Showing papers on "Linear programming published in 1973"
TL;DR: The solution of the Chinese postman problem using matching theory is given and the convex hull of integer solutions is described as a linear programming polyhedron, used to show that a good algorithm gives an optimum solution.
Abstract: The solution of the Chinese postman problem using matching theory is given. The convex hull of integer solutions is described as a linear programming polyhedron. This polyhedron is used to show that a good algorithm gives an optimum solution. The algorithm is a specialization of the more generalb-matching blossom algorithm. Algorithms for finding Euler tours and related problems are also discussed.
963 citations
TL;DR: Convergence was obtained in three iterations for all test systems considered and solution time is small enough to allow the method to be used for on-line dispatching at practical time intervals.
Abstract: The economic dispatch problem is formulated as a quadratic programming problem and solved using Wolfe's algorithm. The method is capable of handling both equality and inequality constraints on p, q, and v and can solve the load flow as well as the economic dispatch problem. The quadratic programming algorithm does not require the use of penalty factors or the determination of gradient step size which can cause convergence difficulties. Convergence was obtained in three iterations for all test systems considered and solution time is small enough to allow the method to be used for on-line dispatching at practical time intervals. Results are presented for 5, 14, 30, 57, and 118 bus test systems.
182 citations
TL;DR: In this paper the applications of goal programming to industrial and economic planning are reviewed, and it is shown how the linear programming framework can be extended to include a series of goals (or objectives).
Abstract: In this paper the applications of goal programming to industrial and economic planning are reviewed, and it is shown how the linear programming framework can be extended to include a series of goals (or objectives). A planning method is suggested in which the decision maker is presented with a list of all the possible attainable goals from which he picks the most appropriate set.
176 citations
TL;DR: A direct application of the linear programming duality theorem leads to a new necessary condition for the existence of hamiltonian circuits; this condition appears to be stronger than the ones previously known.
Abstract: Jack Edmonds developed a new way of looking at extremal combinatorial problems and applied his technique with a great success to the problems of the maximal-weight degreeconstrained subgraphs. Professor C. St. J.A. Nash-Williams suggested to use Edmonds' approach in the context of hamiltonian graphs. In the present paper, we determine a new set of inequalities (the “comb inequalities”) which are satisfied by the characteristic functions of hamiltonian circuits but are not explicit in the straightforward integer programming formulation. A direct application of the linear programming duality theorem then leads to a new necessary condition for the existence of hamiltonian circuits; this condition appears to be stronger than the ones previously known. Relating linear programming to hamiltonian circuits, the present paper can also be seen as a continuation of the work of Dantzig, Fulkerson and Johnson on the traveling salesman problem.
157 citations
TL;DR: It is shown that a linear optimization problem with multiple objective functions can be formulated in this fashion in order to obtain a solution set satisfying all the requirements for an efficient solution of the problem.
Abstract: The purpose of this paper is to develop a useful technique for solving linear programmes involving more than one objective function. Motivation for solving multicriterion linear programmes is given along with the inherent difficulty associated with obtaining a satisfactory solution set. By applying a linear programming approach for the solution of two person–zero sum games with mixed strategies, it is shown that a linear optimization problem with multiple objective functions can be formulated in this fashion in order to obtain a solution set satisfying all the requirements for an efficient solution of the problem. The solution method is then refined to take into account disparities between the magnitude of the values generated by each of the objective functions and solution preferences as determined by a decision-maker. A summary of the technique is then given along with several examples in order to demonstrate its applicability.
136 citations
TL;DR: This paper presents an algorithm, based on the simplex routine, that provides a way to solve a problem in which the objective function is not linear, but rather is represented by a ratio of two linear functions.
Abstract: This paper presents an algorithm, based on the simplex routine, that provides a way to solve a problem in which the objective function is not linear, but rather is represented by a ratio of two linear functions. This algorithm has a computational advantage over two previous ones because it requires neither variable transformations nor the introduction of new variables and constraints.
118 citations
TL;DR: Rules are given that enable the transformation of a0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints.
