Linear Optimization over Permutation Groups

For a permutation group given by a set of generators, the problem of finding "special" group members is NP-hard in many cases. E.g., this is true for the problem of finding a permutation with a minimum number of fixed points or a permutation with a minimal Hamming distance from a given permutation. Many of these problems can be modeled as linear optimization problems over permutation groups. We develop a polyhedral approach to this general problem and derive an exact and practically fast algorithm based on the branch&cut-technique

A branch-and-cut approach to the crossing number problem

The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph in the plane. Extensive research has produced bounds on the crossing number and exact formulae for special graph classes, yet the crossing numbers of graphs such as K_{11} or K_{9,11} are still unknown. Finding the crossing number is NP-hard for general graphs and no practical algorithm for its computation has been published so far. We present an integer linear programming formulation that is based on a reduction of the general problem to a restricted version of the crossing number problem in which each edge may be crossed at most once. We also present cutting plane generation heuristics and a column generation scheme. As we demonstrate in a computational study, a branch-and-cut algorithm based on these techniques as well as recently published preprocessing algorithms can be used to successfully compute the crossing number for small to medium sized general graphs

Enhancement of the bond portfolio Immunization under a parallel shift of the yield curve

Abstract Hedging under a parallel shift of the interest rate curve is well-known for a long date in finance literature. It is based on the use of a duration-convexity approximation essentially pioneered by . However the situation is inaccurately formulated such that the obtained result is very questionable. Motivations and enhancement of such approximation have been performed in our recent working paper JEL classification numbers: G11, G12

The ABACUS System for Branch-and-Cut-and-Price Algorithms in Integer Programming and Combinatorial Optimization

The development of new mathematical theory and its application in software systems for the solution of hard optimization problems have a long tradition in mathematical programming. In this tradition we implemented ABACUS, an object-oriented software framework for branch-and-cut-and-price algorithms for the solution of mixed integer and combinatorial optimization problems. This paper discusses some difficulties in the implementation of branch-and-cut-and-price algorithms for combinatorial optimization problems and shows how they are managed by ABACUS

Problems, Models and Algorithms in One- and Two-Dimensional Cutting

Within such disciplines as Management Science, Information and Computer Science, Engineering, Mathematics and Operations Research, problems of cutting and packing (C&P) of concrete and abstract objects appear under various specifications (cutting problems, knapsack problems, container and vehicle loading problems, pallet loading, bin packing, assembly line balancing, capital budgeting, changing coins, etc.), although they all have essentially the same logical structure. In cutting problems, a large object must be divided into smaller pieces; in packing problems, small items must be combined to large objects. Most of these problems are NP-hard. Since the pioneer work of L.V. Kantorovich in 1939, which first appeared in the West in 1960, there has been a steadily growing number of contributions in this research area. In 1961, P. Gilmore and R. Gomory presented a linear programming relaxation of the one-dimensional cutting stock problem. The best-performing algorithms today are based on their relaxation. It was, however, more than three decades before the first `optimum? algorithms appeared in the literature and they even proved to perform better than heuristics. They were of two main kinds: enumerative algorithms working by separation of the feasible set and cutting plane algorithms which cut off infeasible solutions. For many other combinatorial problems, these two approaches have been successfully combined. In this thesis we do it for one-dimensional stock cutting and two-dimensional two-stage constrained cutting. For the two-dimensional problem, the combined scheme provides mostly better solutions than other methods, especially on large-scale instances, in little time. For the one-dimensional problem, the integration of cuts into the enumerative scheme improves the results of the latter only in exceptional cases. While the main optimization goal is to minimize material input or trim loss (waste), in a real-life cutting process there are some further criteria, e.g., the number of different cutting patterns (setups) and open stacks. Some new methods and models are proposed. Then, an approach combining both objectives will be presented, to our knowledge, for the first time. We believe this approach will be highly relevant for industry

Optimised search heuristics: combining metaheuristics and exact methods to solve scheduling problems

Tese dout., Matemática, Investigação Operacional, Universidade do Algarve, 2009Scheduling problems have many real life applications, from automotive industry to air traffic control. These problems are defined by the need of processing a set of jobs on a shared set of resources. For most scheduling problems there is no known deterministic procedure that can solve them in polynomial time. This is the reason why researchers study methods that can provide a good solution in a reasonable amount of time. Much attention was given to the mathematical formulation of scheduling problems and the algebraic characterisation of the space of feasible solutions when exact algorithms were being developed; but exact methods proved inefficient to solve real sized instances. Local search based heuristics were developed that managed to quickly find good solutions, starting from feasible solutions produced by constructive heuristics. Local search algorithms have the disadvantage of stopping at the first local optimum they find when searching the feasible region. Research evolved to the design of metaheuristics, procedures that guide the search beyond the entrapment of local optima. Recently a new class of hybrid procedures, that combine local search based (meta) heuristics and exact algorithms of the operations research field, have been designed to find solutions for combinatorial optimisation problems, scheduling problems included. In this thesis we study the algebraic structure of scheduling problems; we address the existent hybrid procedures that combine exact methods with metaheuristics and produce a mapping of type of combination versus application and finally we develop new innovative metaheuristics and apply them to solve scheduling problems. These new methods developed include some combinatorial optimisation algorithms as components to guide the search in the solution space using the knowledge of the algebraic structure of the problem being solved. Namely we develop two new methods: a simple method that combines a GRASP procedure with a branch-and-bound algorithm; and a more elaborated procedure that combines the verification of the violation of valid inequalities with a tabu search. We focus on the job-shop scheduling problem