49,388 research outputs found

    Cutting plane methods for general integer programming

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    Integer programming (IP) problems are difficult to solve due to the integer restrictions imposed on them. A technique for solving these problems is the cutting plane method. In this method, linear constraints are added to the associated linear programming (LP) problem until an integer optimal solution is found. These constraints cut off part of the LP solution space but do not eliminate any feasible integer solution. In this report algorithms for solving IP due to Gomory and to Dantzig are presented. Two other cutting plane approaches and two extensions to Gomory's algorithm are also discussed. Although these methods are mathematically elegant they are known to have slow convergence and an explosive storage requirement. As a result cutting planes are generally not computationally successful

    Algorithms for solving linear integer progamming problems

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    This thesis discusses several methods used to solve linear integer programming problems. Chapters 1-3 give the necessary linear programming background. Chapter 4 introduces integer programming and describes, in general, the two classes of solution methods -enumeration and cutting planes. Chapter 5 details two specific cutting plane methods, one all-integer approach and one fractional approach.Chapter 6 describes the branch-and-bound method, one of the enumeration methods. In Chapter 7 the additive algorithm for 0-1 programming problems is discussed. Chapter 8 describes the branch and-cut method, a combination of the cutting plane and branch-and bound approaches. Chapter 9 presents the plant location problem as an example of the integer programming problem. Examples of each method are included in the thesis

    Contemporary Approaches to the Solution of the Integer Programming Problem

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    The purpose of this thesis is to provide analysis of the modem development of the methods for solution to the integer linear programming problem. The thesis simultaneously discusses some main approaches that lead to the development of the algorithms for the solution to the integer linear programming problem. Chapter 1 introduces the Generalized Linear Programming Problem alongside with the properties of a solution to the Linear Programming Problem. The simplex procedure presented to solve the Linear Programming Problem by adding slack variables along with the artificial-basis technique. Chapter 2 refers to the primal-dual simplex procedure. The dual simplex algorithm reflects the dual simplex procedure. Chapter 3 discusses the mixed and alternative formulations of the integer programming problem. Chapter 4 considers the optimality conditions with the imposed relaxations to solve the Linear Programming Relaxation Problem. The methods of the Integer Programming are introduced for the Linear Programming Relaxation. Chapter 5 discusses the concepts of the Branch-and Bound method followed by the direct application of the Branch-and-Bound method. Chapter 6 introduces the fundamental concepts of the cutting method. The main concept of the valid inequalities presented for the Linear Programming Problem as well as for the Integer Programming Problem. Gomory\u27s Fractional cutting plane Algorithm represents the desired step to obtain the solution for the Integer Programming Problem. Furthermore, the mixed integer cuts generalizes the concepts to provide the corresponding solution for the Integer Programming Problem. Chapter 7 describes the Gomory method for the pure Integer Program followed by the Gomory method for the mixed Integer Program. In the Appendix the computer program LINDO is used. Throughout the whole thesis this computer program is applied to emphasize the very helpful tool in Linear Programming. All above mentioned chapters include the variety of examples corresponding to the Linear Programming Problem and the Integer Program

    Scheduling aircraft landings - the static case

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    This is the publisher version of the article, obtained from the link below.In this paper, we consider the problem of scheduling aircraft (plane) landings at an airport. This problem is one of deciding a landing time for each plane such that each plane lands within a predetermined time window and that separation criteria between the landing of a plane and the landing of all successive planes are respected. We present a mixed-integer zero–one formulation of the problem for the single runway case and extend it to the multiple runway case. We strengthen the linear programming relaxations of these formulations by introducing additional constraints. Throughout, we discuss how our formulations can be used to model a number of issues (choice of objective function, precedence restrictions, restricting the number of landings in a given time period, runway workload balancing) commonly encountered in practice. The problem is solved optimally using linear programming-based tree search. We also present an effective heuristic algorithm for the problem. Computational results for both the heuristic and the optimal algorithm are presented for a number of test problems involving up to 50 planes and four runways.J.E.Beasley. would like to acknowledge the financial support of the Commonwealth Scientific and Industrial Research Organization, Australia

