A variant of Karmarkar's linear programming algorithm for problems in standard form

This paper presents a variant of Karmarkar's linear programming algorithm that works directly with problems expressed in standard form and requires no a priori knowledge of the optimal objective function value. Rather, it uses a variation on Todd and Burrell's approach to compute ever better bounds on the optimal value, and it can be run as a prima-dual algorithm that produces sequences of primal and dual feasible solutions whose objective function values convege to this value. The only restrictive assumption is that the feasible region is bounded with a nonempty interior; compactness of the feasible region can be relaxed to compactness of the (nonempty) set of optimal solutions.     

An implementation of Karmarkar's algorithm for linear programming

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.     

On the number of iterations of Karmarkar's algorithm for linear programming

Karmarkar's algorithm for linear programming was published in 1984, and it is highly important to both theory and practice. On the practical side some of its variants have been found to be far more efficient than the simplex method on a wide range of very large calculations, while its polynomial time properties are fundamental to research on complexity. These properties depend on the fact that each iteration reduces a potential function by an amount that is bounded away from zero, the bound being independent of all the coefficients that occur. It follows that, under mild conditions on the initial vector of variables, the number of iterations that are needed to achieve a prescribed accuracy in the final value of the linear objective function is at most a multiple ofn, wheren is the number of inequality constraints. By considering a simple example that allowsn to be arbitrarily large, we deduce analytically that the magnitude of this complexity bound is correct. Specifically, we prove that the solution of the example by Karmarkar's original algorithm can require aboutn/20 iterations. Further, we find that the algorithm makes changes to the variables that are closely related to the steps of the simplex method.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.     

On the improvement per iteration in Karmarkar's algorithm for linear programming

We investigate the decrease in potential at an iteration of Karmarkar's projective method for linear programming. For a fixed step length parameter (so that we must have 0 < 1) the best possible guarantee _n () inn dimensional space is essentially ln 2 0.69; and to achieve this we must take about 1. Indeed we show the precise result that _n () equals ln(1 +)-ln(1 –/(n – 1)) forn sufficiently large. If we choose an optimal step length at each iteration then this guarantee increases only to about ^* 0.72. We also shed some light on the remarkable empirical observation that the number of iterations required seems scarcely to grow with the size of the problem.     

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.     

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln ³) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.     

Updating lower bounds when using Karmarkar's projective algorithm for linear programming

We give two results related to Gonzaga's recent paper showing that lower bounds derived from the Todd-Burrell update can be obtained by solving a one-variable linear programming problem involving the centering direction and the affine direction. We show how these results may be used to update the primal solution when using the dual affine variant of Karmarkar's algorithm. This leads to a dual projective algorithm.This research was partially supported by ONR Grant Number N00014-90-J-1714.     

An algorithm for solving a two-person game with information transfer

An algorithm for solving a two-person game with information transfer is proposed. The algorithm is based on a special linear programming problem. An example is given.     

Asymptotic behaviour of Karmarkar's method for linear programming

The asymptotic behaviour of Karmarkar's method is studied and an estimate of the rate of the objective function value decrease is given. Two possible sources of numerical instability are discussed and a stabilizing procedure is proposed.Research supported in part by Republicka zajednica za nauku SR Srbije.     

An extension of the simplex algorithm for semi-infinite linear programming

We present a primal method for the solution of the semi-infinite linear programming problem with constraint index setS. We begin with a detailed treatment of the case whenS is a closed line interval in . A characterization of the extreme points of the feasible set is given, together with a purification algorithm which constructs an extreme point from any initial feasible solution. The set of points inS where the constraints are active is crucial to the development we give. In the non-degenerate case, the descent step for the new algorithm takes one of two forms: either an active point is dropped, or an active point is perturbed to the left or right. We also discuss the form of the algorithm when the extreme point solution is degenerate, and in the general case when the constraint index set lies in ^p . The method has associated with it some numerical difficulties which are at present unresolved. Hence it is primarily of interest in the theoretical context of infinite-dimensional extensions of the simplex algorithm.     

A two-stage successive overrelaxation algorithm for solving the symmetric linear complementarity problem

We propose a two-stage successive overrelaxation method (TSOR) algorithm for solving the symmetric linear complementarity problem. After the first SOR preprocessing stage this algorithm concentrates on updating a certain prescribed subset of variables which is determined by exploiting the complementarity property. We demonstrate that this algorithm successfully solves problems with up to ten thousand variables.This material is based on research supported by National Science Foundation Grants DCR-8420963 and DCR-8521228 and Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the Computer Sciences Department at the University of Wisconsin-Madison, USA.     

