Book Description

Book Description

Mathematical elegance is a constant theme in this treatment of linear programming and matrix games. Condensed tableaus, minimal in size and notation, are employed for the simplex algorithm. In the context of these tableaus the beautiful termination theorem of RG Bland is proven more simply than heretofore, and the important duality theorem becomes almost obvious. Examples and extensive discussions throughout the book provide insight into definitions, theorems, and applications. There is considerable informal discussion on how best to play matrix games.

The book is designed for a one-semester undergraduate course. Readers will need a degree of mathematical sophistication and general tools such as sets, functions, and summation notation. No single college course is a prerequisite, but most students will do better with some prior college mathematics. This thorough introduction to linear programming and game theory will impart a deep understanding of the material and also increase the student's mathematical maturity. ... Read more

Book Description

Integer Optimization addresses a wide spectrum of practically important optimization problems and represents a major challenge for algorithmics. The goal of integer optimization is to solve a system of constraints and optimization criteria over discrete variables. Integer Optimization by Local Search introduces a new approach to domain-independent integer optimization, which, unlike traditional strategies, is based on local search. It develops the central concepts and strategies of integer local search and describes possible combinations with classical methods from linear programming. The surprising effectiveness of the approach is demonstrated in a variety of case studies on large-scale, realistic problems, including production planning, timetabling, radar surveillance, and sports scheduling. The monograph is written for practitioners and researchers from artificial intelligence and operations research. ... Read more

Book Description

This translation of the important Russian text covers the theory and computational methods of modified Lagrangian functions (MLFs)—a new branch of mathematical programming used to solve optimization problems. Providing a thorough analysis for both traditional convex programming and monotone maps, the book shows the advantages of MLFs over classical Lagrangian functions in such practical applications as numerical algorithms, economic modeling, de-composition, and nonconvex local constrained optimization.

Following an overview of convex analysis, the authors introduce MLFs through the more general formalism of weak modified Lagrangian functions (WMLFs). They use the two concepts to develop a theory of duality supported by examples of elementary economic models. Also examined are the benefits of MLFs in the application of dual methods in linear programming and in problems with inconsistent constraints.

This is the first volume in which mono-tone maps are treated broadly, in line with their growing importance in optimization and mathematical economics. Two chapters on monotone maps cover point-to-set maps, propose modifications that would achieve a point-to-point map with improved properties, show how to arrive at new MLF constructions, and detail decomposition methods for convex programming.

A chapter on the saddle gradient method covers convergence properties exhibited by MLFs—making available convergent algorithms of convex programming. Finally, the book shows how MLFs are used to solve smooth mathematical programming problems, and gives the convergence rate for those dual methods based on MLFs.

For mathematicians involved in discrete math and optimization, and for graduate students taking courses in complex analysis and mathematical programming, Modified Lagrangians and Monotone Maps in Optimization serves as an indispensable professional reference and graduate-level text that goes beyond the classical Lagrange scheme, and offers diverse techniques for tackling this field.

How modified Lagrangian functions improve the classical Lagrange scheme—a unique guide for working out optimization problems

This volume presents the theory and applications of modified Lagrangian functions. It offers here, for the first time, a detailed analysis and numerous techniques for this fast-growing branch of mathematical programming. Focusing on two key areas, traditional convex programming and monotone maps, the book explores a number of practical applications for MLFs and shows how MLFs are especially relevant to traditional convex programming.

For mathematicians and graduate students working with optimization problem analysis, this combined text and reference

• Describes the benefits of MLFs in applications such as numerical algorithms for the general convex programming problem, decomposition, economic modeling, nonconvex local constrained optimization, and more
• Uses the concepts of MLFs and WMLFs (weak modified Lagrangian functions) to develop a theory of duality, and illustrates the analysis with an elementary economic model
• Covers convex programming methods that are based on the iterative solution of dual problems generated by MLFs, showing how the proper choice of an MLF can guarantee the smoothness of the results
• Discusses monotone maps in much more detail than has been done to date in the professional literature, and explains how to use new MLF constructions to solve equations associated with monotone maps
• Considers convergence properties in MLFs, and how they relate to the saddle gradient method and to problem solving in convex programming
• Shows how to solve smooth mathematical programming problems, and includes results that relate to the convergence rate of the dual methods based on MLFs

