Showing papers on "Linear-fractional programming published in 1989"

Multiple Criteria Optimization: Theory, Computation, and Application

Abstract: Mathematical Background Topics from Linear Algebra Single Objective Linear Programming Determining all Alternative Optima Comments about Objective Row Parametric Programming Utility Functions, Nondominated Criterion Vectors and Efficient Points Point Estimate Weighted-sums Approach Optimal Weighting Vectors, Scaling and Reduced Feasible Region Methods Vector-Maximum Algorithms Goal Programming Filtering and Set Discretization Multiple Objective Linear Fractional Programming Interactive Procedures Interactive Weighted Tchebycheff Procedure Tchebycheff/Weighted-Sums Implementation Applications Future Directions Index.

Pathways to the optimal set in linear programming

Abstract: This chapter presents continuous paths leading to the set of optimal solutions of a linear programming problem These paths are derived from the weighted logarithmic barrier function The defining equations are bilinear and have some nice primal-dual symmetry properties Extensions to the general linear complementarity problem are indicated

An implementation of Karmarkar's algorithm for linear programming

Abstract: This paper describes the implementation of power series dual affine scaling variants of Karmarkar's algorithm for linear programming. Based on a continuous version of Karmarkar's algorithm, two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested. Linear programs are expressed in an inequality form, which allows for the inexact computation of the algorithm's direction of improvement, resulting in a significant computational advantage. Implementation issues particular to this family of algorithms, such as treatment of dense columns, are discussed. The code is tested on several standard linear programming problems and compares favorably with the simplex codeMinos 4.0.

Speeding-up linear programming using fast matrix multiplication

Abstract: The author presents an algorithm for solving linear programming problems that requires O((m+n)/sup 15/nL) arithmetic operations in the worst case, where m is the number of constraints, n the number of variables, and L a parameter defined in the paper This result improves on the best known time complexity for linear programming by about square root n A key ingredient in obtaining the speedup is a proper combination and balancing of precomputation of certain matrices by fast matrix multiplication and low-rank incremental updating of inverses of other matrices Specializing the algorithm to problems such as minimum-cost flow, flow with losses and gains, and multicommodity flow leads to algorithms whose time complexity closely matches or is better than the time complexity of the best known algorithms for these problems >

An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations

Abstract: This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations. The total number of arithmetic operations is bounded by O(n 3 L), carried on with the same precision as that in Karmarkar’s algorithm. Each iteration updates a penalty multiplier and solves a Newton-Raphson iteration on the traditional logarithmic barrier function using approximated Hessian matrices. The resulting sequence follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.

An Implementation of a Primal-Dual Interior Point Method for Linear Programming

Abstract: The purpose of this paper is to describe in detail an implementation of a primal-dual interior point method for solving linear programming problems. Preliminary computational results indicate that this implementation compares favorably with a comparable implementation of a dual affine interior point method, and with MINOS 5.0, a state-of-the-art implementation of the simplex method. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Linear Programming in Infinite-Dimensional Spaces

Abstract: (1989). Linear Programming in Infinite-Dimensional Spaces. Journal of the Operational Research Society: Vol. 40, No. 1, pp. 109-110.

Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm

Abstract: This paper describes data structures and programming techniques used in an implementation of Karmarkar's algorithm for linear programming. Most of our discussion focuses on applying Gaussian elimination toward the solution of a sequence of sparse symmetric positive definite systems of linear equations, the main requirement in Karmarkar's algorithm. Our approach relies on a direct factorization scheme, with an extensive symbolic factorization step performed in a preparatory stage of the linear programming algorithm. An interpretative version of Gaussian elimination makes use of the symbolic information to perform the actual numerical computations at each iteration of algorithm. We also discuss ordering algorithms that attempt to reduce the amount of fill-in in the LU factors, a procedure to build the linear system solved at each iteration, the use of a dense window data structure in the Gaussian elimination method, a preprocessing procedure designed to increase the sparsity of the linear programming coeffi...

