface 98 4.4. Concluding remarks 101 Chapter 5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage 103 5.1. Phase field modeling of bulk crack and cohesive interfaces 104 5.1.1. Regularized representation of a discontinuous field 104 5.1.2. Energy functional 106 5.1.3. Displacement and phase field problems 108 5.1.4. Finite element discretization and numerical implementation 111 5.2. Topology optimization method 114 5.2.1. Model definitions 114 5.2.2. Sensitivity analysis 116 5.3. Numerical examples 119 5.3.1. Design of a plate with one initial crack under traction 120 5.3.2. Design of a plate without initial cracks for traction loads 123 5.3.3. Design of a square plate without initial cracks in tensile loading 125 5.3.4. Design of a plate with a single initial crack under three-point bending 128 5.3.5. Design of a plate containing multiple inclusions 130 5.4. Concluding remarks 133 Chapter 6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites 135 6.1. Topology optimization model 136 6.2. Numerical examples 138 6.2.1. Design of a periodic composite under three-point bending 138 6.2.2. Design of a periodic composite under non-symmetric three-point bending 146 6.3. Concluding remarks 151 Conclusion 153 References 157 Index 173

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Book SynopsisThe quest for the optimal is ubiquitous in nature and human behavior. The field of mathematical optimization has a long history and remains active today, particularly in the development of machine learning.Classical and Modern Optimization presents a self-contained overview of classical and modern ideas and methods in approaching optimization problems. The approach is rich and flexible enough to address smooth and non-smooth, convex and non-convex, finite or infinite-dimensional, static or dynamic situations. The first chapters of the book are devoted to the classical toolbox: topology and functional analysis, differential calculus, convex analysis and necessary conditions for differentiable constrained optimization. The remaining chapters are dedicated to more specialized topics and applications.Valuable to a wide audience, including students in mathematics, engineers, data scientists or economists, Classical and Modern Optimization contains more than 200 exercises to assist with self-study or for anyone teaching a third- or fourth-year optimization class.

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Book SynopsisCombinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. Concepts of Combinatorial Optimization, is divided into three parts: On the complexity of combinatorial optimization problems, that presents basics about worst-case and randomized complexity; Classical solution methods, that presents the two most-known methods for solving hard combinatorial optimization problems, that are Branch-and-Bound and Dynamic Programming; Elements from mathematical programming, that presents fundamentals from mathematical programming based methods that are in the heart of Operations Research since the origins of this field. Table of ContentsPreface xiii Vangelis Th. PASCHOS PART I. COMPLEXITY OF COMBINATORIAL OPTIMIZATION PROBLEMS 1 Chapter 1. Basic Concepts in Algorithms and Complexity Theory 3 Vangelis Th. PASCHOS 1.1. Algorithmic complexity 3 1.2. Problem complexity 4 1.3. The classes P, NP and NPO 7 1.4. Karp and Turing reductions 9 1.5. NP-completeness 10 1.6. Two examples of NP-complete problems 13 1.7. A few words on strong and weak NP-completeness 16 1.8. A few other well-known complexity classes 17 1.9. Bibliography 18 Chapter 2. Randomized Complexity 21 Jérémy BARBAY 2.1. Deterministic and probabilistic algorithms 22 2.2. Lower bound technique 28 2.3. Elementary intersection problem 35 2.4. Conclusion 37 2.5 Bibliography 37 PART II. CLASSICAL SOLUTION METHODS 39 Chapter 3. Branch-and-Bound Methods 41 Irène CHARON and Olivier HUDRY 3.1. Introduction 41 3.2. Branch-and-bound method principles 43 3.3. A detailed example: the binary knapsack problem 54 3.4. Conclusion 67 3.5. Bibliography 68 Chapter 4. Dynamic Programming 71 Bruno ESCOFFIER and Olivier SPANJAARD 4.1. Introduction 71 4.2. A first example: crossing the bridge 72 4.3. Formalization 75 4.4. Some other examples 79 4.5. Solution 83 4.6. Solution of the examples 88 4.7. A few extensions 90 4.8. Conclusion 98 4.9. Bibliography 98 PART III. ELEMENTS FROM MATHEMATICAL PROGRAMMING 101 Chapter 5. Mixed Integer Linear Programming Models for Combinatorial Optimization Problems 103 Frédérico DELLA CROCE 5.1. Introduction 103 5.2. General modeling techniques 111 5.3. More advanced MILP models 117 5.4. Conclusions 132 5.5. Bibliography 133 Chapter 6. Simplex Algorithms for Linear Programming 135 Frédérico DELLA CROCE and Andrea GROSSO 6.1. Introduction 135 6.2. Primal and dual programs 135 6.3. The primal simplex method 140 6.4. Bland’s rule 145 6.5. Simplex methods for the dual problem 147 6.6. Using reduced costs and pseudo-costs for integer programming 152 6.7. Bibliography 155 Chapter 7. A Survey of some Linear Programming Methods 157 Pierre TOLLA 7.1. Introduction 157 7.2. Dantzig’s simplex method 158 7.3. Duality 162 7.4. Khachiyan’s algorithm 162 7.5. Interior methods 165 7.6. Conclusion 186 7.7. Bibliography 187 Chapter 8. Quadratic Optimization in 0–1 Variables 189 Alain BILLIONNET 8.1. Introduction 189 8.2. Pseudo-Boolean functions and set functions 190 8.3. Formalization using pseudo-Boolean functions 191 8.4. Quadratic pseudo-Boolean functions (qpBf) 192 8.5. Integer optimum and continuous optimum of qpBfs 194 8.6. Derandomization 195 8.7. Posiforms and quadratic posiforms 196 8.8. Optimizing a qpBf: special cases and polynomial cases 198 8.9. Reductions, relaxations, linearizations, bound calculation and persistence 200 8.10. Local optimum 206 8.11. Exact algorithms and heuristic methods for optimizing qpBfs 208 8.12. Approximation algorithms 211 8.13. Optimizing a quadratic pseudo-Boolean function with linear constraints 213 8.14. Linearization, convexification and Lagrangian relaxation for optimizing a qpBf with linear constraints 220 8.15. -Approximation algorithms for optimizing a qpBf with linear constraints 223 8.16. Bibliography 224 Chapter 9. Column Generation in Integer Linear Programming 235 Irène LOISEAU, Alberto CESELLI, Nelson MACULAN and Matteo SALANI 9.1. Introduction 235 9.2. A column generation method for a bounded variable linear programming problem 236 9.3. An inequality to eliminate the generation of a 0–1 column 238 9.4. Formulations for an integer linear program 240 9.5. Solving an integer linear program using column generation 243 9.6. Applications 247 9.7. Bibliography 255 Chapter 10. Polyhedral Approaches 261 Ali Ridha MAHJOUB 10.1. Introduction 261 10.2. Polyhedra, faces and facets 265 10.3. Combinatorial optimization and linear programming 276 10.4. Proof techniques 282 10.5. Integer polyhedra and min–max relations 293 10.6. Cutting-plane method 301 10.7. The maximum cut problem 308 10.8. The survivable network design problem 313 10.9. Conclusion 319 10.10. Bibliography 320 Chapter 11. Constraint Programming 325 Claude LE PAPE 11.1. Introduction 325 11.2. Problem definition 327 11.3. Decision operators 328 11.4. Propagation 330 11.5. Heuristics 333 11.6. Conclusion 336 11.7. Bibliography 336 List of Authors 339 Index 343 Summary of Other Volumes in the Series 347

