Combinatorial and Algorithmic Mathematics - Baha Alzalg

. .

Résumé :

Detailed review of optimization from first principles, supported by rigorous math and computer science explanations and various learning aids

Supported by rigorous math and computer science foundations, Combinatorial and Algorithmic Mathematics: From Foundation to Optimization provides a from-scratch understanding to the field of optimization, discussing 70 algorithms with roughly 220 illustrative examples, 160 nontrivial end-of-chapter exercises with complete solutions to ensure readers can apply appropriate theories, principles, and concepts when required, and Matlab codes that solve some specific problems. This book helps readers to develop mathematical maturity, including skills such as handling increasingly abstract ideas, recognizing mathematical patterns, and generalizing from specific examples to broad concepts.

Starting from first principles of mathematical logic, set-theoretic structures, and analytic and algebraic structures, this book covers both combinatorics and algorithms in separate sections, then brings the material together in a final section on optimization. This book focuses on topics essential for anyone wanting to develop and apply their understanding of optimization to areas such as data structures, algorithms, artificial intelligence, machine learning, data science, computer systems, networks, and computer security.

Combinatorial and Algorithmic Mathematics includes discussion on:

• Propositional logic and predicate logic, set-theoretic structures such as sets, relations, and functions, and basic analytic and algebraic structures such as sequences, series, subspaces, convex structures, and polyhedra
• Recurrence-solving techniques, counting methods, permutations, combinations, arrangements of objects and sets, and graph basics and properties
• Asymptotic notations, techniques for analyzing algorithms, and computational complexity of various algorithms
• Linear optimization and its geometry and duality, simplex and non-simplex algorithms for linear optimization, second-order cone programming, and semidefinite programming

Combinatorial and Algorithmic Mathematics is an ideal textbook resource on the subject for students studying discrete structures, combinatorics, algorithms, and optimization. It also caters to scientists across diverse disciplines that incorporate algorithms and academics and researchers who wish to better understand some modern optimization methodologies....

Biographie:

Baha Alzalg is a Professor in the Department of Mathematics at the University of Jordan in Amman, Jordan. He has also held the post of visiting associate professor in the Department of Computer Science and Engineering at the Ohio State University in Columbus, Ohio. His research interests include topics in optimization theory, applications, and algorithms, with an emphasis on interior-point methods for cone programming....

Sommaire:

About the Author xiii

Preface xv

Acknowledgments xvii

About the Companion Website xxi

Part I Foundations 1

1 Mathematical Logic 3

1.1 Propositions 3

1.2 Logical Operators 6

1.3 Propositional Formulas 15

1.4 Logical Normal Forms 24

1.5 The Boolean Satisfiability Problem 29

1.6 Predicates and Quantifiers 30

1.7 Symbolizing Statements of the Form All P Are Q 37

2 Set-Theoretic Structures 51

2.1 Induction 51

2.2 Sets 54

2.3 Relations 59

2.4 Partitions 64

2.5 Functions 65

3 Analytic and Algebraic Structures 77

3.1 Sequences 77

3.2 Summations and Series 81

3.3 Matrices, Subspaces, and Bases 87

3.4 Convexity, Polyhedra, and Cones 91

3.5 Farkas' Lemma and Its Variants 95

Part II Combinatorics 103

4 Graphs105

4.1 Basic Graph Definitions 106

4.2 Isomorphism and Properties of Graphs 113

4.3 Eulerian and Hamiltonian Graphs 118

4.4 Graph Coloring 122

4.5 Directed Graphs 125

5 Recurrences 133

5.1 Guess-and-Confirm 133

5.2 Recursion-Iteration 136

5.3 Generating Functions 138

5.4 Recursion-Tree 140

6 Counting149

6.1 Binomial Coefficients and Identities 149

6.2 Fundamental Principles of Counting 154

6.3 The Pigeonhole Principle 161

6.4 Permutations 163

6.5 Combinations 166

Part III Algorithms 179

7 Analysis of Algorithms 181

7.1 Constructing and Comparing Algorithms 182

7.2 Running Time of Algorithms 189

7.3 Asymptotic Notation 199

7.4 Analyzing Decision-Making Statements 211

7.5 Analyzing ProgramsWithout Function Calls 213

7.6 Analyzing Programs with Function Calls 219

7.7 The Complexity Class NP-Complete 224

8 Array and Numeric Algorithms 241

8.1 Array Multiplication Algorithms 241

8.2 Array Searching Algorithms 244

8.3 Array Sorting Algorithms 248

8.4 Euclid's Algorithm 253

8.5 Newton's Method Algorithm 255

9 Elementary Combinatorial Algorithms 267

9.1 Graph Representations 267

9.2 Breadth-First Search Algorithm 270

9.3 Applications of Breadth-First Search 273

9.4 Depth-First Search Algorithm 277

9.5 Applications of Depth-First Search 279

9.6 Topological Sort 283

Part IV Optimization 293

10 Linear Programming 295

10.1 Linear Programming Formulation and Examples 296

10.2 The Graphical Method 302

10.3 Standard Form Linear Programs 309

10.4 Geometry of Linear Programming 311

10.5 The Simplex Method 320

10.6 Duality in Linear Programming 339

10.7 A Homogeneous Interior-Point Method 347

11 Second-Order Cone Programming 363

11.1 The Second-Order Cone and Its Algebraic Structure 363

11.2 Second-Order Cone Programming Formulation 368

11.3 Applications in Engineering and Finance 370

11.4 Duality in Second-Order Cone Programming 375

11.5 A Primal-Dual Path-Following Algorithm 379

11.6 A Homogeneous Self-Dual Algorithm 386

12 Semidefinite Programming and Combinatorial Optimization 395

12.1 The Cone of Positive Semidefinite Matrices 395

12.2 Semidefinite Programming Formulation 399

12.3 Applications in Combinatorial Optimization 401

12.4 Duality in Semidefinite Programming 405

12.5 A Primal-Dual Path-Following Algorithm 408

Exercises 417

Notes and Sources 418

References 418

Appendix A Solutions to Chapter Exercises 421...