Showing papers on "Linear programming published in 2001"
01 Jan 2001
TL;DR: A new approach to fuel-optimal path planning of multiple vehicles using a combination of linear and integer programming and the framework of mixed integer/linear programming is well suited for path planning and collision avoidance problems.
Abstract: This paper presents a new approach to fuel-optimal path planning of multiple vehicles using a combination of linear and integer programming. The basic problem formulation is to have the vehicles move from an initial dynamic state to a final state without colliding with each other, while at the same time avoiding other stationary and moving obstacles. It is shown that this problem can be rewritten as a linear program with mixed integer/linear constraints that account for the collision avoidance. A key benefit of this approach is that the path optimization can be readily solved using the CPLEX optimization software with an AMPL/Matlab interface. An example is worked out to show that the framework of mixed integer/linear programming is well suited for path planning and collision avoidance problems. Implementation issues are also considered. In particular, we compare receding horizon strategies with fixed arrival time approaches.
566 citations
TL;DR: An efficient algorithm that eliminates intron code and a demetic approach to virtually parallelize the system on a single processor are discussed, which show that GP performs comparably in classification and generalization.
Abstract: We introduce a new form of linear genetic programming (GP). Two methods of acceleration of our GP approach are discussed: 1) an efficient algorithm that eliminates intron code and 2) a demetic approach to virtually parallelize the system on a single processor. Acceleration of runtime is especially important when operating with complex data sets, because they are occurring in real-world applications. We compare GP performance on medical classification problems from a benchmark database with results obtained by neural networks. Our results show that GP performs comparably in classification and generalization.
482 citations
[...]
01 Sep 2001
TL;DR: In inverse optimization problems defined as follows, it is proved that if the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also alinear programming problem and inverse versions ofP under the L1 andL8 norms are also polynomially solvable.
Abstract: In this paper, we study inverse optimization problems defined as follows. LetS denote the set of feasible solutions of an optimization problemP, letc be a specified cost vector, andx0 be a given feasible solution. The solutionx0 may or may not be an optimal solution ofP with respect to the cost vectorc. The inverse optimization problem is to perturb the cost vectorc tod so thatx0 is an optimal solution ofP with respect tod and ||d- c || p is minimum, where ||d- c || p is some selectedLp norm. In this paper, we consider the inverse linear programming problem underL1 norm (where ||d- c || p= ? j?Jw j|d j-c j|, withJ denoting the index set of variablesx jandw jdenoting the weight of the variablej) and underL8 norm (where||d- c || p= max j?J{w j|d j-c j|} ). We prove the following results: (i) If the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also a linear programming problem. (ii) If the problemP is a shortest path, assignment or minimum cut problem, then its inverse problem under theL1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problemP is a minimum cost flow problem, then its inverse problem under theL1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problemP is a minimum cost flow problem, then its inverse problem under theL8 norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problemP is polynomially solvable for linear cost functions, then inverse versions ofP under theL1 andL8 norms are also polynomially solvable.
481 citations
Proceedings Article•
03 Jan 2001TL;DR: This work presents a principled and efficient planning algorithm for cooperative multiagent dynamic systems that avoids the exponential blowup in the state and action space and is an efficient alternative to more complicated algorithms even in the single agent case.
Abstract: We present a principled and efficient planning algorithm for cooperative multiagent dynamic systems. A striking feature of our method is that the coordination and communication between the agents is not imposed, but derived directly from the system dynamics and function approximation architecture. We view the entire multiagent system as a single, large Markov decision process (MDP), which we assume can be represented in a factored way using a dynamic Bayesian network (DBN). The action space of the resulting MDP is the joint action space of the entire set of agents. Our approach is based on the use of factored linear value functions as an approximation to the joint value function. This factorization of the value function allows the agents to coordinate their actions at runtime using a natural message passing scheme. We provide a simple and efficient method for computing such an approximate value function by solving a single linear program, whose size is determined by the interaction between the value function structure and the DBN. We thereby avoid the exponential blowup in the state and action space. We show that our approach compares favorably with approaches based on reward sharing. We also show that our algorithm is an efficient alternative to more complicated algorithms even in the single agent case.
479 citations
TL;DR: A heuristic algorithm is developed to solve the problem of generating a timetable for a given network of buses so as to maximize their synchronization, and the efficiency of this algorithm, compared to optimal solutions, is illustrated through a series of examples.
