Showing papers on "Linear programming published in 1975"
TL;DR: A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between ( VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP).
Abstract: We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1/2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.
718 citations
TL;DR: The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals and it is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet.
Abstract: The problem of specifying the optimum quantum detector in multiple hypotheses testing is considered for application to optical communications. The quantum digital detection problem is formulated as a linear programming problem on an infinite-dimensional space. A necessary and sufficient condition is derived by the application of a general duality theorem specifying the optimum detector in terms of a set of linear operator equations and inequalities. Existence of the optimum quantum detector is also established. The optimality of commuting detection operators is discussed in some examples. The structure and performance of the optimal receiver are derived for the quantum detection of narrow-band coherent orthogonal and simplex signals. It is shown that modal photon counting is asymptotically optimum in the limit of a large signaling alphabet and that the capacity goes to infinity in the absence of a bandwidth limitation.
450 citations
TL;DR: An algorithm of this type is developed which is guaranteed to find a schedule satisfying problem constraints, if one exists, and which will accept any of an important class of optimality criteria, not just levelness of reserve.
Abstract: The generator maintenance scheduling problem is formulated as a 0-1 integer linear program. Although previous papers have considered a rigorous integer programming approach intractable, an algorithm of this type is developed which (1) is guaranteed to find a schedule satisfying problem constraints, if one exists; (2) is guaranteed to find the optimal feasible schedules; and (3) will accept any of an important class of optimality criteria, not just levelness of reserve. Particular attention is directed to a new criterion incorporating dollar costs/benefits incurred by delaying or advancing maintenance on a unit.
203 citations
TL;DR: This paper gives a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms, and proves that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps.
Abstract: In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the "improvement process", and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of "improvement processes" which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will in themselves assure convergence to a solution. Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the i th stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point. We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any "improvement process" which searches through a test set at each stage converges to an optimal point in a finite nmnber of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems.
154 citations
TL;DR: In this article, a link performance function is developed to express the loss incurred by platoons traveling through a signal-controlled intersection us a function of link offset, and the optimization problem is formulated as a mixed-integer linear program and a test network is solved by branch and bound techniques using IBM's MPSX pac...
Abstract: Setting traffic signals in a signal-controlled street network involves the determination of cycle time, splits of green time, and offsets. Part I of this paper considers the network coordination problem, i.e., given a common cycle lime and green splits at each intersection, determine offsets for all signals. Part II considers the more general synchronization problem, i.e., determine simultaneously all the control variables for the network including offsets, splits, and cycle time. In Part I, a link performance function is developed to express the loss incurred by platoons traveling through a signal-controlled intersection us a function of link offset. Integer variables enter the formulation because of the periodicity of the traffic lights: The algebraic sum of the offsets around any closed loop of the network must equal an integral number of cycle limes. The optimization problem is formulated as a mixed-integer linear program and a test network is solved by branch-and-bound techniques using IBM's MPSX pac...
154 citations
01 Jan 1975
TL;DR: In this paper, a method for implicitly representing constraints of the form x≤y in linear programs when the variable y may appear in any number of such constraints is developed for problems in linear regression, plant location, and production scheduling.
Abstract: A method is developed for implicitly representing constraints of the form x≤y in linear programs when the variable y may appear in any number of such constraints. Variable x is said to have a variable upper bound (VUB). VUB constraints are common in a number of LP formulations, especially those derived from tightly formulated fixed charge integer programs. For certain of these problems the major portion of the constraints are of the VUB type. Computational experience with the method applied to problems in linear regression, plant location, and production scheduling is presented.
112 citations
TL;DR: In this paper, a procedure for representing competitive and noncompetitive market structures in linear programming (LP) models is presented, where the specification of the objective function follows from the choice of market form to be incorporated in the model.
Abstract: A procedure for representing competitive and noncompetitive market structures in linear programming (LP) models is presented. The specification of the objective function follows from the choice of market form to be incorporated in the model. Development of the function yields demand and expenditure equations and an LP tableau with separable demands. In the event of two or more products that are not separable in demand, the nonlinear demand set can be linearized directly by specification of activity vectors representing points on the demand surface and by incorporating an appropriate convex combination constraint. The specification of commodity demand structures incorporates one characteristic, which makes it particularly convenient for obtaining comparative statics solutions. The demand function for any commodity group can be rotated merely by an appropriate change in the constraint value of the convex combination inequality. A representation of international trade can be incorporated by adding commodity specific importing activities as additional production activities and adding exporting activities as additional selling activities. 21 references.
105 citations
01 Jan 1975
TL;DR: Particular attention is given to two “optimal” scaling methods, giving results on their speed and effectiveness (in terms of their optimality criteria) as well as their influence on the numerical behavior of the problem.