Book Description

This book serves as an introductory text to optimization theory in normed spaces. Topics of this book are existence results, various differentiability notions together with optimality conditions, the contingent cone, a generalization of the Lagrange multiplier rule, duality theory, and the investigation of linear quadratic and time minimal control problems. This textbook presents fundamentals with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory. The reader is expected to have a basic knowledge of linear functional analysis. The book contains examples and several exercises. ... Read more

Book Description

This book/software package uniquely integrates logic and operations research. Its broad coverage provides concepts, templates, and the tools for the task of attacking difficult problems which are repeatedly encountered in decision making. The first part deals with linear programming and the second with search techniques for combinatorially hard problems. The applications discussed include product mix problems, pattern recognition, classical and probabilistic logic, financial planning, and expert systems. ... Read more

Reviews (1)

A fundamental effort in synthesis of OR,AI, DSS
The book provides a sound yet practical introduction to the intersection of three fields operation research, Artificial Intelligence and Decision Support Sytems. The book is based on effective use of constraint logic programming paradigm. There is an associated software package for 2LP language with models and solutions. 2LP is combination of Linear programming and Logic programming. The power of simplex algorithm is used in theorem prover . The language constructs resemble C++; The book is primarily written to show the power of the language in solving various problems; scheduling, pattern recognition, puzzzle solving, assignment problem, task allocation, time table problem, etc. The book does not the discuss about the programming theory in design of the language. The authors discuss about various theoritical aspects of OR and AI. Each cahpter introduces one concept and its relevance explained with some concrete models. All models and solutions are part of the software package. Very good book. ... Read more

Book Description

This collection of papers by leading researchers in the field of finite, nonlinear elasticity concerns itself with the behavior of objects that deform when external forces or temperature gradients are applied. This process is extremely important in many industrial settings, such as aerospace and rubber industries. This book covers the various aspects of the subject comprehensively with careful explanations of the basic theories and individual chapters each covering a different research direction. The authors discuss the use of symbolic manipulation software as well as computer algorithm issues. The emphasis is placed firmly on covering modern, recent developments, rather than the very theoretical approach often found. The book will be an excellent reference for both beginners and specialists in engineering, applied mathematics and physics. ... Read more

Book Description

The workshop on optimization for design and control was made up of specialists in optimization and practitioners in the fields of aerospace engineering, chemical engineering, and fluid and solid mechanics. The major themes included an assessment of the state of the art in optimization algorithms as well as challenging applications in design and control, in the areas of process engineering and systems with partial differential equation models. The papers in this volume represent a balanced selection from the above application areas as well as contributions that survey the state of the art in relevant areas of nonlinear programming. ... Read more

Book Description

This book provides a self-contained exposition of the theory of linear models, including practical aspects of residuals and data analysis. ... Read more

Book Description

This book offers a comprehensive treatment of linear programming as well as of the optimization of linear functions over polyhedra in finite dimensional Euclidean vector spaces. An introduction surveying fifty years of linear optimization is given. The book can serve both as a graduate textbook for linear programming and as a text for advanced topics classes or seminars. Exercises as well as several case studies are included. The book is based on the author's long term experience in teaching and research. For his research work he has received, among other honors, the 1983 Lanchester Prize of the Operations Research Society of America, the 1985 Dantzig Prize of the Mathematical Programming Society and the Society for Industrial Applied Mathematics and a 1989 Alexander-von-Humboldt Senior U.S. Scientist Research Award. ... Read more

Reviews (2)

Computational and Mathematical Excellence
For nearly 30 years, Padberg has been a leader in computational integer programming and in combinatorial optimization theory,
and this book is recommended to serious graduate students (and their professors).

