Showing papers on "Linear-fractional programming published in 1977"
IBM1
TL;DR: An attempt to provide a powerful mathematical programming language, allowing an easy programming of specific studies on medium-size models such as the recursive use of LP or the build-up of algorithms based on the simplex method is described.
Abstract: First, this paper presents the results of experiments with algorithmic techniques for efficiently solving medium and large scale linear and mixed integer programming problems. The techniques presented here are either original or recent.
142 citations
TL;DR: Theoretical results are developed for zero–one linear multiple objective programs for the main problem, having as a feasible set the vertices of the unit hypercube.
Abstract: Theoretical results are developed for zero–one linear multiple objective programs. Initially a simpler program, having as a feasible set the vertices of the unit hypercube, is studied. For the main problem an algorithm, computational experience, parametric analysis and indifference sets are presented. The mixed integer version of the main problem is briefly discussed.
90 citations
TL;DR: This work gives a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution to the Bilinear Programming Problem.
Abstract: The Bilinear Programming Problem is a structured quadratic programming problem whose objective function is, in general, neither convex nor concave. Making use of the formal linearity of a dual formulation of the problem, we give a necessary and sufficient condition for optimality, and an algorithm to find an optimal solution.
84 citations
TL;DR: In this paper, the authors consider a linear programming problem with random aij and bi elements that have known finite mean and variance, but whose distribution functions are otherwise unspecified, and derive deterministic equivalents under different assumptions regarding the stochastic nature of the random parameters.
Abstract: We consider a linear programming problem with random aij and bi elements that have known finite mean and variance, but whose distribution functions are otherwise unspecified. A minimax solution of the stochastic programming model is obtained by solving an equivalent deterministic convex programming problem. We derive these deterministic equivalents under different assumptions regarding the stochastic nature of the random parameters.
45 citations
TL;DR: In this article, the mean and variance of a linear function with arbitrary multivanate randomness in its components are estimated using Tchebycheff-type probability statements, which can be used to accomodate and exploit stochastic dependence.
Abstract: Applications in operations research often employ models which contain linear functions. These linear functions may have some components coefficients and variables which are random. For instance, linear functions in mathematical programming often represent models of processes which exhibit randomness in resource availability, consumption rates, and activity levels. Even when the linearity assumptions of these models is unquestioned, the effects of the randomness in the functions is of concern. Methods to accomodate, or at least estimate for a linear function the implications of randomness in its components typically make several simplifying assumptions. Unfortunately, when components are known to be random in a general, multivariate dependent fashion, concise specification of the randomness exhibited by the linear function is, at best, extremely complicated, usually requiring severe, unrealistic restrictions on the density functions of the random components. Frequent stipulations include assertion of normality of independence-yet, observed data, accepted collateral theory and common sense may dictate that a symmetric distribution with infinite domain limits is inappropriate, or that a dependent structure is definitely present. For example, random resource levels may be highly correlated due to economic conditions, and non-negative for physical reasons. Often, an investigation is performed by discretizing the random components at point quantile levels, or by replacing the random components by their means-methods which give a deterministic “equivalent” model with constant terms, but possibly very misleading results. Outright simulation can be used, but requires considerable time investment for setup and debugging especially for generation of dependent sequences of pseudorandom variates and gives results with high parametric specificity and computation cost. This paper shows how to use elementary methods to estimate the mean and variance of a linear function with arbitrary multivanate randomness in its components. Expressions are given for the mean and variance and are used to make Tchebycheff-type probability statements which can accomodate and exploit stochastic dependence. Simple estimation examples are given which lead to illustrative applications with dependent-stochastic programming models.
31 citations
TL;DR: In this paper, a duality theorem for disjunctive programming has been proved for this class of problems, which generalizes to the class of linear programming problems and generalizes the corresponding result for linear programming.
Abstract: We state a duality theorem for disjunctive programming, which generalizes to this class of problems the corresponding result for linear programming.