Abstract: The scaling of linear programming problems remains a rather poorly understood subject (as indeed it does for linear equations). Although many scaling techniques have been proposed, the rationale behind them is not always evident and very few numerical results are available. This paper considers a number of these techniques and gives numerical results for several real problems. Particular attention is given to two “optimal” scaling methods, giving results on their speed and effectiveness (in terms of their optimality criteria) as well as well as their influence on the numerical behavior of the problem.
79 citations
01 Oct 1975
TL;DR: The Combinatorial Grouping Problem as discussed by the authors is defined as the general problem of computing a set of boolean bounds for an incomplete power set P(U) so that a system of lattice-theoretic filter-ideal intersections (GLB, LUB) results which optimizes a cost function defined on the parent set U. The problem arises as a mathematical formulation of the Group-Technology Problem of Industry.
Abstract: The Combinatorial Grouping Problem is defined as the general problem of computing a set of boolean bounds (GLB, LUB) for an incomplete power set P(U)so that a system of lattice-theoretic filter-ideal intersections (GLB, LUB) results which optimizes a cost function defined on the parent set U. The problem arises as a mathematical formulation of the Group-Technology Problem of Industry. A system of independent Production Cells is required which provides the manufacturing capacity for the commercial life of an industrial product. A group of components is matched with a group of facilities so that there is a maximal degree of scheduling flexibility and a minimal cost of manufacture. A Set-Partitioning Algorithm has been developed which may be adapted to solve a wide class of related combinatorial problems.
57 citations
01 Feb 1975
TL;DR: The probability distribution of the optimum of an integer linear program is discussed in which the elements of the right-hand-side (RHS) are distributed independently and the assumptions of the asymptotic algorithm of Gomory are supposed to hold for each realization of the RHS.
Abstract: The probability distribution of the optimum (Z) of an integer linear program is discussed in which the elements of the right-hand-side (RHS) are distributed independently. The assumptions of the asymptotic algorithm ofGomory are supposed to hold for each realization of the RHS. This algorithm serves also as the theoretic framework of the present communication. Bounds and approximations for the probability function of Z are derived demanding different levels of numerical efforts. The normal distribution is a satisfactory approximation which is asymptotically correct if the elements of the RHS are uniformly distributed within bounds satisfying some requirements and the number of inequalities approaches infinity.
TL;DR: In this paper, the authors present a solution method for finding all regions Ri* that cover K* and do not overlap, based upon an algorithm for a multiparametric problem described in an earlier paper by Gal and Nedoma.
Abstract: The rim multiparametric linear programming problem (RMPLP) is a parametric problem with a vector-parameter in both the right-hand side and objective function (i.e., in the “rim”). The RMPLP determines the region K* ⊂ E* such that the problem, maximize z(λ) = cT(λ)x, subject to Ax = b(λ), x ≧ 0, has a finite optimal solution for all λ ∈ K*. Let Bi be an optimal basis to the given problem, and let Ri*, be a region assigned to Bi such that for all λ ∈ Ri* the basis Bi is optimal. The goal of the RMPLP problem is to cover K* by the Ri* such that the various Ri* do not overlap. The purpose of this paper is to present a solution method for finding all regions Ri* that cover K* and do not overlap. This method is based upon an algorithm for a multiparametric problem described in an earlier paper by Gal and Nedoma.
TL;DR: A mathematical model is presented which addresses one of the administrative tasks concerning administrators in all institutions, the establishment of faculty teaching schedules, using linear programming with two special characteristics: goal programming and mixed-integer programming.
Abstract: The administration of higher education, as with other administrative environments, has become increasingly complex. This paper presents a mathematical model which addresses one of the administrative tasks concerning administrators in all institutions, the establishment of faculty teaching schedules. The assignment technique utilized by the model is linear programming with two special characteristics: goal programming and mixed-integer programming. The goal programming characteristic refers to the provision of explicit stack variables to take on values representing deviations from assignment criteria that may result in resolving conflicts which arise from interactive administrative priorities. The mixed-integer programming characteristic refers to a requirement of the model that certain variables take on only integer values if they appear in the final solution. The model is demonstrated using two different sets of preference orderings for goal achievement.
TL;DR: The Goldfarb's algorithm can be regarded as an extension of linear programming to allow a non-linear objective function.