In practice, Padberg has helped to design and implement "branch-and-cut" methods for finding exact optimal solutions to large traveling salesman problems, and this approach is a method of choice for finding approximately optimal solutions to tough industrial problems.The book provides the mathematical and computational background for understanding branch-and-cut; the established mathematical texts by Nemhauser and Wolsey and by Schrijver are less detailed and more condensed, and omit numerical issues.The treatment of modern simplex algorithms for linear programming---updating LU factorizations and using column- and constraint-generation and -purging---is excellent, and a large bibliography contains recent references.Besides industrial and Berlin-airlift scheduling problems, the book contains TSP examples of circuit-board wiring, U.S. state capitals, and Odysseus!

Three more highlights: The double description algorithm receives a complete description, and this is useful for combinatorial geometers.The discussion of integer-arithmetic and complexity theory is very readable, and these technical topics are slighted by interior-point books (besides Wright's quickie), despite their importance in integer programming and combinatorial optimization. The discussion of interior-point algorithms emphasizes projective geometry, a beautiful theory that has inspired so much of optimization theory---besides Karmarkar's interior-point algorithm,Dantzig's simplex algorithm, Fenchel duality, Davidon's conic algorithm for nonlinear optimization, etc.).

The book is not a comprehensive survey of linear programming,
and lacks a treatment of Nesterov's theory of self-concordant barrier-functions.Also, no treatment is given of pivoting algorithms besides Dantzig's (e.g., Terlaky's criss-cross method, Todd's oriented matroid algorithm).

A good reference for Linear Programming Theory
This book is certainly a very good reference for theoretical topics of linear programming. It covers the Simplex method and the Ellipsoid algorithms. It also covers the geometry of linear programming (polyhedraand polytopes, etc). It certainly covers more topics than most other linearprogramming texts.As expected, a book writen for theoretical topics iscertainly not easy to read, especially for people with no training in doingrigorous mathematical proofs. Also, not many examples or illustrations aregiven in this book, and this might be a problem for some readers. ... Read more

Book Description

Results from the now-classical distribution theory involving convolution and Fourier transformation are extended to cater for Colombeau's generalized functions. Indications are given how these particular generalized functions can be used to investigate linear equations and pseudo differential operators. Furthermore, applications are also given to problems with nonregular data. ... Read more

Book Description

Hardbound. The Haifa 2000 Workshop on "Inherently Parallel Algorithms for Feasibility and Optimization and their Applications" brought together top scientists in this area.The objective of the Workshop was to discuss, analyze and compare the latest developments in this fast growing field of applied mathematics and to identify topics of research which are of special interest for industrial applications and for further theoretical study.

Inherently parallel algorithms, that is, computational methods which are, by their mathematical nature, parallel, have been studied in various contexts for more than fifty years.However, it was only during the last decade that they have mostly proved their practical usefulness because new generations of computers made their implementation possible in order to solve complex feasibility and optimization problems involving huge amounts of data via parallel processing.These led to an accumulation of computational expe ... Read more

Book Description

Over the past thirty years significant progress has been made in the investigation of nonlinear waves--including "soliton equations", a class of nonlinear wave equations that arise frequently in such areas as nonlinear optics, fluid dynamics, and statistical physics. The broad interest in this field can be traced to understanding "solitons" and the associated development of a method of solution termed the inverse scattering transform (IST). The IST technique applies to continuous and discrete nonlinear Schrödinger (NLS) equations of scalar and vector type. This work presents a detailed mathematical study of the scattering theory, offers soliton solutions, and analyzes both scalar and vector soliton interactions. The authors provide advanced students and researchers with a thorough and self-contained presentation of the IST as applied to nonlinear Schrödinger systems. ... Read more

Book Description

The book proposes concepts and a general framework for computer-based modeling. It puts forward a modeling language as a kernel representation for mathematical models. It explores fundamental features of models and defines the notion of mathematical model and other related concepts. It gives a comprehensive overview of the modeling life cycle. The most frequently used methodologies of modeling management systems actually available are reviewed and a new framework in computer-based modeling is proposed. The book not only gives a theoretical foundation of modeling, but presents a concrete implementation using the modeling language LPL. It includes many concrete applications. All models and the complete software can be downloaded from the Web free of charge.Audience: This book is intended for modeling tool designers, as well as students and teachers in mathematical modeling, and for real-live model `practitioners'. ... Read more