26 citations
01 Aug 1977
TL;DR: The fundamental theory and algorithms of quadratic programming and linear complementarity are presented in expository form and Computational experience is reviewed.
Abstract: : The fundamental theory and algorithms of quadratic programming and linear complementarity are presented in expository form. Computational experience is reviewed. (Author)
25 citations
TL;DR: This paper considers heuristic procedures for general mixed integer linear programming with inequality constraints by constructing an “interior path” from which to search for good feasible solutions by constructing piecewise linear paths.
Abstract: This paper considers heuristic procedures for general mixed integer linear programming with inequality constraints. It focuses on the question of how to most effectively initialize such procedures by constructing an “interior path” from which to search for good feasible solutions. These paths lead from an optimal solution for the corresponding linear programming problem (i.e., deleting integrality restrictions) into the interior of the feasible region for this problem. Previous methods for constructing linear paths of this kind are analyzed from a statistical viewpoint, which motivates a promising new method. These methods are then extended to piecewise linear paths in order to improve the direction of search in certain cases where constraints that are not binding on the optimal linear programming solution become particularly relevant. Computational experience is reported.
24 citations
01 Feb 1977
TL;DR: The User's Guide contains an overview of the MINOS System, including descriptions of the theoretical algorithms as well as the details of implementation, and provides complete instructions for the use of MINOS.
Abstract: : MINOS is a Fortran program designed to minimize a linear or nonlinear function subject to linear constraints, where the constraint matrix is in general assumed to be large and sparse. The User's Guide contains an overview of the MINOS System, including descriptions of the theoretical algorithms as well as the details of implementation. The Guide also provides complete instructions for the use of MINOS, and illustrates the diversity of application by several examples. (Author)
18 citations
TL;DR: Using dynamic programming, feedback Stackelberg strategies for the general linear quadratic discrete-time game were derived in this paper, where the game was formulated as a convex combinatorial game.
Abstract: Using dynamic programming, feedback Stackelberg strategies are derived for the general linear quadratic discrete-time game.
15 citations
TL;DR: Results are presented of an investigation as to the computational efficiency of several non-linear programming algorithms on the constrained structural weight optimization of plane trusses and plane stress plates.
Abstract: Results are presented of an investigation as to the computational efficiency of several non-linear programming algorithms on the constrained structural weight optimization of plane trusses and plane stress plates. The optimization algorithms considered are the methods of sequential linear programming, of feasible directions based on Zoutendijk's procedure P1, and of sequential unconstrained minimizations using the methods of Powell, Stewart, Fletcher–Powell, and Newton. Results are presented on computational efficiency as a function of number of design parameters and as a function of the function evaluation effort to derivative evaluation effort ratio.
01 May 1977
TL;DR: In this paper, the authors present a linear programming design technique for constant group delay filters with real and imaginary components of the frequency domain specifications, where the value of the group delay in the specifications is altered until a minimum error is achieved.
Abstract: Previous linear programming design techniques have concentrated on magnitude only specifications, In the technique presented here we design the filter to prescribed real and imaginary components of the frequency domain specifications. Although general phase characteristics have proved difficult to achieve, good results are obtained for constant group delay designs. The technique employed is to alter the value of the group delay in the specifications until a minimum error is achieved. Stable designs can be established by employing usual stability criteria. Results of the design technique are presented and an extension to two dimensional filter designs is discussed.
TL;DR: A substantial reduction of computational burden, the so called curse of dimensionality, in the existing discrete optimal control algorithms, is obtained and an effective multistage decomposition algorithm for distributed-lag models is obtained.
TL;DR: In this paper, the authors discuss the general problem of designing building subsystems which include probabilistic demands thus producing a stochastic problem and show that dynamic programming is the more powerful method both in terms of obtaining the solution and in its post-optimality analysis.
