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Showing papers on "Linear programming published in 1992"


Journal ArticleDOI
TL;DR: A Taylor polynomial of second order is used to approximate a primal-dual trajectory and an adaptive heuristic for estimating the centering parameter is given, which treats primal and dual problems symmetrically.
Abstract: This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show...

1,668 citations


Journal ArticleDOI
TL;DR: This paper presents a new optimization algorithm capable of optimally solving 100-customer problems of the vehicle routing problem with time windows VRPTW and indicates that this algorithm proved to be successful on a variety of practical sized benchmark VRPTw test problems.
Abstract: The vehicle routing problem with time windows VRPTW is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service. In this paper, we present the development of a new optimization algorithm for its solution. The LP relaxation of the set partitioning formulation of the VRPTW is solved by column generation. Feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming. The LP solution obtained generally provides an excellent lower bound that is used in a branch-and-bound algorithm to solve the integer set partitioning formulation. Our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark VRPTW test problems. The algorithm was capable of optimally solving 100-customer problems. This problem size is six times larger than any reported to date by other published research.

1,085 citations


Journal ArticleDOI
TL;DR: A single linear programming formulation is proposed which generates a plane that of minimizes an average sum of misclassified points belonging to two disjoint points sets in n-dimensional real space, without the imposition of extraneous normalization constraints that inevitably fail to handle certain cases.
Abstract: A single linear programming formulation is proposed which generates a plane that of minimizes an average sum of misclassified points belonging to two disjoint points sets in n-dimensional real space. When the convex hulls of the two sets are also disjoint, the plane completely separates the two sets. When the convex hulls intersect, our linear program, unlike all previously proposed linear programs, is guaranteed to generate some error-minimizing plane, without the imposition of extraneous normalization constraints that inevitably fail to handle certain cases. The effectiveness of the proposed linear program has been demonstrated by successfully testing it on a number of databases. In addition, it has been used in conjunction with the multisurface method of piecewise-linear separation to train a feed-forward neural network with a single hidden layer.

771 citations


Journal ArticleDOI
TL;DR: This paper deals with the problem of finding closed form schedules as affine or piecewise affine functions of the iteration vector and presents an algorithm which reduces the scheduling problem to a parametric linear program of small size, which can be readily solved by an efficient algorithm.
Abstract: Programs and systems of recurrence equations may be represented as sets of actions which are to be executed subject to precedence constraints. In may cases, actions may be labelled by integral vectors in some iterations domains, and precedence constraints may be described by affine relations. A schedule for such a program is a function which assigns an execution data to each action. Knowledge of such a schedule allows one to estimate the intrinsic degree of parallelism of the program and to compile a parallel version for multiprocessor architectures or systolic arrays. This paper deals with the problem of finding closed form schedules as affine or piecewise affine functions of the iteration vector. An algorithm is presented which reduces the scheduling problem to a parametric linear program of small size, which can be readily solved by an efficient algorithm.

614 citations


Journal ArticleDOI
01 Nov 1992
TL;DR: In this paper, a grey linear programming (GLP) model is introduced to the civil engineering area, which allows uncertainties in the model inputs to be communicated into the optimization process, and thereby solutions reflecting the inherent uncertainties can be derived.
Abstract: In optimization analysis by linear programming, uncertainties may exist in model coefficients and stipulations (right-hand side constraints). These uncertainties can propagate through the analysis and generate uncertainties in the results. However, among the previous methods dealing with uncertainty, some were too complicated to be applied to actual problems, and some were unable to reflect completely the uncertainties of the input and output information. In this paper, a grey linear programming (GLP) model is introduced to the civil engineering area. This method allows uncertainties in the model inputs to be communicated into the optimization process, and thereby solutions reflecting the inherent uncertainties can be derived. A grey linear programming problem can be solved easily by running a simplex program several times. The modelling approach is applied to a hypothetical problem of waste flow allocation planning within a municipal solid waste management system. The results indicate that reaso...

