Finite algorithm for generalized linear multiplicative programming

The nonconvex problem of minimizing the sum of a linear function and the product of two linear functions over a convex polyhedron is considered. A finite algorithm is proposed which either finds a global optimum or shows that the objective function is unbounded from below in the feasible region. This is done by means of a sequence of primal and/or dual simplex iterations.The first author gratefully acknowledges the research support received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata all' Economia, Universitá di Pisa, Pisa, Italy, Spring 1992.     

On solution-containing ellipsoids in linear programming

Ellipsoids that contain all optimal dual slack solutions and those that contain all optimal primal solutions and that are independent of the algorithm used are derived. Based upon these ellipsoids, two criteria each for detecting optimal basic and nonbasic variables prior to optimality in interior-point methods are obtained. Using these results, we then derive a sufficient condition for a linear program to be feasible.This research was supported in part by NSF Grant DMS-85-12277, DMS-91-06195, CDR-84-21402 and ONR Contract N00014-87-K-0214.     

Sensitivity method for basis inverse representation in multistage stochastic linear programming problems

A version of the simplex method for solving stochastic linear control problems is presented. The method uses a compact basis inverse representation that extensively exploits the original problem data and takes advantage of the supersparse structure of the problem. Computational experience indicates that the method is capable of solving large problems.This research was supported by Programs CPBP02.15 and RPI.02.     

Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming

This paper presents a “standard form” variant of Karmarkar's algorithm for linear programming. The tecniques of using duality and cutting objective are combined in this variant to maintain polynomial-time complexity and to bypass the difficulties found in Karmarkar's original algorithm. The variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value. Some computational results are also reported.     

A quadratically convergent O(\sqrt n L)-iteration algorithm for linear programming

Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.Supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.Supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Supported in part by NSF Grant DMS-9102761 and DOE Grant DE-FG05-91ER25100.     

Steepest-edge simplex algorithms for linear programming

We present several new steepest-edge simplex algorithms for solving linear programming problems, including variants of both the primal and the dual simplex method. These algorithms differ depending upon the space in which the problem is viewed as residing, and include variants in which this space varies dynamically. We present computational results comparing steepest-edge simplex algorithms and approximate versions of them against simplex algorithms that use standard pivoting rules on truly large-scale realworld linear programs with as many as tens of thousands of rows and columns. These results demonstrate unambiguously the superiority of steepest-edge pivot selection criteria to other pivot selection criteria in the simplex method.The research of this author was supported in part by NSF Grants DMS 85-12277, DMS 91-0619 and CDR 84-21402.     

A variation on Karmarkar’s algorithm for solving linear programming problems

The algorithm described here is a variation on Karmarkar’s algorithm for linear programming. It has several advantages over Karmarkar’s original algorithm. In the first place, it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function. The minimum value of the objective function does not have to be known in advance. Secondly, in the absence of degeneracy, the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero. This makes it possible to identify an optimal basis before the algorithm converges.     

Extensions of the potential reduction algorithm for linear programming

In this work, we study several extensions of the potential reduction algorithm that was developed for linear programming. These extensions include choosing different potential functions, generating the analytic center of a polytope, and finding the equilibrium of a zero-sum bimatrix game.     

Reminiscences about the origins of linear programming

The author recalls the early days of linear programming, the contributions of von Neumann, Leontief, Koopmans and others.Linear Programming is viewed as a revolutionary development giving us the ability for the first time to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity.     

A simplex algorithm for piecewise-linear programming III: Computational analysis and applications

The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustarate these arguments; the piecewise-linear simplex algorithm is observed to run 2–6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linear programming algorithm is provided by a survey of many varied applications.This research has been supported in part by the National Science Foundation under grant DMS-8217261.     

求解0-1线性整数规划问题的有界单纯形法

提出了一种求解0-1线性整数规划问题的有界单纯形法, 不仅通过数学论证, 讨论了该方法的合理性, 奠定了其数学理论基础, 而且通过求解无容量设施选址问题, 验证了该方法的可行性. 在此基础上, 就该有界单纯形法的不足和存在的问题, 给出了进一步改进的途径和手段.     

