Showing papers on "Linear programming published in 1980"
2,032 citations
TL;DR: An algorithm for the 0-1 knapsack problem (KP), which relies mainly on three new ideas, one of which is a binary search-type procedure for solving LKP which, unlike earlier methods, does not require any ordering of the variables.
Abstract: We describe an algorithm for the 0-1 knapsack problem (KP), which relies mainly on three new ideas. The first one is to focus on what we call the core of the problem, namely, a knapsack problem equivalent to KP, defined on a particular subset of the variables. The size of this core is usually a small fraction of the full problem size, and does not seem to increase with the latter. While the core cannot be identified without solving KP, a satisfactory approximation can be found by solving the associated linear program (LKP). The second new ingredient is a binary search-type procedure for solving LKP which, unlike earlier methods, does not require any ordering of the variables. The computational effort involved in this procedure is linear in the number of variables. Finally, the third new feature is a simple heuristic which under certain conditions finds an optimal solution with a probability that increases with the size of KP. Computational experience with an algorithm based on the above ideas, on several ...
468 citations
01 Aug 1980
TL;DR: Issues have in general not been well understood, including the exact character of the ellipsoid method and of Khachiyan''s result on polynomiality, its practical significance inlinear programming, its implementation, its potential applicability to problems outside of the domain of linear programming, and its relationship to earlier work.
Abstract: IN February 1979 a note by L.G. Khachiyan indicated how an ellipsoid method for linear programming can be implemented in polynomial time. This result has caused great excitement and stimulated a flood of technical papers. Ordinarily there would be no need for a survey of work so recent. The current circumstances are obviously exceptional. Word of Khachiyan''s result has spread extraordinarily fast, much faster than comprehension of its significance. A variety of issues have in general not been well understood, including the exact character of the ellipsoid method and of Khachiyan''s result on polynomiality, its practical significance in linear programming, its implementation, its potential applicability to problems outside of the domain of linear programming, and its relationship to earlier work. Our aim here is to help clarify these important issues in the context of a survey of the ellipsoid method, its historical antecedents, recent developments, and current research.
411 citations
TL;DR: Pivot and Complement as mentioned in this paper is a heuristic for finding approximate solutions to 0-1 programming problems that uses the fact that a 0 -1 program is equivalent to the associated linear program with the added requirement that all slack variables, other than those in the upper bounding constraints, be basic.
Abstract: Pivot and Complement is a heuristic for finding approximate solutions to 0-1 programming problems. It uses the fact that a 0-1 program is equivalent to the associated linear program with the added requirement that all slack variables, other than those in the upper bounding constraints, be basic. The procedure starts by solving the linear program; then performs a sequence of pivots aimed at putting all slacks into the basis at a minimal cost; finally, it attempts to improve the 0-1 solution obtained in this way by a local search based on complementing certain sets of 0-1 variables. The computational effort involved in the procedure is bounded by a polynomial in the number of constraints and variables. For the 92 test problems with 5 to 200 constraints and 20 to 900 variables on which the procedure was run, the time used to solve the linear program on the average considerably exceeded the time used for everything else. As to the quality of the solutions obtained, in 45 cases, or 49% of the total, the proced...
259 citations
01 Jan 1980
TL;DR: In this article, the authors consider two questions arising in the analysis of heuristic algorithms: 1) is there a general procedure involved when analysing a particular problem heuristic, and 2) how can heuristic procedures be incorporated into optimising algorithms such as branch and bound.
Abstract: We consider two questions arising in the analysis of heuristic algorithms.
(i)
Is there a general procedure involved when analysing a particular problem heuristic?
(ii)
How can heuristic procedures be incorporated into optimising algorithms such as branch and bound?
241 citations
TL;DR: In this paper, the authors consider linear multiple objective programs with coefficients of the criteria given by intervals and present a branch and bound algorithm to test if a feasible extreme point is efficient in the problem considered.
Abstract: In this paper we consider linear multiple objective programs with coefficients of the criteria given by intervals. This class of problems is of practical interest since in many instances it is difficult to determine precisely the coefficients of the objective functions. A subproblem to test if a feasible extreme point is efficient in the problem considered is obtained. A branch and bound algorithm to solve the subproblem as well as computational results are provided. Extensions are discussed.
181 citations
TL;DR: This work identifies a large class of cyclic staffing problems for which special structure permits the ILP to be solved parametrically as a bounded series of network flow problems.
Abstract: A fundamental problem of cyclic staffing is to size and schedule a minimum-cost workforce so that sufficient workers are on duty during each time period. This may be modeled as an integer linear program with a cyclically structured 0-1 constraint matrix. We identify a large class of such problems for which special structure permits the ILP to be solved parametrically as a bounded series of network flow problems. Moreover, an alternative solution technique is shown in which the continuous-valued LP is solved, and the result rounded in a special way to yield an optimum solution to the ILP.
