Showing papers on "Integer programming published in 1998"

Branch-And-Price: Column Generation for Solving Huge Integer Programs

Abstract: We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

Robust Solutions to Uncertain Semidefinite Programs

Abstract: In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Holder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

Voltage scheduling problem for dynamically variable voltage processors

Abstract: This paper presents a model of dynamically variable voltage processors and basic theorems for power-delay optimization. A static voltage scheduling problem is also proposed and formulated as an integer linear programming (ILP) problem. In the problem, we assume that a core processor can vary its supply voltage dynamically, but can use only a single voltage level at a time. For a given application program and a dynamically variable voltage processor, a voltage scheduling which minimizes energy consumption under an execution time constraint can be found.

Effective Continuous-Time Formulation for Short-Term Scheduling. 1. Multipurpose Batch Processes

Abstract: During the last decade, the problem of production scheduling has been realized to be one of the most important problems in industrial plant operations especially when multipurpose/multiproduct batch processes are involved. This paper presents a novel mathematical formulation for the short-term scheduling of batch plants. The proposed formulation is based on a continuous time representation and results in a mixed integer linear programming (MILP) problem. The novel elements of the proposed formulation are (i) the decoupling of the task events from the unit events, (ii) the time sequencing constraints, and (iii) its linearity. In contrast to the previously presented continuous-time scheduling formulations, the proposed approach leads to smaller and simpler mathematical models which exhibit fewer binary and continuous variables, have smaller integrality gaps, require fewer constraints, need fewer linear programming relaxations, and can be solved in significantly less CPU time. Several examples are presented ...

Railway Timetabling Using Lagrangian Relaxation

Abstract: We present a novel optimization approach for the timetabling problem of a railway company, i.e., scheduling of a set of trains to obtain a profit maximizing timetable, while not violating track capacity constraints. The scheduling decisions are based on estimates of the value of running different types of service at specified times. We model the problem as a very large integer programming problem. The model is flexible in that it allows for general cost functions. We have used a Lagrangian relaxation solution approach, in which the track capacity constraints are relaxed and assigned prices, so that the problem separates into one dynamic program for each physical train. The number of dual variables is very large. However, it turns out that only a sm all fraction of these are nonzero, which one may take advantage of in the dual updating schemes. The approach has been tested on a realistic example suggested by the Swedish National Railway Administration. This example contains 18 passenger trains and 8 freight trains to be scheduled during a day on a stretch of single track, consisting of 17 stations. The computation times are rather modest and the obtained timetables are within a few percent of optimality.

An Updated Mixed Integer Programming Library: MIPLIB 3.0

Abstract: An Up datedMixed Integer Programming LibraryMIPLIB Rob ert E BixbyDepartment of Computational and Applied MathematicsRice UniversityHouston TX CPLEX Optimization IncSebastian CeriaSchool of BusinessColumbia UniversityNew York NY Cassandra M McZealDepartment of Computational and Applied MathematicsRice UniversityHouston TX Martin WPSavelsb erghSchool of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlanta GA March Intro ductionIn resp onse to the needs of researchers for access to challenging mixed integer programsBixby et al created MIPLIB an electronically available library of b oth pure andmixed integer programs most of which arise from realworld applicationsSince its intro duction MIPLIB has b ecome a standard test set for comparing thep erformance of mixed integer optimization co des Its availability has provided an imp ortant stimulus for researchers in this very active area As technology has progressedhowever there have b een signi cantimproements in stateoftheart optimizers andcomputing machineryConsequentlyseveral instances have b ecome to o easy and aneed has emerged for more dicult instances Also it has b een observed that certaintyp es of problems are overrepresented in MIPLIB and others underrepresented Theseconsiderations have prompted the present up dateSince mixed integer programming is such an active research area and the p erformance of optimizers keeps improving weanticipate that this up date will not b e the lastSubsequent up dates are planned on a yearly basis We encourage b oth researchers andpractitioners in integer programming to submit realworld instances for considerationand p ossible inclusion in MIPLIB

Solving Steiner Tree Problems in Graphs to Optimality

Abstract: In this paper, we present the implementation of a branch-and-cut algorithm for solving Steiner tree problems in graphs. Our algorithm is based on an integer programming formulation for directed graphs and comprises preprocessing, separation algorithms, and primal heuristics. We are able to solve nearly all problem instances discussed in the literature to optimality, including one problem that—to our knowledge—has not yet been solved. We also report on our computational experiences with some very large Steiner tree problems arising from the design of electronic circuits. All test problems are gathered in a newly introduced library called SteinLib that is accessible via the World Wide Web. © 1998 John Wiley & Sons, Inc. Networks 32: 207–232, 1998

Large Scale Linear and Integer Optimization: A Unified Approach

Abstract: 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.