Abstract: This paper addresses the problem of generating a timetable for a given network of buses so as to maximize their synchronization. It attempts to maximize the number of simultaneous bus arrivals at the connection (transfer) nodes of the network. Transit schedulers, taking into account the satisfaction and convenience of the system's users, appreciate the importance of creating a timetable with maximal synchronization, which enables the transfer of passengers from one route to another with minimum waiting time at the transfer nodes. In this paper, the problem is formulated as a mixed integer linear programming problem, and a heuristic algorithm is developed to solve the problem in polynomial time. The efficiency of this algorithm, compared to optimal solutions, is illustrated through a series of examples.
295 citations
TL;DR: In this paper, the authors describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming, by identifying a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables.
276 citations
TL;DR: An interval linear programming problem is defined as an extension of the classicallinear programming problem to an inexact environment on the basis of a comparative study on ordering interval numbers, inequality constraints involving interval coefficients are reduced in their satisfactory crisp equivalent forms and a satisfactory solution of the problem isdefined.
254 citations
TL;DR: Numerical experiences show that the solution technique is computationally efficient, simple, and suitable for decision support of short-term hydro operations planning and can be easily extended for scheduling applications in deregulated environments.
Abstract: This paper describes experiences with mixed integer linear programming (MILP) based approaches on the short-term hydro scheduling (STHS) function. The STHS is used to determine the optimal or near-optimal schedules for the dispatchable hydro units in a hydro-dominant system for a user-definable study period at each time step while respecting all system and hydraulic constraints. The problem can be modeled in detail for a hydro system that contains both conventional and pumped-storage units. Discrete and dynamic constraints such as unit startup/shutdown and minimum-up/minimum-down time limits are also included in the model for hydro unit commitment (HUC). The STHS problem is solved with a state-of-the-art package which includes an algebraic modeling language and a MILP solver. The usefulness of the proposed solution algorithm is illustrated by testing the problem with actual hydraulic system data. Numerical experiences show that the solution technique is computationally efficient, simple, and suitable for decision support of short-term hydro operations planning. In addition, the proposed approaches can be easily extended for scheduling applications in a deregulated environment.
251 citations
TL;DR: It is proved that, under certain conditions, having equality degree constraints with multiple edges allowed in the design of logical topologies does not affect congestion and helps in reducing the dimensionality of the search space and hence speeds up the search for an optimal solution of the linear formulation.
Abstract: We consider the problem of constructing logical topologies over a wavelength-routed optical network with no wavelength changers. We present a general linear formulation which considers routing traffic demands, and routing and assigning wavelengths to lightpaths, as a combined optimization problem. The formulation also takes into account the maximum number of hops a lightpath is permitted to take, multiple logical links in the logical topology, multiple physical links in the physical topology, and symmetry/asymmetry restrictions in designing logical topologies. The objective is to minimize congestion. We show by examples how equality and inequality logical degree constraints have a bearing on congestion. We prove that, under certain conditions, having equality degree constraints with multiple edges allowed in the design of logical topologies does not affect congestion. This helps in reducing the dimensionality of the search space and hence speeds up the search for an optimal solution of the linear formulation. We solve the linear formulation for small examples and show the tradeoff between congestion, number of wavelengths available and the maximum number of hops a lightpath is allowed to take. For large networks, we solve the linear formulation by relaxing the integer constraints. We develop topology design algorithms for large networks based on rounding the solutions obtained by solving the relaxed problem. Since the whole problem is linearizable, the solution obtained by relaxation of the integer constraints yields a lower bound on congestion. This is useful in comparing the efficiency of our heuristic algorithms. Following Bienstock and Gunluk (1995), we introduce a cutting plane which helps in obtaining better lower bounds on congestion and also enables us to reduce the previously obtained upper bounds on congestion.
224 citations
TL;DR: Semi-infinite programming (SIP) as discussed by the authors is an optimization problem in which finitely many variables appear in infinitely many constraints, and it naturally arises in an abundant number of applications in different fields of mathematics, economics and engineering.
213 citations
25 Nov 2001
TL;DR: It is shown that subcarrier allocation in this approach can be optimized by the linear programming (LP) relaxation of the IP.