558 citations


Journal ArticleDOI
TL;DR: An algorithm to solve the economic lot sizing problem in O(n log n) time is presented and it is shown how the Wagner-Whitin case can even be solved in linear time.
Abstract: We consider the n-period economic lot sizing problem, where the cost coefficients are not restricted in sign. In their seminal paper, H. M. Wagner and T. M. Whitin proposed an O(n2) algorithm for the special case of this problem, where the marginal production costs are equal in all periods and the unit holding costs are nonnegative. It is well known that their approach can also be used to solve the general problem, without affecting the complexity of the algorithm. In this paper, we present an algorithm to solve the economic lot sizing problem in O(n log n) time, and we show how the Wagner-Whitin case can even be solved in linear time. Our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure. Furthermore, we show how Wagner and Whitin's and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem.

490 citations


Journal ArticleDOI
TL;DR: In this paper, a branch-and-bound algorithm for linear bilevel programming is proposed, where necessary optimality conditions expressed in terms of tightness of the follower's constraints are used to fathom or simplify subproblems, branch and obtain penalties similar to those used in mixed-integer programming.
Abstract: A new branch-and-bound algorithm for linear bilevel programming is proposed. Necessary optimality conditions expressed in terms of tightness of the follower’s constraints are used to fathom or simplify subproblems, branch and obtain penalties similar to those used in mixed-integer programming. Computational results are reported and compare favorably to those of previous methods. Problems with up to 150 constraints, 250 variables controlled by the leader, and 150 variables controlled by the follower have been solved.

433 citations


Journal ArticleDOI
TL;DR: This paper is concerned with the development of an algorithm for general bilinear programming problems, and develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm.
Abstract: This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This “RLT” process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.

400 citations


Journal ArticleDOI
TL;DR: In this article, an auxiliary multiple objective linear programming model is proposed to solve a linear programming problem with imprecise objective and/or constraint coefficients, where the strategy is to maximize the most possible value of the imprecising profit.

394 citations


Journal ArticleDOI
TL;DR: A survey of branch-and-bound algorithms for the problem of finding the minimum cost assignment of jobs to agents as discussed by the authors shows that most approaches seem to be based on branch and bound with bound supplied through heuristics and through relaxations of the primal problem formulation.

378 citations


Journal ArticleDOI
TL;DR: A unified treatment of algorithms is described for linear programming methods based on the central path, which is a curve along which the cost decreases, and that stays always far from the centre.
Abstract: In this paper a unified treatment of algorithms is described for linear programming methods based on the central path. This path is a curve along which the cost decreases, and that stays always far...

Journal ArticleDOI
TL;DR: Mehrotra as mentioned in this paper described a predictor-corrector variant of the primal-dual interior-point algorithm for linear programming, with extensions for solving problems with free variables and problems with bounds on primal variables.
Abstract: Mehrotra [Tech. Report 90-03, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1990] recently described a predictor–corrector variant of the primal–dual interior-point algorithm for linear programming. This paper describes a full implementation of this algorithm, with extensions for solving problems with free variables and problems with bounds on primal variables. Computational results on the NETLIB test set are given to show that this new method almost always improves the performance of the primal–dual algorithm and that the improvement increases dramatically as the size and complexity of the problem increases. A numerical instability in using Schur complements to remove dense columns is identified, and a numerical remedy is given.

Journal ArticleDOI
Margaret H. Wright1
TL;DR: A self-contained survey of major themes in both classical material and recent developments related to the theory and practice of interior methods for linear programming is presented.
Abstract: Interior methods for optimization were widely used in the 1960s, primarily in the form of barrier methods. However, they were not seriously applied to linear programming because of the dominance of the simplex method. Barrier methods fell from favour during the 1970s for a variety of reasons, including their apparent inefficiency compared with the best available alternatives. In 1984, Karmarkar's announcement of a fast polynomial-time interior method for linear programming caused tremendous excitement in the field of optimization. A formal connection can be shown between his method and classical barrier methods, which have consequently undergone a renaissance in interest and popularity. Most papers published since 1984 have concentrated on issues of computational complexity in interior methods for linear programming. During the same period, implementations of interior methods have displayed great efficiency in solving many large linear programs of ever-increasing size. Interior methods have also been applied with notable success to nonlinear and combinatorial problems. This paper presents a self-contained survey of major themes in both classical material and recent developments related to the theory and practice of interior methods.

ReportDOI
01 Dec 1992
TL;DR: It is shown how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo sampling techniques.
Abstract: For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages.

Journal ArticleDOI
Roman A. Polyak1
TL;DR: The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a “hot” start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.
Abstract: The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces. The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies. Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed. The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods. The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a "hot" start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.