Progress in Mathematical Programming

Abstract: : Most of the progress reported at the conference was on the theoretical side. Several new polynomial algorithms for linear programming were presented. The common feature to most of the new polynomial algorithms is the path-following aspect. The method of McCormick-Sofer for convex programming also follows a path. Efforts in the theoretical analysis of algorithms was also reported. Of special interest, although not in the main direction discussed at the conference, was the report by Rinaldi on the practical solution of some large traveling salesman problems. At the time of the conference it was still not clear weather the new algorithms developed since Karmarkar's algorithm would replace the simplex method in practice. Alan Hoffman presented results on conditions under which linear programming problems can be solved by greedy algorithms. In other presentations, Fourer-Gay-Kernighan presented a programming language (AMPL) for mathematical programming, David Gay presented graphic illustrations of the performance of Karmarkar's algorithm, and James Ho discussed embedding of linear programming in commonly used spreadsheets.

On the ball spanned by balls

Abstract: The procedure for linear programming in linear time in fixed dimension is extended to solve in linear time certain nonlinear problems Examples are the problem of finding the smallest ball enclosingn given balls, and the weighted-center problem in fixed dimension

Modelling and strong linear programs for mixed integer programming

Abstract: Mixed integer programming modeling is considered from two points of view: getting the model correctly generated in an understandable form, and formulating or reformulating the model so that the problem can be solved. For the former considerations, a relational approach is presented. For the latter, three techniques are discussed: preprocessing, constraint generation, and column generation. For all three techniques, mixed integer problems are considered. For column generation, two problem classes (cutting stock and crew scheduling) for which column generation techniques are classical, are presented in a unified framework and then clustering problems are discussed in the same framework. In the constraint generation section, some constraints based on mixed 0–1 implication graphs are presented.

Implementation of a Dual Affine Interior Point Algorithm for Linear Programming

Abstract: The dual affine interior point method is extended to handle variables with simple upper bounds as well as free variables. During execution, variables which appear to be going to zero are fixed at zero, and rows with slack variables bounded away from zero are removed. A variant of the big-M artificial variable method to attain feasibility is derived. The simplex method is used to recover an optimal basis upon completion of the algorithm, and the effects of scaling are discussed. Computational experience on a variety of problems is presented. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A DSS oriented method for multiobjective linear programming problems

Abstract: Most of practical linear programming problems involve multiple and conflicting objectives. The paper presents an interactive method to approach this kind of problems. The main original aspect of this method lies in the fact that it combines the advantageous features of both the paradigms of Satisfactory Goals and Multiattribute Utility Assesment. It is a DSS oriented approach providing a ‘two level’ interaction: 1. (1) interactive assessment of the decision maker's utility function using the UTA ordinal regression model; 2. (2) interactive modification of the satisfaction levels. Piecewise linear optimazation techniques are used to determine, at each iteration, a new compromise solution over the set of efficient solutions.

An interior point algorithm for semi-infinite linear programming

Abstract: We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.

On the structure and properties of linear multilevel programming problem

Abstract: Many decision-making situations involve multiple planners with different, and sometimes conflicting, objective functions. One type of model that has been suggested to represent such situations is the linear multilevel programming problem. However, it appears that theoretical and algorithmic results for linear multilevel programming have been limited, to date, to the bounded case or the case of when only two levels exist. In this paper, we investigate the structure and properties of a linear multilevel programming problem that may be unbounded. We study the geometry of the problem and its feasible region. We also give necessary and sufficient conditions for the problem to be unbounded, and we show how the problem is related to a certain parametric concave minimization problem. The algorithmic implications of the results are also discussed.

Possibilistic Linear Programming with Measurable Multiattribute Value Functions

Abstract: In this paper, a possibilistic linear program is formulated when a measurable multiattribute value function is given. The possibilistic linear program in this paper is an unconstrained linear program with several objective functions whose coefficients are represented by possibility distributions. A possibility measure and a necessity measure are derived from a possibility distribution. Using fuzzy integrals of the measurable multiattribute value function with respect to the possibility measure and the necessity measure, the possible value and the necessary value are defined respectively. In an analogy of the expected utility, the principles of maximizing the possible value and the necessary value are considered as decision procedures under a possibility distribution. The possibilistic linear program is formulated based on these decision procedures and reduced to a nonlinear program. A solution method using linear programming technique is proposed. INFORMS Journal on Computing, ISSN 1091-9856, was publishe...