TL;DR: In this article, the authors defined pseudo-autonomous linear differential equations with bounded and bounded coefficient matrices and showed that a linear differential equation with bounded coefficient matrix is almost reducible if and only if it is reducible.
Abstract: Pseudo-autonomous linear differential equations are defined. A linear differential equation with bounded coefficient matrix is pseudo-autonomous if and only if it is almost reducible. A linear differential equation with recurrent coefficient matrix is pseudo-autonomous if and only if it has pure point spectrum.
TL;DR: In this article, a special purpose linear programming algorithm for obtaining least-absolute value estimators in a linear model with dummy variables is presented, which employs a compact basis inverse procedure and incorporates the advanced basis exchange techniques available in specialized algorithms for the general linear least absolute value problem.
Abstract: Dummy (0, 1) variables are frequently used in statistical modeling to represent the effect of certain extraneous factors. This paper presents a special purpose linear programming algorithm for obtaining least-absolute-value estimators in a linear model with dummy variables. The algorithm employs a compact basis inverse procedure and incorporates the advanced basis exchange techniques available in specialized algorithms for the general linear least-absolute-value problem. Computational results with a computer code version of the algorithm are given.
TL;DR: An error-free computational approach is employed for finding the integer solution to a system of linear equations, using finite-field arithmetic, to find the optimum solution for linear inequalities such as those arising in interval linear programming probloms.
Abstract: An error-free computational approach is employed for finding the integer solution to a system of linear equations, using finite-field arithmetic. This approach is also extended to find the optimum solution for linear inequalities such as those arising in interval linear programming probloms.
TL;DR: It is shown that, in terms of I j irrs classification scheme, the consolidation of linear programming models should be treated as a case of constrained consistency.
Abstract: In his excellent survey of aggregation theory, Ijiri treats the consolidation of linear programming models as an exercise in total consistency [3, p 771] Also, he reports that for this problem only sufficient conditions are available in the literature We wish to show that, in terms of I j irrs classification scheme, the consolidation of linear programming models should be treated as a case of constrained consistency When this is done, the derivation of the necessary conditions becomes rather direct For the benefit of the reader, we summarize first the problem under study Let the ith microsystem be described by the canonical program
TL;DR: The solution of a set of overdetermined linear equations in the $l_1 $ norm is proved to be of a particular form in case the set of equations is a Chebyshev system.
Abstract: Efficient linear programming algorithms for the solution of a set of overdetermined linear equations in the $l_1 $ norm are employed in order to obtain a minimum $l_\infty $ solution to a set of consistent linear equations.In addition, the solution is proved to be of a particular form in case the set of equations is a Chebyshev system.
01 Jun 1977
TL;DR: In this paper, the duality models in fractional linear programming presented in the last years by Swamy, Kaska, Sharma and Swamy and other authors are dealt with.
Abstract: This paper deals with the duality models in fractional linear programming presented in the last years bySwarup, Kaska, Sharma andSwarup and other authors.
01 Jan 1977
TL;DR: In this paper, the authors extended the application of linear programming techniques to elastoplastic arches under variable repeated loadings by adopting linearized models for the arch shape and the yield criteria of its sections.
Abstract: The use of mathematical programming methods proves to be of interest in the formulation and the solution of some rather tedious problems of structural plasticity. This paper extends the application of linear programming techniques to elastoplastic arches under variable repeated loadings by adopting linearized models for the arch shape and the yield criteria of its sections. The concept of M and M + N mechanisms proves convenient in formulating the equilibrium equations by the static approach. With nonlinear yield conditions the formulations presented will still hold, but the solution of corresponding nonlinear programming problems will likely become more complex. In addition to their practical applications, the linear programming approaches suggested and illustrated have the advantage of allowing the incremental collapse load to be derived by a direct, algorithmic procedure, rather than by the trial and error procedures suggested in earlier investigations.
TL;DR: Several implications of the L 2 norm are revealed which are especially important in constraint deletion and rate of reduction of the hypervolume of the residual search region K.