174 citations
TL;DR: In this article, the use of linear prograrming (LP) in power system reactive power control calculations is presented, where solutions are constrained within limits set by busbar voltage levels and reactive power flows.
Abstract: The use of linear prograrming (LP) in power system reactive power control calculations is presented in this paper. Ihe method incorporates the usual reactive power control devices and is particularly successful in rescheduling tap-changing transfornmers. Solutions are constrained within limits set by busbar voltage levels and network reactive power flows. Results are given which indicate a high degree of reliability and versatility, with solution times which are suitable for real-time applications on large systems.
169 citations
01 Jan 1980
TL;DR: The empirical results lend convincing support to the hypothesis that inequalities defining facets of the convex hull of tours are of substantial computational value in the solution of this difficult combinatorial problem.
Abstract: The symmetric travelling salesman problem has been formulated by Dantzig, Fulkerson and Johnson in 1954 as a linear programming problem in zero-one variables. We use this formulation and report the results of a computational study addressing itself to the problem of proving optimality of a particular tour. The empirical results based on a total of 74 problems of sizes ranging from 15-cities to 318-cities lend convincing support to the hypothesis that inequalities defining facets of the convex hull of tours are of substantial computational value in the solution of this difficult combinatorial problem.
162 citations
TL;DR: It is established that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.
Abstract: We establish that in the worst case, the computational effort required for solving a parametric linear program is not bounded above by a polynomial in the size of the problem.
146 citations
TL;DR: A method for generating the entire efficient set for a multiple objective linear program is developed, based on two characterizations of maximal efficient faces.
Abstract: A method for generating the entire efficient set for a multiple objective linear program is developed. The method is based on two characterizations of maximal efficient faces. The first characterization is used to determine the set of maximal efficient faces incident to a given efficient vertex, and the second characterization ensures that previously generated maximal efficient faces are easily recognized (and not regenerated). The efficient set is described as the union of maximal efficient faces. An alternate implicit description of the efficient set as the set of all optimal vectors for a finite set of linear programs is also provided.
TL;DR: An algorithm which converts a linear program min{cx ∣ Ax = b, x ≥ 0} to a network flow problem, using elementary row operations and nonzero variable-scaling, or shows that such a conversion is impossible.
Abstract: We describe an algorithm which converts a linear program min{cx ∣ Ax = b, x ≥ 0} to a network flow problem, using elementary row operations and nonzero variable-scaling, or shows that such a conversion is impossible. If A is in standard form, the computational effort required is bounded by Orn, where r is the number of rows and n is the number of nonzero entries of A. A method for determining whether a “binary matroid” is “graphic” plays an important role in the algorithm.
01 Jan 1980
TL;DR: It is reported how the shortest roundtrip through 120 German cities was found using a commercial linear programming code and adding facetial cutting planes in an interactive way.
Abstract: The polytope associated with the symmetric travelling salesman problem has been intensively studied recently (cf. [7, 10, 11]). In this note we demonstrate how the knowledge of the facets of this polytope can be utilized to solve large-scale travelling salesman problems. In particular, we report how the shortest roundtrip through 120 German cities was found using a commercial linear programming code and adding facetial cutting planes in an interactive way.
TL;DR: Techniques that can be used to develop groups of representative solutions from a given set are addressed, and how the methods discussed can be coordinated with one another so as to form an interactive procedure is outlined.
TL;DR: A sufficient condition for optimality is presented which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon.
Abstract: A dynamic model for the optimal control of traffic flow over a network is considered. The model, which treats congestion explicitly in the flow equations, gives rise to nonlinear, nonconvex mathematical programming problems. It has been shown for a piecewise linear version of this model that a global optimum is contained in the set of optimal solutions of a certain linear program. This paper presents a sufficient condition for optimality which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon. Computational results are reported to indicate the efficiency of this approach.
Book•
01 Jan 1980
TL;DR: This chapter discusses the generation of Deep Cuts using the Fundamental Disjunctive Inequality and Finitely Convergent Algorithms for Facialdisjunctive Programs with Applications to the Linear Complementarity Problem.