Abstract: Goldfarb's algorithm, which is one of the most successful methods for minimizing a function of several variables subject to linear constraints, uses a single matrix to keep second derivative information and to ensure that search directions satisfy any active constraints. In the original version of the algorithm this matrix is full, but by making a change of variables so that the active constraints become bounds on vector components, this matrix is transformed so that the dimension of its non-zero part is only the number of variablesless the number of active constraints. It is shown how this transformation may be used to give a version of the algorithm that usually provides a good saving in the amount of computation over the original version. Also it allows the use of sparse matrix techniques to take advantage of zeros in the matrix of linear constraints. Thus the method described can be regarded as an extension of linear programming to allow a non-linear objective function.
TL;DR: A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories.
Abstract: A modified technique of separable programming was used to maximize the squared correlation ratio of weighted responses to partially ordered categories. The technique employs a polygonal approximation to each single-variable function by choosing mesh points around the initial approximation supplied by Nishisato's method. The major characteristics of this approach are: (i) it does not require any grid refinement; (ii) the entire process of computation quickly converges to the acceptable level of accuracy, and (iii) the method employs specific sets of mesh points for specific variables, whereby it reduces the number of variables for the separable programming technique. Numerical examples were provided to illustrate the procedure.
TL;DR: For the class of parametric linear programs with continuous dependence of coefficients on parameters, the following theorem is proven: for any parameter vectort ≥ 0 in the domain of definition of the maximum, if the set of optimal solutions is bounded, then the maximum is upper semicontinuous att ≥ 0 as mentioned in this paper.
Abstract: For the widest class of parametric linear programs with continuous dependence of coefficients on parameters, the following theorem is proven: for any parameter vectort
0 in the domain of definition of the maximum, if the set of optimal solutions is bounded, then the maximum is upper semicontinuous att
0. If the same proviso is met also in the dual program, then the maximum must be continuous att
0.
TL;DR: In this article, it was shown that the optimal policy set behaves continuously if the constraint vector changes on the set for which the program has a solution, which implies that there exists a continuous optimal policy function for which a construction is indicated.
Abstract: For a linear program it is shown that the optimal policy set behaves continuously if the constraint vector changes on the set for which the program has a solution. The result implies that there exists a continuous optimal policy function for which a construction is indicated.
Journal Article•
TL;DR: An algorithm for solving posynomial geometric programs is presented which uses a modification of the concave simplex method to solve the dual program which has a nondifferentiable objective function.
Abstract: An algorithm for solving posynomial geometric programs is presented. The algorithm uses a modification of the concave simplex method to solve the dual program which has a nondifferentiable objective function. The method permits simultaneous changes in certain blocks of dual variables. A convergence proof follows from the convergence proof of the concave simplex method. Some computational results on problems with up to forty degrees of difficulty are included.
TL;DR: It has been shown by G. Roodman that useful postoptimization capabilities for the 0-1 integer programming problem can be obtained from an implicit enumeration algorithm modified to classify and collect all fathomed partial solutions.
Abstract: : It has been shown by G. Roodman that useful postoptimization capabilities for the 0-1 integer programming problem can be obtained from an implicit enumeration algorithm modified to classify and collect all fathomed partial solutions. The paper extends the approach as follows: (1) Improved parameter ranging formulae are obtained by higher resolution classification criteria. (2) Parameters may be changed so as to tighten the original problem, in adddition to relaxing it. (3) An efficient storage structure is presented to cope with difficult data collection task implicit in this approach. (4) Finally, computer implementation is facilitated by the elaboration of a unified set of algorithms. (Author)
TL;DR: A mathematical programming model is described for the economic planning of generation and transmission systems by iterating between the simulation of operating conditions and the combined costs model the number of constraints are reduced by up to one hundredth of those required for early linear programming models.
Abstract: A mathematical programming model is described for the economic planning of generation and transmission systems. By iterating between the simulation of operating conditions and the combined costs model the number of constraints are reduced by up to one hundredth of those required for early linear programming models.
TL;DR: In this article, the NIP problem is reformulated into a 0-1 linear programming (ZOLP) problem and a one-to-one correspondence is shown between this NIP and the ZOLP problem.
Abstract: Mathematical models for reliability of a redundant system with two classes of failure modes are usually formulated as a nonlinear integer programming (NIP) problem. This paper reformulates the NIP problem into a 0-1 linear programming (ZOLP) problem and a one-to-one correspondence is shown between this NIP problem and the ZOLP problem. A NIP example treated by Tillman is formulated into a ZOLP problem and optimal solutions, identical to Tillman's are obtained by an implicit enumeration method. Calculating the new coefficients of the objective function and the constraints in the ZOLP are straight forward. There are not many constraints or variables in the proposed ZOLP. Consequently, the computation (CPU) time is less.
TL;DR: A linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory, is used, and it is obtained convergence in nearly every case, and quadratic convergence in most cases.