Abstract: This paper gives rules that enable the transformation of a 0-1 polynomial programming problem into a 0-1 linear programming problem to be effected with reduced numbers of constraints. Rules are also given that provide reduced numbers of variables when the true variables of interest are not individual cross-product terms, but sums of such terms or polynomials of the form âxjp.
118 citations
TL;DR: A new method of generating all vertices of a given convex polytope is described, which embeds the givenpolytope in a one-higher-dimensional space and associates a number with each interior point that facilitates the construction of a spanning tree for all of the interior points.
Abstract: This paper describes a new method of generating all vertices of a given convex polytope. Additionally, irrelevant constraints are easily identified without the necessity of enumerating any of the vertices of the given convex polytope. The method embeds the given polytope in a one-higher-dimensional space. The projection of the additional vertices formed by the embedding process into the original space lie in the interior of the polytope and have a tree structure for one and two polytopes. For higher dimensions, the embedding process associates a number with each interior point that facilitates the construction of a spanning tree for all of the interior points. The interior points added can be efficiently generated by a variant of the simplex method. The vertices of the original polytope can be generated easily from these internal points by analyzing the appropriate simplex tableaux.
115 citations
Book•
01 Jan 1973
TL;DR: Theorem 1.1. Convex Cones and linear inequalities over cones as mentioned in this paper, and the complementarity problem of linear programming over cones are discussed in Section 2.1 of the glossary.
Abstract: 1. Convex Cones and linear Inequalities.- 1. Separation theorems.- 2. Cones and duals.- 3. Linear inequalities over cones.- 4. Theorems of the alternative.- 2. Mathematical Programming over Cones.- 5. Linear programming.- 6. Quadratic programming.- 7. The complementarity problem.- 8. Non-linear programming.- 3. Cones in Matrix Theory.- 9. Cones of matrices.- 10. Lyapunov type theorems.- 11. Cone monotonicity.- 12. Iterative methods for linear systems.- References.- Glossary of Notations.
TL;DR: Bounds for the displacement in the solution set of a system of perturbed linear inequalities are developed and results are applied to find estimates for changes in the Solution Set of a perturbedlinear program.
TL;DR: It is shown how a variety of problems may be restructured as multi-stage linear programs to which this algorithm may be applied, and some computational experience is reported.
Abstract: A multi-stage linear program is defined with linking variables that connect consecutive stages. Optimality conditions for the composite problem are partitioned into local and linking conditions. When the Dantzig-Wolfe decomposition scheme is applied with the first stage as the master, the subproblem is also a MLP with one' less stage. The same decomposition is then applied to the subproblem, giving rise to a nested decomposition scheme, in which each stage acts as a master for the following stage and a subproblem for the preceding. Optimizing a single stage problem results in satisfying the “local” optimality conditions. A very general rule is given for selecting the next subproblem to optimize, and finite convergence to a solution satisfying all linking conditions is demonstrated. Procedures for extracting the optimal primal solution at the end of the main algorithm and for initialization are given. Particular rules for selecting the next subproblem and for generating additional proposal vectors are discussed. Finally, it is shown how a variety of problems may be restructured as multi-stage linear programs to which this algorithm may be applied, and some computational experience is reported.
TL;DR: In this article, an iterative procedure for determining the joint positions corresponding to a minimum mass space truss is presented, which includes a sequence of linear programming problems (SLP, with move-limits), and for each of these problems only the nearby constraints are considered.
Abstract: An iterative procedure for determining the joint positions corresponding to a minimum mass space truss is presented. Displacement constraints and nonconstant stress constraints (stability) are taken into account. The truss is presumed to carry consecutively a large number of different systems of forces. The iteration includes a sequence of linear programming problems (SLP, with move-limits), and for each of these problems only the nearby constraints are considered. Analytical expressions are given for the gradients describing the linear problems. A dome is optimized using different constraints.
TL;DR: For 54 unimodular linear programming problems it is shown that either (i) the objective function is unbounded, or (ii) the problem is infeasible, or the problem can be solved by solving a related transportation problem.