Proceedings ArticleDOI
01 Jul 1992
TL;DR: Efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem are presented.
Abstract: We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem. Without any constraint violation, the e-approximation problem for many problems of this type is itself NP-hard. Our methods provide polynomial-time e-approximations while attempting to minimize the packing constraint violation.Our methods lead to the first known approximation algorithms with provable performance guarantees for the s-median problem, the tree prunning problem, and the generalized assignment problem. These important problems have numerous applications to data compression, vector quantization, memory-based learning, computer graphics, image processing, clustering, regression, network location, scheduling, and communication. We provide evidence via reductions that our approximation algorithms are nearly optimal in terms of the packing constraint violation. We also discuss some recent applications of our techniques to scheduling problems.

Book ChapterDOI
13 Feb 1992
TL;DR: A simple randomized algorithm which solves linear programs with n constraints and d variables in expected O(d32 d n) time, and holds for any input.
Abstract: We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected O(d32 d n) time. The expectation is over the internal randomizations performed by the algorithm, and holds for any input.

01 Jun 1992
TL;DR: Efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem are presented.
Abstract: We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem. Without any constraint violation, the epsilon-approximation problem for many problems of this type is itself NP-hard. Our methods provide polynomial-time epsilon-approximations while attempting to minimize the packing constraint violation. Our methods lead to the first known approximation algorithms with provable performance guarantees for the s-median problem, the tree pruning problem, and the generalized assignment problem. These important problems have numerous applications to data compression, vector quantization, memory-based learning, computer graphics, image processing, clustering, regression, network location, scheduling, protocol testing, and communication. We provide evidence via reductions that our approximation algorithms are nearly optimal in terms of the packing constraint violation. We also discuss some recent applications of our techniques to scheduling problems.

Journal ArticleDOI
TL;DR: This work presents several new steepest-edge simplex algorithms for solving linear programming problems, including variants of both the primal and the dual simplex method.
Abstract: We present several new steepest-edge simplex algorithms for solving linear programming problems, including variants of both the primal and the dual simplex method. These algorithms differ depending upon the space in which the problem is viewed as residing, and include variants in which this space varies dynamically. We present computational results comparing steepest-edge simplex algorithms and approximate versions of them against simplex algorithms that use standard pivoting rules on truly large-scale realworld linear programs with as many as tens of thousands of rows and columns. These results demonstrate unambiguously the superiority of steepest-edge pivot selection criteria to other pivot selection criteria in the simplex method.

Journal ArticleDOI
TL;DR: This paper deals with an application of a variant of Karmarkar's projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem, based on a column generation technique defining a sequence of primal linear programming maximization problems.
Abstract: This paper deals with an application of a variant of Karmarkar's projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem. This problem is closely related to the Dantzig-Wolfe decomposition technique used in large-scale convex programming. The proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems. Associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm. When a point close to the minimum of the potential function is reached, a corresponding point in the dual space is constructed, which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem. An admissible cut of the polytope, corresponding to a new supporting hyperplane of the epigraph of the function to minimize, is then generated at this approximate analytic center. In the primal space this new cut translates into a new column for the associated linear programming problem. The algorithm has performed well on a set of convex nondifferentiable programming problems.

Journal Article
TL;DR: A linear programming based interactive decision making method to derive the satisficing solution of the decision maker for the formulated multiobjective programming problems is developed.
Abstract: Fuzzy linear regression models, where both input data and output data are fuzzy numbers, are introduced by using three indices for equalities between fuzzy numbers. By considering the conflict between the fuzzy threshold for the three indices and the fuzziness of the fuzzy linear regression model, three types of multiobjective programming problems for obtaining fuzzy linear regression models are formulated corresponding to the three indices. Then a linear programming based interactive decision making method to derive the satisficing solution of the decision maker for the formulated multiobjective programming problems is developed. A numerical example demonstrates the appropriateness and efficiency of the proposed method.