L -shaped decomposition of two-stage stochastic programs with integer recourse

Abstract: We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.

Numerical Experience with Lower Bounds for MIQP Branch-And-Bound

Abstract: The solution of convex mixed-integer quadratic programming (MIQP) problems with a general branch-and-bound framework is considered It is shown how lower bounds can be computed efficiently during the branch-and-bound process Improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming (QP) problems that have to be solved The branch-and-bound approach is also shown to be superior to other approaches in solving MIQP problems Numerical experience is presented which supports these conclusions

Minimum cost capacity installation for multicommodity network flows

Abstract: Consider a directed graphG = (V,A), and a set of traffic demands to be shipped between pairs of nodes inV. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho, On feasibility conditions of multicommodity flows in networks, IEEE Transactions on Circuit Theory, CT-18 (4) (1971) 425---429.), and uses a formulation with only |A| variables. The second uses an aggregated multicommodity flow formulation and has |V||A| variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real-life problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Dimensioning of survivable WDM networks

Abstract: In this paper routing, planning of working capacity, rerouting, and planning of spare capacity in wavelength division multiplexing (WDM) networks are investigated. Integer linear programming (ILP) and simulated annealing (SA) are used as solution techniques. A complex cost model is presented. The spare capacity assignment is optimized with respect to three restoration strategies. The benefit of wavelength conversion, the choice of the fiber line system, and the influence of cost parameter values are discussed, with respect to the different restoration strategies and solution techniques. Wavelength conversion is found to be of limited importance, whereas tunability at the end points of the connections has substantial benefits.

Practical solutions for QoS-based resource allocation problems

Abstract: The QoS based Resource Allocation Model (Q-RAM) proposed by R. Rajkumar et al. (1998) presented an analytical approach for satisfying multiple quality of service dimensions in a resource constrained environment. Using this model, available system resources can be apportioned across multiple applications such that the net utility that accrues to the end users of those applications is maximized. We present several practical solutions to allocation problems that were beyond the limited scope of Q-RAM. We show that the Q-RAM problem of finding the optimal resource allocation to satisfy multiple QoS dimensions is NP hard. We then present a polynomial solution for this resource allocation problem which yields a solution within a provably fixed and short distance from the optimal allocation. Secondly, Q-RAM dealt mainly with the problem of apportioning a single resource to satisfy multiple QoS dimensions. We study the converse problem of apportioning multiple resources to satisfy a single QoS dimension. In practice, this problem becomes complicated, since a single QoS dimension perceived by the user can be satisfied using different combinations of available resources. We show that this problem can be formulated as a mixed integer programming problem that can be solved efficiently to yield an optimal resource allocation. We also present the run times of these optimizations to illustrate how these solutions can be applied in practice. A good understanding of these solutions will yield insights into the general problem of apportioning multiple resources to satisfy simultaneously multiple QoS dimensions of multiple concurrent applications.

A Two-Phase Exact Algorithm for MAX-SAT and Weighted MAX-SAT Problems

Abstract: We describe a two-phase algorithm for MAX-SAT and weighted MAX-SAT problems. In the first phase, we use the GSAT heuristic to find a good solution to the problem. In the second phase, we use an enumeration procedure based on the Davis-Putnam-Loveland algorithm, to find a provably optimal solution. The first heuristic stage improves the performance of the algorithm by obtaining an upper bound on the minimum number of unsatisfied clauses that can be used in pruning branches of the search tree. We compare our algorithm with an integer programming branch-and-cut algorithm. Our implementation of the two-phase algorithm is faster than the integer programming approach on many problems. However, the integer programming approach is more effective than the two-phase algorithm on some classes of problems, including MAX-2-SAT problems.