An algorithm of sequential systems of linear equations for nonlinear optimization problems with arbitrary initial point

For current sequential quadratic programming (SQP) type algorithms, there exist two problems: (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using ε-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented. This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above. Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.     

Parallel pivotal algorithm for solving the linear complementarity problem

We propose a parallel implementation of the classical Lemke's algorithm for solving the linear complementarity problem. The algorithm is designed for a loosely coupled network of computers which is characterized by relatively high communication costs. We provide an accurate prediction of speedup based on a simple operation count. The algorithm produces speedup nearp, wherep is the number of processors, when tested on large problems as demonstrated by computational results on the CRYSTAL token-ring multicomputer and the Sequent Balance 21000 multiprocessor.This material is based on research supported by National Science Foundation Grants DCR-84-20963 and DCR-850-21228 and by Air Force Office of Scientific Research Grants AFSOR-86-0172 and AFSOR-86-0255 while the author was at the University of Wisconsin, Madison, Wisconsin.     

A new heuristic algorithm solving the linear ordering problem

The linear ordering problem is an NP-hard combinatorial problem with a large number of applications. Contrary to another very popular problem from the same category, the traveling salesman problem, relatively little space in the literature has been devoted to the linear ordering problem so far. This is particularly true for the question of developing good heuristic algorithms solving this problem.In the paper we propose a new heuristic algorithm solving the linear ordering problem. In this algorithm we made use of the sorting through insertion pattern as well as of the operation of permutation reversal. The surprisingly positive effect of the reversal operation, justified in part theoretically and confirmed in computational examples, seems to be the result of a unique property of the problem, called in the paper the symmetry of the linear ordering problem. This property consists in the fact that if a given permutation is an optimal solution of the problem with the criterion function being maximized, then the reversed permutation is a solution of the problem with the same criterion function being minimized.     

Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value

Variants of Karmarkar's algorithm are given for solving linear programs with unknown optimal objective valuez ^*. These new methods combine the approach of Goldfarb and Mehrotra for relaxing the requirement that certain projections be computed exactly with the approach of Todd and Burrell for generating an improving sequence of lower bounds forz ^* using dual feasible solutions. These methods retain the polynomial-time complexity of Karmarkar's algorithm.This research was supported in part by NSF Grants DMS-85-12277 and CDR-84-21402, and ONR Contract N0014-87-K0214.     

An error-free algorithm to solve a linear system of polynomial equations

The paper presents an error-free algorithm to solve a system of linear equations with polynomial coefficients. Modular arithmetic in residual polynomial class and in residual numeric class is employed. The algorithm is iterative and well suited for implementation for computers with vector operations and fast and error-free convolutors.     

Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems

We show that a particular pivoting algorithm, which we call the lexicographic Lemke algorithm, takes an expected number of steps that is bounded by a quadratic inn, when applied to a random linear complementarity problem of dimensionn. We present two probabilistic models, both requiring some nondegeneracy and sign-invariance properties. The second distribution is concerned with linear complementarity problems that arise from linear programming. In this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem, and independent of the larger. Similar results have been obtained by Adler and Megiddo.Research partially funded by a fellowship from the Alfred Sloan Foundation and by NSF Grant ECS82-15361.     

A new algorithm for solving the general quadratic programming problem

For the general quadratic programming problem (including an equivalent form of the linear complementarity problem) a new solution method of branch and bound type is proposed. The branching procedure uses a well-known simplicial subdivision and the bound estimation is performed by solving certain linear programs.     

基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

Exact algorithm for solving a special fixed-charge linear programming problem

A branch-and-bound algorithm (A) for solving a fixed-charge linear programming problem (P) involving identical fixed charges, one equality constraint, and explicit bounds on the variables is presented. Problem (P) can serve as a mathematical model for profit optimization in sawn timber production. Some theoretical considerations upon a fixed-charge problem (P), arising from (P) by permitting the fixed charges to be different for each variable, are carried out. A basic algorithm (A₀) is stated, and it is proved that Algorithm (A₀) finds an optimal solution of Problem (P) [resp., (P)] within a finite number of steps. Algorithm (A₀), combined with bounds developed with regard to Problem (P), yields Algorithm (A), which operates on a subset of all vertices of the feasible region. Finally, computational results concerning the numerical solution of Problem (P) by Algorithm (A) are stated.A part of this work was carried out in connection with the project Optimierung der Schnittholzproducktion auf Zerspaneranlagen, which was done at the Institute of Mathematics of the University of Klagenfurt in cooperation with the firm J. Offner, Holzindustrie GmbH, Wolfsberg. This project was partially supported by Forschungsförderungsfonds für die gewerbliche Wirtschaft. The author would like to thank Professor H. Stettner, C. Nowak, and H. Woschitz for their support and G. Stoiser for his help in achieving the numerical results.