Hypercube implementation of the simplex algorithm

Abstract: Large, sparse, linear systems of equations arise frequently when constructing mathematical models of natural phenomena. Most often, these linear systems are fully constrained and can be solved via direct or iterative techniques. However, one important problem class requires solutions to underconstrained linear systems that maximize some objective function. These linear optimization problems are natural formulations of many business plans and often contain hundreds of equations with thousands of variables. Historically, linear optimization problems have been solved via the simplex method. Despite the excellent performance of the simplex method, the size of the optimization problems and the frequency of their solution make linear optimization a computationally taxing endeavor. This paper examines the performance of parallel variants of the simplex algorithm on the Intel iPSC, a message-based parallel system. Linear optimization test data are drawn from commercial sources and represent realistic problems. Analysis shows that the speedup obtained is sensitive to both the structure of the underlying data and the data partitioning.

Dynamic Programming, Integral Polyhedra and Horn Clause Knowledge Base

Abstract: We show how the structure of proofs in a Horn clause knowledge base is completely described by certain of the extreme solutions to a suitable “dual” linear constraint set. The extreme points of this linear system are integer vectors, and give the count of the number of times that a given proposition is used in a proof of a “target” proposition. The “primal” to this dual provides the pointwise maximum vector which solves a set of dynamic programming recursions, and the latter recursions are derived from the Horn clause knowledge base. Variation in the facts or in the rules of the knowledge base corresponds to changes in only the criterion vector of the dual linear program. This linear programming approach to inference in expert systems also allows for the detection of “near proofs.” The latter are, by definition, proof structures which would become valid, if only exactly one more fact were known to be true. The first author, Robert G. Jeroslow, deceased August 31, 1988. INFORMS Journal on Computing, ISSN 1...

Chapter II Linear programming

Abstract: Publisher Summary This chapter discusses the linear programming. The linear programming problems involve the optimization of a linear function, called the “objective function,” subject to linear constraints, which may be either equalities or inequalities. The chapter discusses the new methods for solving linear programs problems and providing a development of the simplex method and the basic theory of linear programming. The geometry of linear programming models is presented and relevant geometrical concepts are algebraically characterized. The simplex method from a geometric point of view is developed. This development produces, as a by-product, various basic theorems concerning conditions for optimality and unboundedness, and leads in a natural way to the revised version of the simplex method. Duality theory and sensitivity analysis are also covered in the book. The chapter discusses the complexity of linear programming and two new methods—the ellipsoid method and Karmarkar's projective method. These are distinguished from the simplex method and have the desirable theoretical property of polynomial-time boundedness.

A linear-time algorithm for linearL 1 approximation of points

Abstract: In this paper we present a linear-time algorithm for approximating a set ofn points by a linear function, or a line, that minimizes theL1 norm. The algorithmic complexity of this problem appears not to have been investigated, although anO(n3) naive algorithm can be easily obtained based on some simple characteristics of an optimumL1 solution. Our linear-time algorithm is optimal within a constant factor and enables us to use linearL1 approximation of many points in practice. The complexity ofL1 linear approximation of a piecewise linear function is also touched upon.

Algorithmic Enhancements to the Method of Centers for Linear Programming Problems

Abstract: Interior point algorithms for solving linear programming problems are considered. The techniques are derived from a continuous version of Huard's method of centers that yields a family of trajectories in the feasible region that all converge to an optimal solution. The tangential direction of these trajectories is the dual affine direction. Deficiencies in some of these trajectories are discussed, and the need to recenter is argued. Several new algorithms that use the dual affine direction and a recentering direction in a multidirection approach are then derived. The most promising of these algorithms is based on minimizing the cost function on a sequence of two-dimensional cross sections of the feasible region. Numerical results are presented. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A unified approach to projective algorithms for linear programming

Abstract: In this paper it is shown that a projective algorithm based on the minimization of a potential function by a constrained Newton method is general enough to include other known projective methods for linear programming and fractional linear programming It also provides a framework to analyze affine interior point methods and to relate them to projective methods