A feasible direction method for linear programming

We discuss a finite method of a feasible direction for linear programming problems. The method begins with a feasible basic vector for the problem, constructs a profitable direction to move using the updated column vectors of the nonbasic variables eligible to enter this basic vector. It then moves in this direction as far as possible, while retaining feasibility. This move in general takes it though the relative interior of a face of th set of a feasible solutions. The final point, $\text{x}$, obtained at the end of this move will not in general be a basic solution. Using $\text{x}$ the method then constructs a basic feasible solution at which the objective value is better than, or the same as that at $\text{x}$. The whole process repeats with the new basic feasible solution. We show that this method can be implemented using basis inverses. Initial computer runs of this method in comparison with the usual edge following primary simplex algorithms are very encouraging.     

A simplex algorithm for piecewise-linear programming II: Finiteness,feasibility and degeneracy

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. Part I of this paper has developed a general and direct simplex algorithm for piecewise-linear programming, under convenient assumptions that guarantee a finite number of basic solutions, existence of basic feasible solutions, and nondegeneracy of all such solutions. Part II now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise-linear simplex algorithm. The theory of piecewise-linear programming is thereby extended, and numerous features of linear programming are generalized or are seen in a new light. An analysis of the algorithm's computational requirements and a survey of applications will be presented in Part III.This research has been supported in part by the National Science Foundation under grant DMS-8217261.     

Feasibility issues in a primal-dual interior-point method for linear programming

A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a big-M, and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.     

A primal interior point method for the linear semidefinite programming problem

The linear semidefinite programming problem is examined. A primal interior point method is proposed to solve this problem. It extends the barrier-projection method used for linear programs. The basic properties of the proposed method are discussed, and its local convergence is proved.     

The simplex method and unrestricted variables

Suppose that the simplex method is applied to a linear programming problem havingm equality constraints andr unrestricted variables. We give a method of performing the steps of the simplex method which reduces the arithmetic operation count byrm at each iteration. This savings in operations is achieved, since the method does not update the rows of the basis inverse associated with the unrestricted variables. Similar computational savings are achieved when the method is applied to the updating of anLU-factorization of the basis matrix.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189 and by a postgraduate fellowship.     

A “build-down” scheme for linear programming

We propose a build-down scheme for Karmarkar's algorithm and the simplex method for linear programming. The scheme starts with an optimal basis candidate set including all columns of the constraint matrix, then constructs a dual ellipsoid containing all optimal dual solutions. A pricing rule is developed for checking whether or not a dual hyperplane corresponding to a column intersects the containing ellipsoid. If the dual hyperplane has no intersection with the ellipsoid, its corresponding column will not appear in any of the optimal bases, and can be eliminated from. As these methods iterate, is eventually built-down to a set that contains only the optimal basic columns.     

An infeasible (exterior point) simplex algorithm for assignment problems

The so called Modified Hung—Rom Algorithm, based upon theoretical considerations of Hirsch-paths, seems to be one of the most efficient algorithms for assignment problems. Since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region, development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest. This paper presents an algorithm of this kind for assignment problems.     

Unrestricted variables in linear programming

There are a number of ways of dealing with a linear programming problem in which some variables are allowed to take on negative values. One of these methods, which is either ignored or mentioned only incidentally in most textbooks, requires only one additional variable to be introduced regardless of how many unrestricted variables the original problem has. In this note, this method is interpreted geometrically and an application to the computation of extreme points is given.     

Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems

The aim of this article is to introduce a formulation of fuzzy linear programming problems involving the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy numbers as parameters. Indeed, such a formulation is the general form of trapezoidal fuzzy number linear programming problems. Then, it is demonstrated that study of the sensitivity analysis for the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy number linear programming problems gives rise to the same expected results as those obtained for trapezoidal fuzzy number linear programming problems.