Integer and Combinatorial Optimization

New valid inequalities for the fixed-charge and single-node flow polytopes

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Gunluk (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

TheTraveling Salesman Problemor TSP is a fundamental and well‐known problem in combinatorial optimization. At present, the most successful algorithms for solving large‐scale instances of the TSP to proven (near‐)optimality are based on integer programming. This entry introduces the main theoretical and algorithmic tools involved. Topics covered include: formulations of the TSP, polyhedral theory, separation routines, and exact TSP algorithms.

Preface. Part I: Motivation. 1. Linear and Integer Linear Optimization. Part II: Theory. 2. Linear Systems and Projection. 3. Linear Systems and Inverse Projection. 4. Integer Linear Systems: Projection and Inverse Projection. Part III: Algorithms. 5. The Simplex Algorithm. 6. More on Simplex. 7. Interior Point Algorithms: Polyhedral Transformations. 8. Interior Point Algorithms: Barrier Methods. 9. Integer Programming. Part IV: Solving Large Scale Problems: Decomposition Methods. 10. Projection: Benders's Decomposition Methods. 11. Inverse Projection: Dantzig-Wolfe Decomposition. 12. Lagrangian Methods. Part V: Solving Large Scale Problems: Using Special Structure. 13. Sparse Methods. 14. Network Flow Linear Programs. 15. Large Integer Programs: Preprocessing and Cutting Planes. 16. Large Integer Programs: Projection and Inverse Projection. Part VI: Appendix. A. Polyhedral Theory. B. Complexity Theory. C. Basic Graph Theory. D. Software and Test Problems. E. Notation. Bibliography. References. Author Index. Topic Index.

Given an initial marking and final marking for a Petri net model, an optimal firing sequence problem is defined as the problem to find an optimal transition firing sequence to minimize the objective function. For the purpose of analysis of general integer programming problems, we propose a Petri net representation and reachability analysis of integer programming problems. In the proposed method, an integer programming problem is converted into the optimal firing sequence problem of Petri nets. By utilizing the proposed algorithm, integer programming problems can be visualized and analyzed by the Petri net theory. We apply the proposed methodology to the scheduling problems of dual armed cluster tools. The valid inequalities are derived from the reachability analysis. Numerical results show that those valid inequalities can significantly reduce the computational time of the original integer programming problem.

Contributors

Contributors

Supplier selection is a challenging decision that has strategic importance for organizations. Cost is no longer the sole factor in the selection of suppliers, and the complexity of this issue arises from the interplay of several situation-specific criteria (such as total cost, CO2e emissions, development time, lead time) as well as the combinatorial nature of this problem. This paper proposes an approach based on combinatorial optimization (integer linear programming) combined with multi-criteria value analysis to establish priorities and trade-offs among the defined criteria for combinatorial bidding. The approach was employed in a real-world decision, the selection of a supplier for a cosmetics packaging set for a new product line. The obtained solution is compared against standard multi-criteria optimization (without a combinatorial auction formulation) and also against single criterion optimization. The paper also reports on the challenges and advantages of applying the framework in the case study.

A partitioning algorithm for the network loading problem

The Scope of Integer and Combinatorial Optimization

General integer programming (IP) applications and solutions are briefly reviewed in this chapter. Section 4.1 outlines general formulation techniques for IP. Section 4.2 discusses state of the art solutions of general IP problems including classical enumerative and heuristic approaches (ie. simulated annealing). Recent successes in polyhedral approaches to solving partially structured IPs are outlined in section 4.3. Finally the definition and partial structure of the node packing problem (the focus of architectural synthesis) is given in section 4.4. (The notation for a graph is G=(V,E), where V is the set of vertices and E is the set of edges).KeywordsBipartite GraphInteger ProgrammingTravel Salesman ProblemKnapsack ProblemInteger VariableThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preface 1. Clutters 2. T-Cuts and T-Joins 3. Perfect Graphs and Matrices 4. Ideal Matrices 5. Odd Cycles in Graphs 6. 0,+1 Matrices and Integral Polyhedra 7. Signing 0,1 Matrices to Be Totally Unimodular or Balanced 8. Decomposition by k-Sum 9. Decomposition of Balanced Matrices 10. Decomposition of Perfect Graphs Bibliography Index.

In this paper, we propose a new hybrid social spider algorithm with simplex Nelder-Mead method in order to solve integer programming and minimax problems. We call the proposed algorithm a Simplex Social Spider optimization (SSSO) algorithm. In the the proposed SSSO algorithm, we combine the social spider algorithm with its powerful capability of performing exploration, exploitation, and the Nelder-Mead method in order to refine the best obtained solution from the standard social spider algorithm. In order to investigate the general performance of the proposed SSSO algorithm, we test it on 7 integer programming problems and 10 minimax problems and compare against 10 algorithms for solving integer programming problems and 9 algorithms for solving minimax problems. The experiments results show the efficiency of the proposed algorithm and its ability to solve integer and minimax optimization problems in reasonable time.

The constrained shortest path tour problem (CSPTP) is an NP‐hard combinatorial optimization problem defined on a connected directed graph , where V is the set of nodes and A is the set of nonnegative weighted arcs. Given two distinct nodes , an integer value , and node disjoint subsets , , the CSPTP aims at finding the shortest trail from s to t while visiting at least one node in every subset , in this order. In this paper, we perform a comparative analysis between two integer programming (IP) models for the problem. We also propose valid inequalities and a Lagrangian‐based heuristic framework. Branch‐and‐bound algorithms from the literature, as well as a metaheuristic approach, are used for comparison. Extensive computational experiments carried out on benchmark data sets show the effective use of valid inequalities and the quality of bounds obtained by the Lagrangian framework. Because benchmark instances do not require a great computational effort of IP models in the sense that their optimality is reached at the root node of the CPLEX branch‐and‐cut search tree, we introduce new challenging CSPTP instances for which our solution approaches outperform existing ones for the problem.

Induction of number series is a typical task included in intelligence tests. It measures the ability to detect regular patterns and to generalize over them, which is assumed to be crucial for general intelligence. There are some computational approaches to solve number problems. Besides special-purpose algorithms, applicability of general purpose learning algorithms to number series prediction was shown for E-generalization and artificial neural networks (ANN). We present the applicability of the analytical inductive programming system Igor2 to number series problems. An empirical comparison of Igor2 shows that Igor2 has comparable performance on the test series used to evaluate the ANN and the E-generalization approach. Based on findings of a cognitive analysis of number series problems by Holzman et al. (1982, 1983) we conducted a detailed case study, presenting Igor2 with a set of number series problems where the complexity was varied over different dimensions identified as sources of cognitive complexity by Holzman. Our results show that performance times of Igor2 correspond to the cognitive findings for most dimensions.