Network Optimization: Continuous and Discrete Models

Preface:

Network optimization lies in the middle of the great divide that separates the two major types of optimization problems, continuous and discrete. The ties between linear programming and combinatorial optimization can be traced to the representation of the constraint polyhedron as the convex hull of its extreme points. When a network is involved, however, these ties become much stronger because the extreme points of the polyhedron are integer and represent solutions of combinatorial problems that are seemingly unrelated to linear programming. Because of this structure and also because of their intuitive character, network models provide ideal vehicles for explaining many of the fundamental ideas in both continuous and discrete optimization.

Aside from their interesting methodological characteristics, network models are also used extensively in practice, in an ever expanding spectrum of applications. Indeed collectively, network problems such as shortest path, assignment, max-flow, transportation, transhipment, spanning tree, matching, traveling salesman, generalized assignment, vehicle routing, and multicommodity flow constitute the most common class of practical optimization problems. There has been steady progress in the solution methodology of network problems, and in fact the progress has accelerated in the last fifteen years thanks to algorithmic and technological advances.

The purpose of this book is to provide a fairly comprehensive and up-to-date development of linear, nonlinear, and discrete network optimization problems. The interplay between continuous and discrete structures has been highlighted, the associated analytical and algorithmic issues have been treated quite extensively, and a guide to important network models and applications has been provided.

Regarding continuous network optimization, we focus on two ideas, which are also fundamental in general mathematical programming: {\it duality} and {\it iterative cost improvement\/}. We provide an extensive treatment of iterative algorithms for the most common linear cost problem, the minimum cost flow or transhipment problem, and for its convex cost extensions. The discussion of duality is comprehensive: it starts with linear network programming duality, and culminates with Rockafellar's development of monotropic programming duality.

Regarding discrete network optimization, we illustrate problem formulation through major paradigms such as traveling salesman, generalized assignment, spanning tree, matching, and routing. This is essential because the structure of discrete optimization problems is far less streamlined than the structure of their continuous counterparts, and familiarity with important types of problems is important for modeling, analysis, and algorithmic solution. We also develop the main algorithmic approaches, including branch-and-bound, Lagrangian relaxation, Dantzig-Wolfe decomposition, heuristics, and local search methods.

This is meant to be an introductory book that covers a very broad variety of topics. It is thus inevitable that some topics have been treated in less detail than others. The choices made reflect in part personal taste and expertise, and in part a preference for simple models that can help most effectively the reader develop insight. At the same time, our analysis and presentation aims to enhance the reader's mathematical modeling ability in two ways: by delineating the range of problems for which various algorithms are applicable and efficient, and by providing many examples of problem formulation.

The chapter-by-chapter description of the book follows:

**Chapter 1:** This is an introductory chapter that establishes
terminology and basic notions about graphs, discusses some examples of network
models, and provides some orientation regarding linear network optimization
algorithms.

**Chapter 2:** This chapter provides an extensive treatment of
shortest path problems. It covers the major methods, and discusses their
theoretical and practical performance.

**Chapter 3:** This chapter focuses on the max-flow problem and
develops the class of augmenting path algorithms for its solution. In addition to
the classical variants of the Ford-Fulkerson method, a new algorithm based on
auction ideas is discussed.

**Chapter 4:** The minimum cost flow problem (linear cost, single
commodity, no side constraints) and its equivalent variants are introduced here.
Subsequently, the basic duality theory for the problem is developed and
interpreted.

**Chapter 5:** This chapter focuses on simplex methods for the
minimum cost flow problem. The basic results regarding integrality of solutions
are developed here constructively, using the simplex method.

**Chapter 6:** This chapter develops dual ascent methods, including
primal-dual, sequential shortest path, and relaxation methods.

**Chapter 7:** This chapter starts with the auction algorithm
for the assignment problem, and proceeds to show how this algorithm can be extended to
more complex problems. In this way, preflow-push methods for the max-flow problem and
the $\e$-relaxation method for the minimum cost flow problem are obtained. Several
additional variants of auction algorithms are developed.

**Chapter 8:** This is an important chapter that marks the transition
from linear to nonlinear network optimization. Both continuous (convex) and
discrete (integer) problems are discussed with an emphasis on the structure that
ties them to linear problems. This chapter also provides an introduction to the
broad spectrum of nonlinear and combinatorial network methodology, and sets the
stage for the remaining two chapters.

**Chapter 9:** This chapter deals with separable convex problems,
discusses their connection with classical network equilibrium problems, and
develops their rich theoretical structure. The salient features of this structure
are a particularly sharp duality theory, and a combinatorial connection of
descent directions with the finite set of elementary vectors of the subspace
defined by the conservation of flow constraints. Besides treating convex
separable network problems, this chapter provides an introduction to monotropic
programming, which is the largest class of nonlinear programming problems that
possess the strong duality and combinatorial properties of linear programs. This
chapter also develops auction algorithms for convex separable problems and
provides an analysis of their running time.

**Chapter 10:** This chapter deals with the basic methodological
approaches of integer-constrained problems. We treat exact methods such as
branch-and-bound, and the associated methods of Lagrangian relaxation, subgradient
optimization, cutting plane, and decomposition. We also describe
approximate methods based on local search, such as genetic algorithms, tabu search, and
simulated annealing. Finally, we discuss rollout algorithms, a relatively new and broadly
applicable class of approximate methods, which can be used in place of, or in conjunction
with local search.

The book can be used for a course on network optimization or for part of a course on introductory optimization at the first-year graduate level. The prerequisites are fairly elementary. The main one is a certain degree of mathematical maturity, as provided for example by a rigorous mathematics course beyond the calculus level. One may cover most of the book in a course on linear and nonlinear network optimization. A shorter version of this course may consist of Chapters 1-5, and 8. Alternatively, one may teach a course that focuses on linear and discrete network optimization, using Chapters 1-5, a small part of Chapter 8, and Chapter 10.

The book contains a large number of examples and exercises, which should enhance its suitability for classroom instruction. Some of the exercises are theoretical in nature and supplement substantially the main text. Solutions to a subset of these (as well errata and additional material) will be posted and periodically updated on the book's web page.

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