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Book SynopsisCombinatorial optimization is a multidisciplinary scientific area, lying in the interface of three major scientific domains: mathematics, theoretical computer science and management. The three volumes of the Combinatorial Optimization series aims to cover a wide range of topics in this area. These topics also deal with fundamental notions and approaches as with several classical applications of combinatorial optimization. “Paradigms of Combinatorial Optimization” is divided in two parts: • Paradigmatic Problems, that handles several famous combinatorial optimization problems as max cut, min coloring, optimal satisfiability tsp, etc., the study of which has largely contributed to both the development, the legitimization and the establishment of the Combinatorial Optimization as one of the most active actual scientific domains; • Classical and New Approaches, that presents the several methodological approaches that fertilize and are fertilized by Combinatorial optimization such as: Polynomial Approximation, Online Computation, Robustness, etc., and, more recently, Algorithmic Game Theory.Trade Review"Finally, the essay is useful for researchers and scientists in diverse fields (mathematics, programmers, engineers, etc.) as well as post-graduate students (and even undergraduates)." (Contemporary Physics, 19 August 2011) Table of ContentsPreface xvii Vangelis Th. PASCHOS PART I. PARADIGMATIC PROBLEMS 1 Chapter 1. Optimal Satisfiability 3 Cristina BAZGAN Chapter 2. Scheduling Problems 33 Philippe CHRÉTIENNE and Christophe PICOULEAU Chapter 3. Location Problems 61 Aristotelis GIANNAKOS Chapter 4. MiniMax Algorithms and Games 89 Michel KOSKAS Chapter 5. Two-dimensional Bin Packing Problems 107 Andrea LODI, Silvano MARTELLO, Michele MONACI and Daniele VIGO Chapter 6. The Maximum Cut Problem 131 Walid BEN-AMEUR, Ali Ridha MAHJOUB and José NETO Chapter 7. The Traveling Salesman Problem and its Variations 173 Jérôme MONNOT and Sophie TOULOUSE Chapter 8. 0–1 Knapsack Problems 215 Gérard PLATEAU and Anass NAGIH Chapter 9. Integer Quadratic Knapsack Problems 243 Dominique QUADRI, Eric SOUTIF and Pierre TOLLA Chapter 10. Graph Coloring Problems 265 Dominique DE WERRA and Daniel KOBLER PART II. NEW APPROACHES 311 Chapter 11. Polynomial Approximation 313 Marc DEMANGE and Vangelis Th. PASCHOS Chapter 12. Approximation Preserving Reductions 351 Giorgio AUSIELLO and Vangelis Th. PASCHOS Chapter 13. Inapproximability of Combinatorial Optimization Problems 381 Luca TREVISAN Chapter 14. Local Search: Complexity and Approximation 435 Eric ANGEL, Petros CHRISTOPOULOS and Vassilis ZISSIMOPOULOS Chapter 15. On-line Algorithms 473 Giorgio AUSIELLO and Luca BECCHETTI Chapter 16. Polynomial Approximation for Multicriteria Combinatorial Optimization Problems 511 Eric ANGEL, Evripidis BAMPIS and Laurent GOURVÈS Chapter 17. An Introduction to Inverse Combinatorial Problems 547 Marc DEMANGE and Jérôme MONNOT Chapter 18. Probabilistic Combinatorial Optimization 587 Cécile MURAT and Vangelis Th. PASCHOS Chapter 19. Robust Shortest Path Problems 615 Virginie GABREL and Cécile MURAT Chapter 20. Algorithmic Games 641 Aristotelis GIANNAKOS and Vangelis PASCHOS List of Authors 675 Index 681 Summary of Other Volumes in the Series 689

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