Abstract: For 54 unimodular linear programming problems it is shown that either (i) the objective function is unbounded, or (ii) the problem is infeasible, or (iii) the problem can be solved by solving a related transportation problem. The related transportation problem is obtained by adding at the most two new constraints to the original problem.
TL;DR: The formulations as well as the proofs and the transformations provided by the general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases.
Abstract: : A complete analysis and explicit solution is presented for the problem of linear fractional programming with interval programming constraints whose matrix is of full row rank. The analysis proceeds by simple transformation to canonical form, exploitation of the Farkas-Minkowski lemma and the duality relationships which emerge from the Charnes-Cooper linear programming equivalent for general linear fractional programming. The formulations as well as the proofs and the transformations provided by our general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases. The augmentation developing the explicit solution is presented, for clarity, in an algorithmic format.
TL;DR: The feasibility of simulating the overall system problem, including load-frequency control of equivalent generation, using a combined analogue-digital process computer system linked to a large scientific digital computer by data link is investigated.
Abstract: Optimum scheduling of generation subject to constraints on maximum and minimum levels and rates of change of generation, spare capacity and line flow limits, is studied. A linear programming formulation for system constraints is used with a quadratic cost function for generation and transmission line loss. An optimal solution for real power dispatch is obtained by quadratic programming, and optimum allocation of reactive power is based on a gradient technique which minimises transmission loss subject to nodal voltage and reactive power limits. An ac load flow analysis is incorporated together with a load prediction program based on a spectral analysis of past load data. The feasibility of simulating the overall system problem, including load-frequency control of equivalent generation, using a combined analogue-digital process computer system linked to a large scientific digital computer by data link is investigated.
01 May 1973
TL;DR: In this article, a 2nd-order approximation to the power-generation-cost function is derived by the derivation of linear constraints through the system sensitivity relations and by the use of a two-dimensional approximation of the power generation cost function.
Abstract: Mathematical programming offers attractive advantages as an optimising technique. Unfortunately, the optimisation of economic dispatch in power systems is a nonlinear problem, and so it is, in principle, beyond the reach of mathematical programming. In the paper, this difficulty is resolved by the derivation of linear constraints through the system sensitivity relations and by the use of a 2nd-order approximation to the power-generation-cost function. Quadratic programming is employed to solve the problem, and, with only one application of the algorithm, the results are comparable to those obtained from gradient techniques. The use of quadratic programming and the change of type of control variables during optimisation obviate the need for penalty functions. The computing times taken by the algorithm when it is applied to test systems are encouragingly short. Security constraints can be easily incorporated, and, if required, the minimum-reactive-power problem can be solved. A solution of the minimum-loss problem with linear programming is also illustrated.
TL;DR: The zero-one composition of the constraint matrix and the right-hand side of ones suggested an algorithm in which dual simplex iterations are performed whenever unit pivots are available and Gomory all integer cuts are adjoined when they are not.
Abstract: Computational experience with a modified linear programming method for the inequality or equality set covering problem (i.e. minimize cx subject to Ex ≥ e or Ex = e, xi = 0 or 1, where E is a zero-one matrix, e is a column of ones, and c is a nonnegative integral row) is presented. The zero-one composition of the constraint matrix and the right-hand side of ones suggested an algorithm in which dual simplex iterations are performed whenever unit pivots are available and Gomory all integer cuts are adjoined when they are not. Applications to enumerative and heuristic schemes are also discussed.
TL;DR: This paper studies the linear programming problem in which all coefficients are rational functions of a single parameter t called “time,” and provides an algorithm that can solve problems of the following two types: steady-state behavior and sensitivity analysis.
Abstract: This paper studies the linear programming problem in which all coefficients (even those of the stipulations matrix) are rational functions of a single parameter t called “time,” and provides an algorithm that can solve problems of the following two types: (1) Steady-state behavior [the algorithm can be used to determine the functional form x(t) of the optimal solution as a function of t, this form being valid for all “sufficiently large” values of t], and (2) sensitivity analysis [if a value t0 of “time” is given, the algorithm can be used to determine the two possible functional forms of the optimal solution for all values of t “sufficiently close” to t0 (the first functional form valid for t < t0, the second for t < t0)] In addition, the paper gives certain qualitative information regarding steady-state behavior, including the following result: If for some one of the properties of consistency, boundedness, or bounded constraint set, there exists a sequence tn ↗ +∞ such that the linear program at n has
TL;DR: In this article, a new method of kineto-elastodynamic design is developed and illustrated with examples, which is achieved by first performing the kinematic synthesis of the mechanism considering its links to be rigid and then proportioning the areas of cross-section of the links optimally to account for the kineto elastodynamic effects.
