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AN EXTERNAL RECONSTRUCTION APPROACH (ERA) TO linear-programming

COMPUTERS & OPERATIONS RESEARCH 1986年第1期13卷 95-100页

作者： ZELENY, M Graduate School of Business Fordham University at Lincoln Center New York NY 10023 U.S.A.

In recent years, there has been a marked revival of interest in linear programming, searching for new, nonsimplex ways of dealing with the old linear programming (LP) problem. The present analysis presents a simple, computationally attractive approach of considerable practical promise and theoretical importance. An externally initiated optimum-searching procedure for large-scale LP optimization and design is developed. It is potentially useful for LP problems characterized by a large number of constraints relative to the number of variables. Although the approach, called External Reconstruction Approach (ERA), still relies on LP subproblems (and their simplex-type solution procedures), its global philosophy is clearly nonsimplex and nonellipsoid in nature. In addition to its computational potential, ERA's main advantages are: 1. renewed flexibility, 2. built-in parallelism, 3. suitability to design and optimization, and 4. the ability to handle mixed (both soft and hard) constraints. A numerical example is provided.

关键词： mathematical programming, linear

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BOUNDS ON THE EFFECT OF AGGREGATING VARIABLES IN linear-PROGRAMS

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OPERATIONS RESEARCH 1980年第2期28卷 403-418页

作者： ZIPKIN, PH Columbia University New York New York

This paper explores the effects of aggregating variables in large linear programs. We define a reasonable criterion for the resulting loss in accuracy, and derive bounds on this quantity. A posteriori bounds may be calculated after solving the aggregated problem, and a priori bounds before. Also, we show that standard iterative methods can be used to improve the accuracy of a given aggregated problem. A numerical example illustrates the results.

关键词： mathematical programming, linear

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NOTE ON EXPLICIT SOLUTION IN linear FRACTIONAL programming

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NAVAL RESEARCH LOGISTICS 1976年第1期23卷 161-167页

作者： CHARNES, A GRANOT, D GRANOT, F UNIV TEXAS AUSTINTX 78705 UNIV BRITISH COLUMBIA VANCOUVER V6T 1WSBRITISH COLUMBICANADA

The fractional interval programming problem (FIP) is analyzed and all its optimal solutions are found explicitly. The proofs and analysis provided are considerably simpler than those of A. Charnes and W. W. Cooper (Na... 详细信息

关键词： mathematical programming, linear

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AN ADVANCED START ALGORITHM FOR ALL-INTEGER programming

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COMPUTERS & OPERATIONS RESEARCH 1985年第3期12卷 301-309页

作者： HANNA, ME AUSTIN, LM TEXAS TECH UNIV COLL BUSINESS ADMLUBBOCKTX 79409

An all-integer cutting plane technique - the Advanced Start Algorithm (ASA) - is presented for solving the all-integer (otherwise linear) programming (IP) problem. This is the 7th algorithm in an ongoing research program aimed at reviving use of the elegant all-integer cutting plane. The approach does not suffer from computer round-off error. A good advanced primal-infeasible start is developed based on the optimal solution to the linear programming relaxation, and a 2-stage dual/primal algorithm is employed to obtain the optimal solution to IP. Operation of the ASA is illustrated on 3 small problems; computational results are shown on a set of standard test problems.

关键词： mathematical programming, linear

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SUCCESSIVE APPROXIMATION IN SEPARABLE programming - AN IMPROVED PROCEDURE FOR CONVEX SEPARABLE PROGRAMS

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NAVAL RESEARCH LOGISTICS 1986年第2期33卷 325-358页

作者： THAKUR, LS Yale School of Management Yale University New Haven Connecticut 06520

We implement a solution procedure for general convex separable programs where a series of relatively small piecewise linear programs are solved as opposed to a single large one, and where, based on bound calculations developed in [13] and [14], the ranges of linearization are systematically reduced for successive programs. The procedure inherits ε‐convergence to the global optimum in a finite number of steps, but perhaps its most distinct feature is the rigorous way in which ranges containing an optimal solution are reduced from iteration to iteration. This paper describes the procedure, called successive approximation , discusses its convergence, tightness of the bounds, bound‐calculation overhead, and its robustness. It presents a computer implementation to demonstrate its effectiveness for general problems and compares it (1) with the more standard separable programming approach and (2) with one of the recent augmented Lagrangian methods [10] included in a comprehensive study of nonlinear programming codes [12]. It seems clear from over 130 cases resulting from 80 distinct problems studied here that significant savings in terms of computational effort can be realized by a judicious use of the procedure, and the ease with which it can be used is appreciably increased by the robustness it shows. Moreover, for most of these problems, the advantage increases as the size, nonlinearity, and the degree of desired accuracy increase. Other important benefits include significantly smaller storage requirements, the ability to estimate the error in the current solution, and to terminate the algorithm as soon as the acceptable level of accuracy has been achieved. Problems requiring up to about 10,000 nonzero elements in their specification and about 45,000 nonzero elements in the generated separable programs resulting from up to 70 original nonlinear variables and 70 nonlinear constraints are included in the computations.