Branch-and-Price Algorithms for the One-Dimensional Cutting Stock Problem

Abstract: We compare two branch-and-price approaches for the cutting stock problem. Each algorithm is based on a different integer programming formulation of the column generation master problem. One formulation results in a master problem with 0–1 integer variables while the other has general integer variables. Both algorithms employ column generation for solving LP relaxations at each node of a branch-and-bound tree to obtain optimal integer solutions. These different formulations yield the same column generation subproblem, but require different branch-and-bound approaches. Computational results for both real and randomly generated test problems are presented.

Mesh refinement in direct transcription methods for optimal control

Abstract: SUMMARY The direct transcription method for solving optimal control problems involves the use of a discrete approximation to the original problem. This paper describes a technique for changing the discretization in order to improve the accuracy of the approximation. An integer programming technique is used to minimize the maximum error during the refinement iterations. The eƒciency of the method is illustrated for an application with path inequality constraints. ( 1998 John Wiley & Sons, Ltd.

Plant location with minimum inventory

Abstract: We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality.

A Bilevel Decomposition Algorithm for Long-Range Planning of Process Networks

Abstract: The solution of the multiperiod MILP model for long-range planning of process networks by Sahinidis et al. (Comput. Chem. Eng. 1989, 13, 1049) is addressed in this paper. The model determines the optimal selection and expansion of processes over a long-range planning horizon, incorporating multiple scenarios for varying forecasts for demands and prices of chemicals. A rigorous bilevel decomposition algorithm is proposed to reduce the computational cost in the multiperiod MILP model. The decomposition algorithm solves a master problem in the reduced space of binary variables to determine a selection of processes and an upper bound to the net present value. A planning model is then solved for the selected processes to determine the expansion policy and a lower bound to the objective function. Numerical examples are presented to illustrate the performance of the algorithm and to compare it with a full-space branch and bound method.

Note. Optimal Balancing of U-Shaped Assembly Lines

Abstract: This note presents an integer programming formulation for determining the optimal balance for the U-line line balancing problem. It is shown that this model can optimally solve larger problems than previously reported.

Branch and Infer: a Unifying Framework for Integer and Finite Domain Constraint Programming

Abstract: We introduce branch and infer, a unifying framework for integer linear programming and finite domain constraint programming. We use this framework to compare the two approaches with respect to their modeling and solving capabilities, to introduce symbolic constraint abstractions into integer programming, and to discuss possible combinations of the two approaches.

Solving the Uncapacitated Network Design Problem by a Lagrangean Heuristic and Branch-And-Bound

Abstract: The network design problem is a multicommodity minimal cost network flow problem with fixed costs on the arcs, i.e., a structured linear mixed-integer programming problem. It has various applications, such as construction of new links in transportation networks, topological design of computer communication networks, and planning of empty freight car transportation on railways. We present a Lagrangean heuristic within a branch-and-bound framework as a method for finding the exact optimal solution of the uncapacitated network design problem with single origins and destinations for each commodity (the simplest problem in this class, but still NP-hard). The Lagrangean heuristic uses a Lagrangean relaxation as subproblem, solving the Lagrange dual with subgradient optimization, combined with a primal heuristic (the Benders subproblem) yielding primal feasible solutions. Computational tests on problems of various sizes (up to 1000 arcs, 70 nodes and 138 commodities or 40 nodes and 600 commodities) and of several different structures lead to the conclusion that the method is quite powerful, outperforming for example a state-of-the-art mixed-integer code, both with respect to problem size and solution time.

Resource Leveling of Linear Schedules Using Integer Linear Programming

Abstract: Since the early 1960s many techniques have been developed to plan and schedule linear construction projects. However, one, the critical path method (CPM), overshadowed the others. As a result, CPM developed into the powerful and effective tool that it is today. However, research has indicated that CPM is ineffective for linear construction. Linear construction projects are typified by activities that must be repeated in different locations such as highways, pipelines, and tunnels. Recently, there has been renewed interest in linear scheduling. Much of this interest has involved a technique called the linear scheduling method (LSM). Only recently has there been the ability to calculate the controlling activities of a linear schedule, independent of network analysis. Additional research needs to be done to develop some of the techniques available in CPM into comparable ones for linear scheduling. One of these techniques is resource leveling. This paper uses the vehicle of a highway construction project to present an integer linear programming formulation to level the resources of linear projects.