Abstract: I: Introduction.- 1.1 Basic Concepts.- 1.2 Special Cases of Disjunctive Programs and Their Applications.- 1.3 Notes and References.- II: Basic Concepts and Principles.- 2.1 Introduction.- 2.2 Surrogate Constraints.- 2.3 Pointwise-Supremal Cuts.- 2.4 Basic Disjunctive Cut Principle.- 2.5 Notes and References.- III: Generation of Deep Cuts Using the Fundamental Disjunctive Inequality.- 3.1 Introduction.- 3.2 Defining Suitable Criteria for Evaluating the Depth of a Cut.- 3.3 Deriving Deep Cuts for DC1.- 3.4 Deriving Deep Cuts for DC2.- 3.5 Other Criteria for Obtaining Deep Cuts.- 3.6 Some Standard Choices of Surrogate Constraint Multipliers.- 3.7 Notes and References.- IV: Effect of Disjunctive Statement Formulation on Depth of Cut and Polyhedral Annexation Techniques.- 4.1 Introduction.- 4.2 Illustration of the Tradeoff Between Effort for Cut Generation and the Depth of Cut.- 4.3 Some General Comments with Applications to the Generalized Lattice Point and the Linear Complementarity Problem.- 4.4 Sequential Polyhedral Annexation.- 4.5 A Supporting Hyperplane Scheme for Improving Edge Extensions.- 4.6 Illustrative Example.- 4.7 Notes and References.- V: Generation of Facets of the Closure of the Convex Hull of Feasible Points.- 5.1 Introduction.- 5.2 A Linear Programming Equivalent of the Disjunctive Program.- 5.3 Alternative Characterization of the Closure of the Convex Hull of Feasible Points.- 5.4 Generation of Facets of the Closure of the Convex Hull of Feasible Points.- 5.5 Illustrative Example.- 5.6 Facial Disjunctive Programs.- 5.7 Notes and References.- VI: Derivation and Improvement of Some Existing Cuts Through Disjunctive Principles.- 6.1 Introduction.- 6.2 Gomory's Mixed Integer Cuts.- 6.3 Convexity or Intersection Cuts with Positive Edge Extensions.- 6.4 Reverse Outer Polar Cuts for Zero-One Programming.- 6.5 Notes and References.- VII: Finitely Convergent Algorithms for Facial Disjunctive Programs with Applications to the Linear Complementarity Problem.- 7.1 Introduction.- 7.2 Principal Aspects of Facial Disjunctive Programs.- 7.3 Stepwise Approximation of the Convex Hull of Feasible Points.- 7.4 Approximation of the Convex Hull of Feasible Points Through an Extreme Point Characterization.- 7.5 Specializations of the Extreme Point Method for the Linear Complementarity Problem.- 7.6 Notes and References.- VIII: Some Specific Applications of Disjunctive Programming Problems.- 8.1 Introduction.- 8.2 Some Examples of Bi-Quasiconcave Problems.- 8.3 Load Balancing Problem.- 8.4 The Segregated Storage Problem.- 8.5 Production Scheduling on N-Identical Machines.- 8.6 Fixed Charge Problem.- 8.7 Project Selection/Portfolio Allocation/Goal Programming.- 8.8 Other Applications.- 8.9 Notes and References.- Selected References.
TL;DR: In this paper, game theoretic relations between families of games and linear programs are established by means of a family of games related to a linear programming problem, and the games-to-programming relations also open new possibilities for further relations.
TL;DR: Five problems of finding efficient vectors as a subset of a finite set of vectors are shown to be related and a common methodology based on the simplex method of linear programming is developed for solving all of them.
Abstract: Five problems of finding efficient vectors as a subset of a finite set of vectors are shown to be related and a common methodology based on the simplex method of linear programming is developed for solving all of them. Randomly generated problems for one of the five types are solved using the method, and the implications regarding computational requirements are discussed.
TL;DR: A network investment application is given which includes as a special case a coal transportation problem which is exploited to solve two stage linear programs under uncertainty where the first stage variables are 0–1.
Abstract: Stochastic programs with continuous variables are often solved using a cutting plane method similar to Benders' partitioning algorithm. However, mixed 0–1 integer programs are also solved using a similar procedure along with enumeration. This similarity is exploited in this paper to solve two stage linear programs under uncertainty where the first stage variables are 0–1. Such problems often arise in capital investment. A network investment application is given which includes as a special case a coal transportation problem.
TL;DR: Methods for assessing the loss in accuracy resulting from aggregation are developed, and several reasonable measures of "accuracy loss" for this case are defined, and the bounds on these quantities derived.
Abstract: Most applied linear programs reflect a certain degree of aggregation-either explicit or implicit-of some larger, more detailed problem. This paper develops methods for assessing the loss in accuracy resulting from aggregation. We showed previously that, when columns only are aggregated, a feasible solution to the larger problem can be recovered. This may not be the case under row-aggregation. Several reasonable measures of "accuracy loss" for this case are defined, and the bounds on these quantities derived. These results enable the modeler to compare and evaluate alternative approximate models of the same problem.