    Integer programming, lattices, and results in fixed dimension

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    We review and describe several results regarding integer programming problems in fixed dimension. First, we describe various lattice basis reduction algorithms that are used as auxiliary algorithms when solving integer feasibility and optimization problems. Next, we review three algorithms for solving the integer feasibility problem. These algorithms are based on the idea of branching on lattice hyperplanes, and their running time is polynomial in fixed dimension. We also briefly describe an algorithm, based on a different principle, to count integer points in an integer polytope. We then turn the attention to integer optimization. Again, we describe three algorithms: binary search, a linear algorithm for a fixed number of constraints, and a randomized algorithm for a varying number of constraints. The topic of the next part of our chapter is how to use lattice basis reduction in problem reformulation. Finally, we review cutting plane results when the dimension is fixe

    Improving the Accuracy and Efficiency of MAP Inference for Markov Logic

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    In this work we present Cutting Plane Inference (CPI), a Maximum A Posteriori (MAP) inference method for Statistical Relational Learning. Framed in terms of Markov Logic and inspired by the Cutting Plane Method, it can be seen as a meta algorithm that instantiates small parts of a large and complex Markov Network and then solves these using a conventional MAP method. We evaluate CPI on two tasks, Semantic Role Labelling and Joint Entity Resolution, while plugging in two different MAP inference methods: the current method of choice for MAP inference in Markov Logic, MaxWalkSAT, and Integer Linear Programming. We observe that when used with CPI both methods are significantly faster than when used alone. In addition, CPI improves the accuracy of MaxWalkSAT and maintains the exactness of Integer Linear Programming

    Cjelobrojno programiranje

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    Cjelobrojno linearno programiranje se bavi problemom optimizacije linearnog funkcionala uz linearne uvjete tipa jednakosti i nejednakosti gdje je dodatno uveden zahtjev cjelobrojnosti na neke (ili sve) varijable. Područje primjene cjelobrojnog linearnog programiranja je široki : proizvodnja, transport i distribucija, marketing, financijsko ulaganje i planiranje, raspored zaposlenika… Najčešće metode pri rješavanju problema cjelobrojnog programiranja su metoda grananja i ograđivanja i metoda odsijecajućih ravnina. U ovom radu smo istražili metodu odsijecajućih ravnina. U prvom i drugom poglavlju smo se upoznali s teorijom potrebnom da bi se dokazao glavni teorem za metodu odsijecajućih ravnina, a to je da se u konačno mnogo koraka odsijecanjem poliedarskog skupa P može dobiti njegova cjelobrojna ljuska i pokazali smo ocjenu broja tih koraka. U trećem poglavlju smo uveli algoritam za metodu odsijecajućih ravnina te smo pokazali dva primjera problema cjelobrojnog linearnog programiranja riješenih metodom odsijecajućih ravnina.Integer linear programming deals with the optimisation problem of linear functional, concerning linear constraints of equality and inequality, where some (or all) variables are additionally restricted to be integer. Scope of integer linear programming usage is wide: production, transport and distribution, marketing, financial and investment planning, employees scheduling etc. The most common methods for solving integer programming problems are branch and bound method and cutting plane method. In this paper we studied the cutting plane method. In the first two chapters we’ve met the theory necessary to prove the main cutting plane method theorem, which says that the integer hull of polyhedral set P can be obtained by cutting its parts in finite number of times. Furthermore, we showed the rating of this finite steps numbers. In the third chapter, we’ve introduced the algorithm of cutting plane method and demonstrated two integer linear programming problem examples, solved with cutting plane method

    Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products

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    The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs is developed based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. A detailed computational study based on implementations in two solvers evaluates the performance impact of the proposed methods.Comment: 16 pages, 0 figures, submitted to the 24th Conference on Integer Programming and Combinatorial Optimizatio
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