Abstract: We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.
TL;DR: An algorithm for handling nonlinear minimum-cost multicommodity flow problems and applies it to a specific large-scale network, which is, in fact, a series of shortest route problems.
Abstract: This paper develops an algorithm for handling nonlinear minimum-cost multicommodity flow problems and applies it to a specific large-scale network. The commodities will be imports and exports; the cost functions will be quadratic and convex. The setting will be a Port Planning Model which will seek to find optimal simultaneous routings through the network while fulfilling requirements both at foreign ports and at domestic hinterlands. The computer program written solves such a problem. The algorithm involves linearizing the cost function and solving the resulting linear program, which is, in fact, a series of shortest route problems. Negative cycles are studied in depth.
TL;DR: In this paper, the authors present a framework which allows uncertainties in the matrix elements of the linear program to be taken into account without requiring detailed knowledge of the statistical characterstics of these uncertainties.
Abstract: In most industrial applications the linear model used for optimization by linear programming involves significant uncertainties and inaccuracies in the model parameters. This paper presents a framework which allows uncertainties in the matrix elements of the linear program to be taken into account without requiring detailed knowledge of the statistical characterstics of these uncertainties. Three cases for the inequality constrained problem are considered: independent variations in the array elements, column dependent variations, and row dependent variations. In each case the problem can still be solved as a linear program. In the first two cases, the problem size is doubled, while for the row dependent case a finitely terminating cutting plane algorithm is constructed.
TL;DR: An improved dual simplex algorithm for the solution of the discrete linear L 1 approximation problem is described, which has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution.
Abstract: An improved dual simplex algorithm for the solution of the discrete linear L 1 approximation problem is described. In this algorithm certain intermediate iterations are skipped. This method is comparable with an improved simplex method due to Barrodale and Roberts, in both speed and number of iterations. It also has the advantage that in case of ill-conditioned problems, the basis matrix can lend itself to triangular factorization and can thus ensure a stable solution. Numerical results are given.
TL;DR: In this paper, the linearized load flow model is written in compact matrix notation and generation constraints are considered to be multivariate normal, and under these conditions, the simultaneous interchange capacity becomes a random variable.
Abstract: Recently, a new algorithm for calculating simultaneous power interchange capacity for a power system has appeared in the literature [1]: the method exploits a linearized load flow model appealing to linear programming to maximize the simultaneous power interchange. Realistic constraints in such a calculation must include limitations on available generation via interchanges. The latter is non-deterministic, and it is desirable to use a stochastic model in representing generation margin constraints for interchange studies. In this paper, the linearized load flow model is written in compact matrix notation and generation constraints are considered to be multivariate normal. Under these conditions, the simultaneous interchange capacity becomes a random variable. The statistics of the simultaneous interchange capacity are obtained using information from the Simplex tableau used in the linear programming solution. In a wide variety of cases, virtually no additional calculation is required over the deterministic model in order to obtain these statistics.
TL;DR: A graph-theoretic approach is used to define the basis as an one-forest consisting of one-trees (a tree with an extra edge) and algorithmic development of the pivot-step is presented by the representation of a two-tree ( a tree with two extra edges).
Abstract: : The paper gives a new organization of the theoretical results of the Generalized Transportation Problem with capacity constraints. A graph-theoretic approach is used to define the basis as an one-forest consisting of one-trees (a tree with an extra edge). Algorithmic development of the pivot-step is presented by the representation of a two-tree (a tree with two extra edges). Constructive procedures and proofs leading to an efficient computer code are provided. The basic definition of an operator theory which leads to the discussion of various operators is also given. In later papers the authors will present additional results on the operator theory for the generalized transportation problem based on the results in the present paper. (Author)
TL;DR: Two algorithms are described for the general problem and a single algorithm for the bottleneck transportation problem which minimizes a bottleneck type objective.
Abstract: We consider here the problem of finding a non-negative solution to a set of linear equations which minimizes a bottleneck type objective. Two algorithms are described for the general problem and a single algorithm for the bottleneck transportation problem.
TL;DR: An iterative algorithm based on linear programming for solving the minimum-time minimum-effort discrete control problem for linear time-invariant systems is presented and some interesting properties of the iterative linear programming algorithm are shown.
Abstract: An iterative algorithm based on linear programming for solving the minimum-time minimum-effort discrete control problem for linear time-invariant systems is presented. The algorithm is based on transforming the minimum-effort problem to an equivalent linear programming one. Some interesting properties of the iterative linear programming algorithm are shown which result in reducing the computational time and storage memory required. Numerical examples are provided to indicate the efficiency of the proposed method.