TL;DR: A constructive algorithm for solving systems of linear inequalities (LI) with at most two variables per inequality with time complexity O(mn^3 I) on a random access machine.
Abstract: We present a constructive algorithm for solving systems of linear inequalities (LI) with at most two variables per inequality. The time complexity of the algorithm is $O(mn^3 I)$ on a random access machine, where m is the number of inequalities, n the number of variables, and I the size of the binary encoding of the input. The LI problem is of importance in complexity theory because it is polynomial time (Turing) equivalent to linear programming. The subclass of LI treated in this paper is of practical interest in mechanical verification systems.
01 Jan 1980
TL;DR: In this paper, the authors proposed a model for the constraints of the trip distribution problem, and the Gravity Model as the optimal solution of the Entropy Constrained Aggregate Linear Program (ECALP).
Abstract: 1 The Transportation Planning Process.- 1.1. Goal formulation, generation and evaluation of alternatives.- 1.2. Descriptive approaches.- 1.3. Optimizing approaches.- 1.4. Welfare theory and measures of efficiency.- 1.5. The use of existing traffic patterns in the planning process.- 1.6. Outline of the paper.- I On Entropy.- 2 Entropy as a Measure of Dispersion.- 3 Some Comments upon Entropy Maximizing.- II The Doubly Constrained Trip Distribution Problem.- 4 A Model for the Constraints.- 5 The Objective Function and Our Minimization Problem.- 6 The Gravity Model as the Optimal Solution of the Entropy Constrained Aggregate Linear Program.- 7 Sensitivity and the Dual Program.- 8 Interactivity and Entropy.- 9 Benefit Measures and the Gravity Model.- III Modal Split and Assignment.- 10 Modal Split.- 11 Assignment to the Network.- IV Maximizing Total Utility.- 12 An Utility Approach to the Original Trip Distribution Problem.- 13 Entropy Constrained Aggregate Linear Program.- References.
13 Oct 1980
TL;DR: The ellipsoid method for linear programming is applied to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities.
Abstract: We show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any NP-complete combinatorial optimization problem unless NP = co-NP -- a very unlikely event. We also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities.
TL;DR: It is shown that important problems arising in process synthesis can be tackled effectively using mixed-integer linear programming techniques.
01 Jan 1980
TL;DR: An economic interpretation is given and properties of the fuzzy dual problems are derived, leading to a pair of “fuzzy dual” optimization problems.
Abstract: In classical duality theory of linear programming the saddlepoint of the Lagrangian is the solution of the max-min problem as well as of the min-max problem. In using the theory of fuzzy sets these problems are interpreted in a new sense — leading to a pair of “fuzzy dual” optimization problems. An economic interpretation is given and properties of the fuzzy dual problems are derived.
TL;DR: In this article, the shape of the cross section is defined by 5th degree polynomial which is completely determined by the boundary conditions and four design variables and the stress analysis of the disk is carried out by finite element method using isoparametric elements.
TL;DR: L. G. Khachiyan's polynomial time algorithm for determining whether a system of linear inequalities is satisfiable is presented together with a proof of its validity and can be used to solve linear programs in polynometric time.
TL;DR: In this article, a new approach to time-of-day control is presented based on the solution of a linear programming optimization problem and makes freeway volume hold the capacity constraints for the total time of control operation.
Abstract: This paper presents a new approach to time-of-day control. While time-of-day control strategies presented up-to-now are only optimal under steady-state conditions, the control algorithm derived in this paper takes into account the evolution of traffic flow according to the time delay between a volume change at a ramp and its subsequent disturbance at a freeway point downstream. The new control strategy is based on the solution of a linear programming optimization problem and makes freeway volume hold the capacity constraints for the total time of control operation. In order to reduce the computational effort a simplified version of the new algorithm is also discussed. Simulation results obtained by use of two different traffic flow models show that control derived through the new algorithm can avoid congestion and ensure operation with peak performance even if a steady-state condition is never attained.
TL;DR: In this paper, it was shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytoope defined by only the linear constraints.
Abstract: A constraintg(x)⩾0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.
TL;DR: In this paper, a fast linear programming relaxation of the MCPK problem is presented, where the running time of the algorithm is bounded by 0J log Jmax + 0N. Under certain conditions it is possible to reduce this bound to 0N steps on the average.
Abstract: A fast algorithm is presented for the linear programming relaxation of the Multiple Choice Knapsack Problem. If N is the total number of variables and J and Jmax denote the total number of multiple choice variables and the cardinality of the largest multiple choice set, respectively, the running time of the algorithm is then bounded by 0J log Jmax + 0N. Under certain conditions it is possible to reduce this bound to 0N steps on the average. Possible further improvements are also discussed.