TL;DR: The results of the study indicate that the Remes algorithm and the differential correction algorithm III are the most satisfactory methods to use in practice.
TL;DR: The generalized lattice-point problem, posed by Charnes and studied by M. L. Kirby, is a linear program whose solutions are constrained to be extreme points of a specified polytope and how to exploit this and more general problems by convexity or intersection cut strategies without resorting to standard problem-augmenting techniques.
Abstract: The generalized lattice-point problem, posed by Charnes and studied by M. J. L. Kirby, H. Love, and others, is a linear program whose solutions are constrained to be extreme points of a specified polytope. We show how to exploit this and more general problems by convexity or intersection cut strategies without resorting to standard problem-augmenting techniques such as introducing 0-1 variables. In addition, we show how to circumvent "degeneracy" difficulties inherent in this problem without relying on perturbation which provides uselessly shallow cuts by identifying nondegenerate subregions relative to which cuts may be defined effectively. Finally, we give results that make it possible to obtain strengthening cuts for problems with special structures.
TL;DR: In this article, a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty is presented, in which the m-dimensional space in which each combination of the discrete values is a lattice point is used to delete infeasible points from the space.
Abstract: This paper presents a solution procedure for discrete stochastic programs with recourse linear programs under uncertainty. It views the m stochastic elements of the requirements vector as an m-dimensional space in which each combination of the discrete values is a lattice point. For a given second-stage basis, certain of the lattice points are feasible. A procedure is presented to delete infeasible points from the space. Thus, the aggregate probability associated with points feasible for this basis can be enumerated, and used to weight the vector of dual variables defined by the basis. Finally, the paper presents a systematic procedure for changing optimal bases so that a feasible and optimal basis is found for every lattice point.
TL;DR: In this paper, the problem of optimizing the prestressing force and the tendon configuration for an indeterminate prestressed structure with prescribed cross-sectional dimensions is formulated in linear programming form, the structure is subjected to multiple load conditions and constraints are related to the structural behaviour and tendon configuration.
Abstract: The problem of optimizing the prestressing force and the tendon configuration for an indeterminate prestressed structure with prescribed cross-sectional dimensions is formulated in linear programming form, The structure is subjected to multiple load conditions and constraints are related to the structural behaviour and to the tendon configuration.
It is shown that the number of behaviour constraints at each point in the structure can be reduced, as they represent parallel hyperplanes in the design space. Necessary conditions for feasible solutions are derived. The method based on transformation of the design variables, is suitable for beams, frames, grids and plates. Its application is illustrated for the case of a prestressed bridge.
01 Jul 1973
TL;DR: The approximate solution reveals also the structure of the exact solution and makes its construction possible and the optimal solution is shown to be a function of bounded variation.
Abstract: A method for finding approximate solutions for continuous linear programming problems is suggested. The required conditions to be met are: a) the matrix associated with the integrals in the constraints is constant; b) all functions involved are of bounded variation; c) the matrices involved satisfy certain boundedness conditions, and d) there exist feasible solutions.
TL;DR: In this article, it was shown that a linear fractional program can have a fractional linear program as its dual, under certain restrictions, under the assumption that the program is linear.
TL;DR: This paper shows that the linear programming formulation of the two-commodity network flow problem leads to a direct derivation of the known results concerning this problem.
Abstract: This paper shows that the linear programming formulation of the two-commodity network flow problem leads to a direct derivation of the known results concerning this problem. An algorithm for solving the problem is given which essentially consists of two applications of the Ford—Fulkerson max flow computation. Moreover, the algorithm provides constructive proofs for the results. Some new facts concerning feasible integer flows are also given.