Abstract: The dilemma of when to cease analysis of a subproblem and go on to the next subproblem in convex programming is an especially difficult one. This paper views convex programming as an extension of linear programming and focuses attention on the analysis of a given subproblem. Specifically, if K = {x∣Ax ≤ b} is bounded with a nonempty interior and represents a residual unsearched region in some convex programming algorithm, what is (are) the “best” trial point(s) which can be selected? The generalized slack variable linear program (GSVLP) provides a mechanism for implementing trial point selection based on many strategies. Several implications of the L2 norm are revealed which are especially important in constraint deletion and rate of reduction of the hypervolume of the residual search region K.
TL;DR: In this paper, the authors make use of a general selection theorem to answer the first question positively, and obtain an answer to the second question at least theoretically at the same time.
Abstract: The “distribution problem” of stochastic linear programming consists in answering the following two questions: Is the optimal value of a given stochastic linear program—regarded as a function—measurable, and if so, what is its distribution? In the present note we make use of a general selection theorem to answer the first question positively. By this approach, an answer to the second question is obtained—at least theoretically—at the same time.
TL;DR: It is shown that the regularity condition is so weak that it is automatically satisfied in linear programming (prototype posynomial) geometric programming, quadratic programming (with either linear or quadRatic constraints),lp-regression analysis, optimal location, roadway network analysis, and chemical equilibrium analysis.
Abstract: F.E. Clark has shown that if at least one of the feasible solution sets for a pair of dual linear programming problems is nonempty then at least one of them is both nonempty and unbounded. Subsequently, M. Avriel and A.C. Williams have obtained the same result in the more general context of (prototype posynomial) geometric programming. In this paper we show that the same result is actually false in the even more general context of convex programming — unless a certain regularity condition is satisfied.
Book•
01 Jan 1977
TL;DR: Data Flow Analysis can be used to find some of the errors in a computer program and gives as output a set of dubious paths of the investigated program, which have to be checked for executability.
Abstract: Data Flow Analysis can be used to find some of the errors in a computer program and gives as output a set of dubious paths of the investigated program, which have to be checked for executability. This can be done by solving a system of inequalities in order to obtain input data for that path. This article discusses how to obtain a reliable solution of this system in the linear case, when rounding effects are taken into account. The method is based on the simplex algorithm from linear programming, and returns a solution in the middle of the feasible region. The general nonlinear case is much more difficult to handle.
TL;DR: This paper describes how a water resources system, including both water quantity and water quality, can be modeled to form a non-linear programming problem, which is solved by two techniques: a Generalized Reduced Gradient method, and a conjugate gradient projection method.
Abstract: Optimization of a water resources system necessarily must appropriately mesh the modeling of the system with the optimization technique used. If the system model is linear, many effective optimization techniques exist. But if the model is non-linear in the objective function and/or constraints, very few effective optimization methods exist. This paper describes how a water resources system, including both water quantity and water quality, can be modeled to form a non-linear programming problem. The latter is solved by two techniques: (a) a Generalized Reduced Gradient method, and (b) a conjugate gradient projection method. The relationship between the model and the formulation of the non-linear programming problem is discussed, and computational experience with each of the algorithms is described.
TL;DR: In this paper, the authors analyse parametric linear fractional programming problems, where all coefficients of the numerator and (or) of the denominator of the objective function and all right hand sides are parameters.
Abstract: In this article we analyse parametric linear fractional programming problems. In these problems all coefficients of the numerator and (or) of the denominator of the objective function and (or) all right hand sides are parameters. The analysis deals with the characterization of the set of those parameters, for which the supremum of the objective function of such a problem is finite resp. the problem is solvable. In further deals with the form and the qualities of local stability sets, of a unique partition of the solvability set and qualities of the supremum-function and the solution-function. All researches we made without any algorithm for linear fractional programming problems.