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Algebraic and Geometric Ideas in the Theory of Discrete Optimization
In recent years, many new techniques have emerged in the mathematical theory of discrete optimization that have proven to be effective in solving a number of hard problems. This book presents these recent advances, particularly those that arise from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside of the standard curriculum in optimization. These new techniques, all of which are presented with minimal prerequisites, provide a transition from linear to nonlinear discrete optimization. This book can be used as a textbook for advanced undergraduates or first-year graduate students in mathematics, computer science or operations research. It is also appropriate for mathematicians, engineers, and scientists engaged in computation who wish to gain a deeper understanding of how and why algorithms work.
Triangulations
Triangulations presents the first comprehensive treatment of the theory of secondary polytopes and related topics. The text discusses the geometric structure behind the algorithms and shows new emerging applications, including hundreds of illustrations, examples, and exercises.
Discrete Geometry and Algebraic Combinatorics
This volume contains the proceedings of the AMS Special Session on Discrete Geometry and Algebraic Combinatorics held on January 11, 2013, in San Diego, California. The collection of articles in this volume is devoted to packings of metric spaces and related questions, and contains new results as well as surveys of some areas of discrete geometry. This volume consists of papers on combinatorics of transportation polytopes, including results on the diameter of graphs of such polytopes; the generalized Steiner problem and related topics of the minimal fillings theory; a survey of distance graphs and graphs of diameters, and a group of papers on applications of algebraic combinatorics to packings of metric spaces including sphere packings and topics in coding theory. In particular, this volume presents a new approach to duality in sphere packing based on the Poisson summation formula, applications of semidefinite programming to spherical codes and equiangular lines, new results in list decoding of a family of algebraic codes, and constructions of bent and semi-bent functions.
Computing the Continuous Discretely
This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summat...
An Introduction to Convexity, Optimization, and Algorithms
This concise, self-contained volume introduces convex analysis and optimization algorithms, with an emphasis on bridging the two areas. It explores cutting-edge algorithms—such as the proximal gradient, Douglas–Rachford, Peaceman–Rachford, and FISTA—that have applications in machine learning, signal processing, image reconstruction, and other fields. An Introduction to Convexity, Optimization, and Algorithms contains algorithms illustrated by Julia examples and more than 200 exercises that enhance the reader’s understanding of the topic. Clear explanations and step-by-step algorithmic descriptions facilitate self-study for individuals looking to enhance their expertise in convex analysis and optimization. Designed for courses in convex analysis, numerical optimization, and related subjects, this volume is intended for undergraduate and graduate students in mathematics, computer science, and engineering. Its concise length makes it ideal for a one-semester course. Researchers and professionals in applied areas, such as data science and machine learning, will find insights relevant to their work.
Introduction to Nonlinear Optimization
Built on the framework of the successful first edition, this book serves as a modern introduction to the field of optimization. The author’s objective is to provide the foundations of theory and algorithms of nonlinear optimization as well as to present a variety of applications from diverse areas of applied sciences. Introduction to Nonlinear Optimization gradually yet rigorously builds connections between theory, algorithms, applications, and actual implementation. The book contains several topics not typically included in optimization books, such as optimality conditions in sparsity constrained optimization, hidden convexity, and total least squares. Readers will discover a wide array o...
Computing and Combinatorics
The book is aimed at graduate students, researchers, engineers and physicists involved in fluid computations. An up-to-date account is given of the present state of the art of numerical methods employed in computational fluid dynamics. The underlying numerical principles are treated with a fair amount of detail, using elementary methods. Attention is given to the difficulties arising from geometric complexity of the flow domain. Uniform accuracy for singular perturbation problems is studied, pointing the way to accurate computation of flows at high Reynolds number. Unified methods for compressible and incompressible flows are discussed. A treatment of the shallow-water equations is included. A basic introduction is given to efficient iterative solution methods. Many pointers are given to the current literature, facilitating further study.
Evaluation Complexity of Algorithms for Nonconvex Optimization
A popular way to assess the “effort” needed to solve a problem is to count how many evaluations of the problem functions (and their derivatives) are required. In many cases, this is often the dominating computational cost. Given an optimization problem satisfying reasonable assumptions—and given access to problem-function values and derivatives of various degrees—how many evaluations might be required to approximately solve the problem? Evaluation Complexity of Algorithms for Nonconvex Optimization: Theory, Computation, and Perspectives addresses this question for nonconvex optimization problems, those that may have local minimizers and appear most often in practice. This is the firs...
Handbook of Satisfiability
Propositional logic has been recognized throughout the centuries as one of the cornerstones of reasoning in philosophy and mathematics. Over time, its formalization into Boolean algebra was accompanied by the recognition that a wide range of combinatorial problems can be expressed as propositional satisfiability (SAT) problems. Because of this dual role, SAT developed into a mature, multi-faceted scientific discipline, and from the earliest days of computing a search was underway to discover how to solve SAT problems in an automated fashion. This book, the Handbook of Satisfiability, is the second, updated and revised edition of the book first published in 2009 under the same name. The handb...
Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition
This second edition provides an enhanced exposition of the long-overlooked Hadamard semidifferential calculus, first introduced in the 1920s by mathematicians Jacques Hadamard and Maurice René Fréchet. Hadamard semidifferential calculus is possibly the largest family of nondifferentiable functions that retains all the features of classical differential calculus, including the chain rule, making it a natural framework for initiating a large audience of undergraduates and non-mathematicians into the world of nondifferentiable optimization. Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition builds upon its prior editions foundations in Hadamard semidifferent...