关键词： mathematical programming, linear

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PROBABILISTIC APPROACH TO ASSESSMENT OF HEURISTICS IN 0-1 linear-programming

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CYBERNETICS 1987年第1期23卷 1-7页

作者： SERGIENKO, IV SHILO, VP

The ability to exploit the specific features of the problem for its solution is of major importance in applications of 0-1 linear programming. The efficiency of the solution procedure generally depends on how correctly and to what extent the particular algorithm exploits the specific features of the problem. One of the possible techniques for 'tuning' the algorithm for the particular problem is by developing some heuristics, or bounding rules, which usually show good performance when solving problems from the class for which they were originally designed. Heuristics are very popular for the solution of many applied problems. Always every optimizer has taken advantage of his personal experience to invent heuristics which perform with varying degrees of success, and it is therefore important to be able to choose from among this multitude a good heuristic for a specific application. In this article, we attempt to solve this problem for some heuristics. Our approach is based on probabilistic analysis of the heuristics and the corresponding applications.

关键词： mathematical programming, linear

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HEURISTICS AND REDUCTION METHODS FOR MULTIPLE CONSTRAINTS 0-1 linear-programming PROBLEMS

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1986年第2期24卷 206-215页

作者： FREVILLE, A PLATEAU, G UNIV LILLE 1 UER IEEA INFORMATBAT M3F-59655 VILLENEUVE DASCQFRANCE

This work extends the efficient results relative to the 0-1 knapsack problem to the multiple inequality constraints 0-1 linear programming problems. The two crucial phases for the solving of this type of problems are presented: (i) Two linear expected time complexity greedy algorithms are proposed for the determination of a lower bound on the optimal value by using a cascade of surrogate relaxations of the original problem whose sizes are decreasing step by step. A comparative study with the best known heuristic methods is reported; it concerned the accuracy of the approximate solutions and the practical computational times. (ii) This greedy algorithm is inserted in an efficient reduction framework. Variables and constraints are eliminated by the conjunction of tests applied to Lagrangean and surrogate relaxations of the original problem. A lot of computational results are summarized by considering test problems of the literature. [ABSTRACT FROM AUTHOR]

关键词： mathematical programming, linear

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A COMPUTATIONAL COMPARISON OF GOMORY AND KNAPSACK CUTS

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COMPUTERS & OPERATIONS RESEARCH 1987年第6期14卷 449-456页

作者： JENKINS, L PETERS, D Department of Engineering Management Royal Military College of Canada Kingston Ontario K7K 5L0 Canada

A widespread method of solving integer programs is to solve the problem first as a linear program (LP) and then add constraints that cut off noninteger solutions from the set of LP feasible solutions. The method of Gomory can generate a variety of different cuts, but there are few reports on the systematic testing of the effectiveness of different cuts. Computational comparisons are reported between a number of different cuts, including a new and successful one. Gomory cuts can be unsuccessful because of slow convergence with the accompanying difficulties of computer round-off error. The knapsack method of cuts has been proposed for generating, for zero to one integer problems, cuts that are generally tighter than Gomory and thus give faster convergence. The 2 methods are compared. Knapsack is found to be superior to Gomory for zero to one problems having dense constraint matrices but only sightly better for problems with sparse matrices. Knapsack cuts applied to general integer problems that are reformulated as zero to one are less successful than Gomory cuts applied to the original integer problem.

关键词： mathematical programming, linear

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PRACTICAL linear-programming IN MACROECONOMIC PLANNING - THE CASE OF HUNGARY

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1985年第2期22卷 204-215页

作者： AUGUSZTINOVICS, M BOD, P HUNGARIAN ACAD SCI INST MATHV REALTANODA U 13-15POB 127H-1364 BUDAPESTHUNGARY

Discusses practical linear programming in macroeconomic planning by studying the case of Hungary. Models built for long term economic planning of Hungary; Approaches adopted for dealing with non-linearity of the model due to variant parameters affecting long-term production planning; Economic assumptions of the study; Description of the operations of the planning system.

关键词： mathematical programming, linear

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PERTURBATION ANALYSIS OF GENERAL LP MODELS - A UNIFIED APPROACH TO SENSITIVITY, PARAMETRIC, TOLERANCE, AND MORE-FOR-LESS ANALYSIS

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mathematical AND COMPUTER MODELLING 1990年第8期13卷 79-102页

作者： ARSHAM, H OBLAK, M Department of Information and Quantitative Sciences University of Baltimore Baltimore MD 21201 U.S.A.

This paper develops an alternative approach to postoptimality analysis for general linear programming (LP) problems that provides a simple framework for the analysis of any single or simultaneous change of right-hand side (RHS) or cost coefficient terms for which the current basis remains optimal by solving the nominal LP problem with perturbed RHS terms. Postoptimality analysis of a row or column of the matrix coefficients is also discussed. The goal is a theoretical unification, as well as an advancement in the practical implementation of postoptimality analysis. Some common applications, such as ordinary sensitivity, the 100% rule, and parametric analysis, as well as extensions of recent developments such as tolerance analysis and the more-for-less paradox, are discussed in the context of numerical examples.

关键词： mathematical programming, linear

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