Abstract: Adaptive subcarrier allocation and adaptive modulation for multiuser orthogonal frequency division multiplexing (OFDM) is considered. The optimal subcarrier and bit allocation problems, that have been formulated in Wong et al., (1999), and Rhee et al., (2000), as nonlinear optimizations, are converted into linear ones and solved by integer programming (IP). A suboptimal approach that separately performs subcarrier allocation and bit loading is proposed. It is shown that subcarrier allocation in this approach can be optimized by the linear programming (LP) relaxation of the IP. Comparison through computer simulation indicates that performance of the suboptimal approach can be close to that of the optimal.
Book•
01 Jan 2001
TL;DR: Linear optimisation basic concepts Dantzig's simplex method duality and optimality sensitivity analysis karmarkar's interior path method integer linear optimisation linear network models computational complexity issues model building, case studies, and advanced techniques solutions to selected exercises.
Abstract: Linear optimisation basic concepts Dantzig's simplex method duality and optimality sensitivity analysis karmarkar's interior path method integer linear optimisation linear network models computational complexity issues model building, case studies, and advanced techniques solutions to selected exercises Appendices: linear algebra convexity graph theory optimisation theory computer package INTPM
TL;DR: In this article, an integrated fuzzy-stochastic linear programming model is developed and applied to municipal solid waste management, with the objective of minimizing system costs over the planning horizon.
Abstract: In this study, an integrated fuzzy-stochastic linear programming model is developed and applied to municipal solid waste management. Methods of chance-constrained programming and fuzzy linear programming are incorporated within a general interval-parameter mixed-integer linear programming framework. It improves upon the existing optimization methods with advantages in uncertainty reflection, data availability, and computational requirement. The model can be used for answering questions related to types, times and sites of solid waste management practices, with the objective of minimizing system costs over the planning horizon. The model can effectively reflect dynamic, interactive, and uncertain characteristics of municipal waste management systems. In its solution process, the model is transformed into two deterministic submodels, corresponding to upper and lower bounds of the desired objective function values under a given significance level, based on an interactive algorithm. Results of the method's application to a hypothetical case indicate that reasonable outputs have been obtained. It demonstrates the practical applicability of the proposed methodology.
TL;DR: It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
Abstract: We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.
TL;DR: The proposed model is more appropriate than the unfuzzy problem formulation in terms of reflecting a realistic situation and the information costs are generally decreased.
14 Oct 2001
TL;DR: This work proposes a heuristic for allocation in combinatorial auctions that can provide excellent solutions for problems with over 1000 items and 10,000 bids and achieves an average approximation error of less than 1%.
Abstract: We propose a heuristic for allocation in combinatorial auctions. We first run an approximation algorithm on the linear programming relaxation of the combinatorial auction. We then run a sequence of greedy algorithms, starting with the order on the bids determined by the approximate linear program and continuing in a hill-climbing fashion using local improvements in the order of bids. We have implemented the algorithm and have tested it on the complete corpus of instances provided by Vohra and de Vries as well as on instances drawn from the distributions of Leyton-Brown, Pearson, and Shoham. Our algorithm typically runs two to three orders of magnitude faster than the reported running times of Vohra and de Vries, while achieving an average approximation error of less than 1%. This algorithm can provide, in less than a minute of CPU time, excellent solutions for problems with over 1000 items and 10,000 bids. We thus believe that combinatorial auctions for most purposes face no practical computational hurdles.
14 Oct 2001
TL;DR: The authors give the first constant factor approximation algorithm for the facility location problem with nonuniform, hard capacities by presenting a local-search heuristic that yields an approximation guarantee of 9 + /spl epsi/ for the case of non uniform hard capacities.
Abstract: The authors give the first constant factor approximation algorithm for the facility location problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of these algorithms are based on linear programming, but the LP techniques developed thus far have been unsuccessful in dealing with hard capacities. A local-search based approximation algorithm (M. Korupolu et al., 1998; F.A. Chudak and D.P. Williamson, 1999) is known for the special case of hard but uniform capacities. We present a local-search heuristic that yields an approximation guarantee of 9 + /spl epsi/ for the case of nonuniform hard capacities. To obtain this result, we introduce new operations that are natural in this context. Our proof is based on network flow techniques.
TL;DR: This paper studies an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max-product composition and captures some special characteristics of its feasible domain and the optimal solutions.
