https://leeds-faculty.colorado.edu/glover/TS%20-%20Future%20Paths%20for%20Integer%20Programming.pdf

Future paths for integer programming and links to artificial intelligence

A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.

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Metaheuristics: From Design to Implementation

Integer optimization problems are concerned with the efficient allocation of limited resources to meet a desired objective when some of the resources in question can only be divided into discrete parts. In such cases, the divisibility constraints on these resources, which may be people, machines, or other discrete inputs, may restrict the possible alternatives to a finite set. Nevertheless, there are usually too many alternatives to make complete enumeration a viable option for instances of realistic size. For example, an airline may need to determine crew schedules that minimize the total operating cost; an automotive manufacturer may want to determine the optimal mix of models to produce in order to maximize profit; or a flexible manufacturing facility may want to schedule production for a plant without knowing precisely what parts will be needed in future periods. In today’s changing and competitive industrial environment, the difference between ad hoc planning methods and those that use sophisticated mathematical models to determine an optimal course of action can determine whether or not a company survives.

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Integer and Combinatorial Optimization

This is a survey of "Iterated Local Search", a general purpose metaheuristic for finding good solutions of combinatorial optimization problems. It is based on building a sequence of (locally optimal) solutions by: (1) perturbing the current solution; (2) applying local search to that modified solution. At a high level, the method is simple, yet it allows for a detailed use of problem-specific properties. After giving a general framework, we cover the uses of Iterated Local Search on a number of well studied problems.

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Iterated local search

JuMP is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. JuMP takes advantage of advanced features of the Julia programming language to offer unique functionality while achieving performance on par with commercial modeling tools for standard tasks. In this work we will provide benchmarks, present the novel aspects of the implementation, and discuss how JuMP can be extended to new problem classes and composed with state-of-the-art tools for visualization and interactivity.

JuMP: A Modeling Language for Mathematical Optimization

This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home to where you began. Though seemingly modest, this exercise has inspired studies by mathematicians, chemists, and physicists. Teachers use it in the classroom. It has practical applications in genetics, telecommunications, and neuroscience. The authors of this book are the same pioneers who for nearly two decades have led the investigation into the traveling salesman problem. They have derived solutions to almost eighty-six thousand cities, yet a general solution to the problem has yet to be discovered. Here they describe the method and computer code they used to solve a broad range of large-scale problems, and along the way they demonstrate the interplay of applied mathematics with increasingly powerful computing platforms. They also give the fascinating history of the problem--how it developed, and why it continues to intrigue us.

The Traveling Salesman Problem: A Computational Study

Following the theoretical studies of J.B. Robinson and H.W. Kuhn in the late 1940s and the early 1950s, G.B. Dantzig, R. Fulk- erson, and S.M. Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hun- dred cities; over the years, it has evolved further through the work of M. Grotschel, S. Hong, M. Junger, P. Miliotis, D. Naddef, M. Padberg, W.R. Pulleyblank, G. Reinelt, G. Rinaldi, and others. We enumerate some of its refinements that led to the solution of a 13,509-city instance. 1991 Mathematics Subject Classification: 90C10 90C27 Keywords and Phrases: traveling salesman; cutting planes The traveling salesman problem, or TSP for short, is easy to state: given a finite number of "cities" along with the cost of travel between each pair of them, find the cheapest way of visiting all of the cities and returning to your starting point. The travel costs are symmetric in the sense that traveling from city X to city Y costs just as much as traveling from Y to X; the "way of visiting all the cities" is simply the order in which the cities are visited. The simplicity of this problem, coupled with its apparent intractability, makes it an ideal platform for exploring new algorithmic ideas. Surveys of work on the TSP can be found in Bellmore and Nemhauser (1968), Lawler, Lenstra, Rinnooy Kan, and Shmoys (1985), Reinelt (1994), and Junger, Reinelt, and Rinaldi (1995). The origins of the TSP are obscure. In the 1920's, the mathematician and economist Karl Menger publicized it among his colleagues in Vienna. In the 1930's, the problem reappeared in the mathematical circles of Princeton. In the 1940's, it was studied by statisticians (Mahalanobis (1940), Jessen (1942)) in connection with an agricultural application and the mathematician Merrill Flood popularized it among his colleagues at the RAND Corporation. Eventually, the TSP gained notoriety as the prototype of a hard problem in combinatorial optimization. A breakthrough came when Dantzig, Fulkerson, and Johnson (1954) published a description of a method for solving the TSP and illustrated the power of this method by solving an instance with 49 cities, an impressive size at that time. Riding the wave of excitement over the numerous applications of the simplex

On the Solution of Traveling Salesman Problems

A simple theorem on 3-connectivity

We describe an algorithm which converts a linear program min{cx ∣ Ax = b, x ≥ 0} to a network flow problem, using elementary row operations and nonzero variable-scaling, or shows that such a conversion is impossible. If A is in standard form, the computational effort required is bounded by Orn, where r is the number of rows and n is the number of nonzero entries of A. A method for determining whether a “binary matroid” is “graphic” plays an important role in the algorithm.

Robert E. Bixby

Papers

The Traveling Salesman Problem: A Computational Study

On the Solution of Traveling Salesman Problems

A simple theorem on 3-connectivity

Converting Linear Programs to Network Problems

On Reid's characterization of the ternary matroids