Book Description

This book provides the fundamentals of dynamic optimization which can be used to improve the performance of engineering systems. Most results are derived using the theory of calculus of variations. The methods are illustrated by a number of examples. Numerical implementation of the theory by direct and indirect methods is also described. Computer programs are provided that can be used to work out engineering problems. The book also introduces some new results in dynamic optimization using higher-order approaches. The book is appropriate for undergraduate seniors or first-year graduate students. It should also be of interest to professionals in the areas of automotive, aerospace, or manufacturing engineering. Students and researchers in the areas of systems theory, control, and optimization will benefit from the relatively simple exposition in this book. ... Read more

Book Description

This is a teaching text for the advanced statistics undergraduate or the beginning graduate student of statistics. It is assumed that the user of the text has had at least a full year course in applied or mathematical statistics. The text is intended for a one semester introductory course in the theory of linear statistical models. ... Read more

Book Description

This book deals with decision making in environments of significant data uncertainty, with particular emphasis on operations and production management applications. For such environments, we suggest the use of the robustness approach to decision making, which assumes inadequate knowledge of the decision maker about the random state of nature and develops a decision that hedges against the worst contingency that may arise. Robust Discrete Optimization is a comprehensive mathematical programming framework for robust decision making. This book takes a giant first step in presenting decision support tools and solution methods for generating robust decisions in a variety of interesting application environments such as: linear programming, assignment problems, shortest paths, minimum spanning trees, knapsack problems, resource allocation, scheduling, production planning, location, inventory, layout planning, network design, and international sourcing. Beyond theoretical results, the book provides many suggestions and useful advice to the practitioners of the robustness approach. Emphasis is placed upon the assessment of the decision environment for applicability of the approach, structuring of data uncertainty and the scenario generation process, choice of appropriate robustness criteria, and formulation and solution of robust decision problems.Audience: The book will be of interest to researchers, practitioners and graduate students working in the fields of operations research, management science, industrial and systems engineering, computer science, decision analysis and applied mathematics. ... Read more

Book Description

This monograph focuses on a class of problems that in effect have yet to be solved. Location, scheduling and design problems are assignment type problems with quadratic cost functions and occur in many contexts. Applications of these problems abound in diverse disciplines, such as anthropology, archeology, architecture, chemistry, computer science, economics, electronics, ergonomics, marketing, operations management, political science, statistical physics, zoology, etc.Padberg and Rijal have taken an important step in the solution of these problems. In this monograph they classify mathematical properties for ten classes of assignment problems: Quadratic Assignment Problems, Traveling Salesman Problems, Triangulation Problems, Linear Assignment Problems, VLSI Circuit Layout Design Problems, Multi-Processor Problems, Scheduling Problems with Interaction Costs, Operation-Scheduling Problems, Graph and Clique Partitioning Problems, and Boolean Quadratic Problems. They note that before these problems can be solved computationally, one must know and understand their mathematical properties. After discussing these properties, an integer programming approach is offered for solving them. The computational approach has shown considerable algorithmic success. The heart of this monograph is the theoretical work on assignment problems and the computation results that were produced using algorithms developed at NYU.The authors conclude that implementing a proper branch-and-cut algorithm on these types of problems will push the limits of exact computation far beyond the current ones. ... Read more

Book Description

In many decision processes there is an hierarchy of decision-makers and decisions are taken at different levels in this hierarchy. Multilevel programming focuses on the whole hierarchy structure. In terms of modeling, the constraint domain associated with a multilevel programming problem is implicitly determined by a series of optimization problems which must be solved in a predetermined sequence. The field of multilevel optimization has become a well-known and important research field. Hierarchical structures can be found in scientific disciplines such as environment, ecology, biology, chemical engineering, mechanics, classification theory, databases, network design, transportation, game theory and economics. Moreover, new applications are constantly being introduced. This has stimulated the development of new theory and efficient algorithms. This volume contains 16 chapters written by various leading researchers and presents a cohesive authoritative overview of developments and applications in their emerging field of optimization.Audience: Researchers whose work involves the application of mathematical programming and optimization to hierarchical structures. ... Read more