TL;DR: In this paper, a multi-facility, multi-product and multi-period industrial problem is considered in the form of a network flow problem with relatively few additional 0-1 variables describing the linking constraints between periods.
Abstract: In this paper, we are interested in a multi-facility, multi-product and multi-period industrial problem. In this problem, both production and distribution costs are significant and they are inter-related. Therefore they should be considered simultaneously in a cost optimization problem. We model this combined production–distribution problem in the form of a network flow problem with relatively few additional 0–1 variables describing the linking constraints between periods. Computational experiments show that the real size problems we encountered can be solved in reasonable time using commercial linear programming codes like CPLEX.
Proceedings Article•
04 Aug 2001TL;DR: This work presents a hybrid approach for the 0-1 multidimensional knapsack problem that combines linear programming and Tabu Search and improves significantly on the best known results of a set of more than 150 benchmark instances.
Abstract: We present a hybrid approach for the 0-1 multidimensional knapsack problem The proposed approach combines linear programming and Tabu Search The resulting algorithm improves significantly on the best known results of a set of more than 150 benchmark instances
TL;DR: This paper proposes an extended linear programming model for a similar hybrid approach and shows that the proposed approach finds the better solution in a less number of iterations compared to the approach by Byrne and Bakir.
TL;DR: In this paper the delay management problem is formulated as a mixed integer linear program, and solution approaches based on this formulation are indicated.
03 Jan 2001
TL;DR: This work designs a simple tabu search meta-heuristic that exploits the special properties of different types of neighborhood moves, and creates highly effective candidate list strategies to solve an airport gate assignment problem that dynamically assigns airport gates to scheduled flights.
Abstract: Considers an airport gate assignment problem that dynamically assigns airport gates to scheduled flights based on passengers' daily origin and destination flow data. The objective of the problem is to minimize the overall connection times during which passengers walk to catch their connection flights. We formulate this problem as a mixed 0-1 quadratic integer programming problem and then reformulate it as a mixed 0-1 integer problem with a linear objective function and constraints. We design a simple tabu search meta-heuristic to solve the problem. The algorithm exploits the special properties of different types of neighborhood moves, and create highly effective candidate list strategies. We also address issues of tabu short-term memory, dynamic tabu tenure, aspiration rules, and various intensification and diversification strategies. Preliminary computational experiments are conducted, and the results are presented and analyzed.
TL;DR: The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort.
Abstract: We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort.
TL;DR: The design of scoring functions (or potentials) for threading, differentiating native‐like from non‐native structures with a limited computational cost, is an active field of research and linear programming is used to design optimal scoring functions.
Abstract: The design of scoring functions (or potentials) for threading, differentiating native-like from non-native structures with a limited computational cost, is an active field of research. We revisit two widely used families of threading potentials: the pairwise and profile models. To design optimal scoring functions we use linear programming (LP). The LP protocol makes it possible to measure the difficulty of a particular training set in conjunction with a specific form of the scoring function. Gapless threading demonstrates that pair potentials have larger prediction capacity compared with profile energies. However, alignments with gaps are easier to compute with profile potentials. We therefore search and propose a new profile model with comparable prediction capacity to contact potentials. A protocol to determine optimal energy parameters for gaps, using LP, is also presented. A statistical test, based on a combination of local and global Z-scores, is employed to filter out false-positives. Extensive tests of the new protocol are presented. The new model provides an efficient alternative for threading with pair energies, maintaining comparable accuracy. The code, databases, and a prediction server are available at http://www.tc.cornell.edu/CBIO/loopp. Proteins 2001;45:241–261. © 2001 Wiley-Liss, Inc.
TL;DR: This paper is an examination of two important issues related to formulation of project selection models such as the one presented here, and shows that the solution for the illustrative problem is reasonably robust to rather large variations in the measure of value.