Journal ArticleDOI
TL;DR: In this paper, the sensitivity of the additive model's classifications in data envelopment analysis (DEA) is investigated by means of new DEA formulations focusing on the stability (sensitivity) of an organization's classification (whether efficient or inefficient).
Abstract: In contrast to existing sufficient conditions for preservation of efficiency under special perturbations and matrix structural assumptions, sensitivity of the additive model's classifications in data envelopment analysis (DEA) is investigated by means of new DEA formulations focusing on the stability (sensitivity) of an organization's classification (whether efficient or inefficient). The formulations for the additive model are linear programming problems whose solutions yield a particular region of stability, a ‘cell’, in which an organization's classification remains unchanged. The largest such cell can always be easily computed for each organization and additionally theoretically characterized simply as optimal solutions of particular linear programming problems.

Journal ArticleDOI
TL;DR: The network proposed by M.P. Kennedy and L.O. Chua is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems, such as the least-squares problem.
Abstract: Neural networks for linear and quadratic programming are analyzed. The network proposed by M.P. Kennedy and L.O. Chua (IEEE Trans. Circuits Syst., vol.35, pp.554-562, May 1988) is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems, such as the least-squares problem. For quadratic programming, the network converges either to an equilibrium or to an exact solution, depending on whether the problem has constraints or not. The results also suggest an analytical approach to solve the linear system Bx=b without calculating the matrix inverse. The results are directly applicable to optimization problems with C/sup 2/ convex objective functions and linear constraints. The dynamics and applicability of the networks are demonstrated by simulation. The distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter. >

Journal ArticleDOI
TL;DR: This paper presents approximation algorithms for median problems in metric spaces and fixed-dimensional Euclidean space that use a new method for transforming an optimal solution of the linear program relaxation of the s-median problem into a provably good integral solution.

Journal ArticleDOI
TL;DR: Various circuit architectures of simple neuron-like analog processors are considered for online solving of a system of linear equations with real constant and/or time-variable coefficients and can be used for solving linear and quadratic programming problems.
Abstract: Various circuit architectures of simple neuron-like analog processors are considered for online solving of a system of linear equations with real constant and/or time-variable coefficients. The proposed circuit structures can be used, after slight modifications, in related problems, namely, inversion and pseudo-inversion of matrices and for solving linear and quadratic programming problems. Various ordinary differential equation formulation schemes (generally nonlinear) and corresponding circuit architectures are investigated to find which are best suited for VLSI implementations. Special emphasis is given to ill-conditioned problems. The properties and performance of the proposed circuit structures are investigated by extensive computer simulations. >

Journal ArticleDOI
TL;DR: An improved lower bound of 1.540 is given for the parametric case where all items are smaller than or equal to 1/r, rϵ N +, and this work gives improved lower bounds for the asymptotic worst case ratio.

Journal ArticleDOI
TL;DR: This paper presents an application of fuzzy linear programming to the linear multiobjective transportation problem, which gives efficient solutions as well as an optimal compromise solution.

Journal ArticleDOI
TL;DR: Fuzzy sets are used to modify the linear programming (LP) approach to voltage control and to incorporate some heuristic concepts of the expert system approach.
Abstract: The integration of traditional and heuristic techniques is considered for the reactive power/voltage control program. The steady-state reactive power problem is addressed. Fuzzy sets are used to modify the linear programming (LP) approach to voltage control and to incorporate some heuristic concepts of the expert system approach. Multiple objectives and soft constraints are modeled using fuzzy sets. Piecewise linear convex membership functions for the fuzzy sets are defined. Under this definition, the fuzzy optimization problem is reformulated as a standard linear programming problem. The objective function represents the compromise among the original competing objectives and the soft constraints. In addition, discrete constraints are considered. Numerical examples demonstrate the approach. >

Journal ArticleDOI
TL;DR: A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables.
Abstract: Experience with solving a 12.753.313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX, an interior point method as implemented in OBI, and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Journal ArticleDOI
TL;DR: In this article, a linear programming model for generation rescheduling and minimization of the amount of load shed is presented for correcting the voltage problem, and the composite system is classified into different system states for which probabilistic indices are calculated.
Abstract: An electric power network containing generation and transmission facilities can be divided into several states in terms of the degree to which adequacy and security constraints are satisfied in a reliability evaluation of the composite system. The composite system is classified into different system states for which probabilistic indices are calculated. Both annualized and annual indices using a seven-step load model are presented for two test systems. Selection methods are used to detect problem-creating contingencies. A linear programming model for generation rescheduling and minimization of the amount of load shed is presented. A linear programming model for correcting the voltage problem is presented. >