Abstract: A mathematical formulation of an optimization model designed to select projects for inclusion in an R&D portfolio, subject to a wide variety of constraints (e.g., capital, headcount, strategic intent, etc.), is presented. The model is similar to others that have previously appeared in the literature and is in the form of a mixed integer programming (MIP) problem known as the multidimensional knapsack problem. Exact solution of such problems is generally difficult, but can be accomplished in reasonable time using specialized algorithms. The main contribution of this paper is an examination of two important issues related to formulation of project selection models such as the one presented here. If partial funding and implementation of projects is allowed, the resulting formulation is a linear programming (LP) problem which can be solved quite easily. Several plausible assumptions about how partial funding impacts project value are presented. In general, our examples suggest that the problem might best be formulated as a nonlinear programming (NLP) problem, but that there is a need for further research to determine an appropriate expression for the value of a partially funded project. In light of that gap in the current body of knowledge and for practical reasons, the LP relaxation of this model is preferred. The LP relaxation can be implemented in a spreadsheet (even for relatively large problems) and gives reasonable results when applied to a test problem based on GM's R&D project selection process. There has been much discussion in the literature on the topic of assigning a quantitative measure of value to each project. Although many alternatives are suggested, no one way is universally accepted as the preferred way. There does seem to be general agreement that all of the proposed methods are subject to considerable uncertainty. A systematic way to examine the sensitivity of project selection decisions to variations in the measure of value is developed. It is shown that the solution for the illustrative problem is reasonably robust to rather large variations in the measure of value. We cannot, however, conclude that this would be the case in general. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 18–40, 2001
TL;DR: In this paper, a new method is proposed for long-term reservoir operation planning with stochastic inflows, which is formulated as a two-stage linear program with simple recourse.
Abstract: A new method is proposed for long-term reservoir operation planning with stochastic inflows. In particular, the problem is formulated as a two-stage stochastic linear program with simple recourse. The stochastic inflows are approximated by multiple inflow scenarios, leading to a very large deterministic model which is hard to solve using conventional optimization methods. This paper presents an efficient interior-point optimization algorithm for solving the resulting deterministic problem. It is also shown how exploiting the problem structure enhances the performance of the algorithm. Application to regulation of the Great Lakes system shows that the proposed approach can handle the stochasticity of the inflows as well as the nonlinearity of the operating conditions in a real-world reservoir system.
TL;DR: In this paper, the authors extend from linear programming to nonlinear OPF using the efficient multiple centrality corrections (MCC) technique that was developed by Gondzio.
Abstract: Large scale nonlinear optimal power flow (OPF) problems have been efficiently solved by extensions from linear programming to nonlinear programming of the primal-dual logarithmic barrier interior-point method and its predictor-corrector variant. Motivated by the impressive performance of the nonlinear predictor-corrector extension, in this paper we extend from linear programming to nonlinear OPF the efficient multiple centrality corrections (MCC) technique that was developed by Gondzio. The numerical performance of the proposed MCC algorithm is evaluated on a set of power networks ranging in size from 118 buses to 2098 buses. Extensive computational results demonstrate that the MCC technique is fast and robust, and outperforms the successful predictor-corrector technique.
TL;DR: It is shown that not only the maximum relative congestion is minimized, but the congestion of the edges is distributed equally such that the solution is optimal in a well-defined sense.
Abstract: We show how the new approximation algorithms by Garg and Konemann with extensions due to Fleischer for the multicommodity flow problem can be modified to solve the linear programming relaxation of the global routing problem. Implementation issues to improve the performance, such as a discussion of different functions for the dual variables and how to use the Newton method as an additional optimization step, are given. It is shown that not only the maximum relative congestion is minimized, but the congestion of the edges is distributed equally such that the solution is optimal in a well-defined sense: the vector of the relative congestion of the edges sorted in nonincreasing order is minimal by lexicographic order. This is an important step toward improving signal integrity by extra spacing between wires. Finally, we show how the weighted netlength can be minimized. Our computational results with recent IBM processor chips show that this approach can be used in practice even for large chips and that it is superior on difficult instances where ripup and reroute algorithms fail.
TL;DR: A new approach for solving the airline crew scheduling problem that is based on enumerating hundreds of millions random pairings is developed, which produces solutions that are significantly better than ones found by current practice.
Abstract: The airline crew scheduling problem is the problem of assigning crew itineraries to flights. We develop a new approach for solving the problem that is based on enumerating hundreds of millions random pairings. The linear programming relaxation is solved first and then millions of columns with best reduced cost are selected for the integer program. The number of columns is further reduced by a linear programming based heuristic. Finally an integer solution is obtained with a commercial integer programming solver. The branching rule of the solver is enhanced with a combination of strong branching and a specialized branching rule. The algorithm produces solutions that are significantly better than ones found by current practice.