Linear programming with multiple choice constraints for single chain undiscounted Markov decision problems

This paper develops an efficient LP algorithm for solving single chain undiscounted Markov decision problems. The algorithm imposes, in the framework of the simplex method, the multiple choice constraints that exactly one basic variable be chosen from each Markov state. It is proved that the algorithm converges to an optimal solution in a finite number of steps.     

Towards a practical parallelisation of the simplex method

The simplex method is frequently the most efficient method of solving linear programming (LP) problems. This paper reviews previous attempts to parallelise the simplex method in relation to efficient serial simplex techniques and the nature of practical LP problems. For the major challenge of solving general large sparse LP problems, there has been no parallelisation of the simplex method that offers significantly improved performance over a good serial implementation. However, there has been some success in developing parallel solvers for LPs that are dense or have particular structural properties. As an outcome of the review, this paper identifies scope for future work towards the goal of developing parallel implementations of the simplex method that are of practical value.     

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

The dual simplex algorithm has become a strong contender in solving large scale LP problems. One key problem of any dual simplex algorithm is to obtain a dual feasible basis as a starting point. We give an overview of methods which have been proposed in the literature and present new stable and efficient ways to combine them within a state-of-the-art optimization system for solving real world linear and mixed integer programs. Furthermore, we address implementation aspects and the connection between dual feasibility and LP-preprocessing. Computational results are given for a large set of large scale LP problems, which show our dual simplex implementation to be superior to the best existing research and open-source codes and competitive to the leading commercial code on many of our most difficult problem instances.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Intelligent gradient search in linear programming

The simplex algorithm is still the best known and most frequently used way to solve LP problems. Khachian has suggested a method to solve these problems in polynomial time. The average behaviour of his method, however, is still inferior to the modern simplex based LP codes. A new gradient based approach which also has polynomial worst-case behaviour has been suggested by Karmarkar. This method was modified, programmed and compared with other available LP codes. It is shown that the numerical efficiency of Karmarkar's method compares favourably with other LP codes, particularly for problems with high numbers of variables and few constraints.     

Revised simplex algorithm for finite Markov decision processes

We introduce a revised simplex algorithm for solving a typical type of dynamic programming equation arising from a class of finite Markov decision processes. The algorithm also applies to several types of optimal control problems with diffusion models after discretization. It is based on the regular simplex algorithm, the duality concept in linear programming, and certain special features of the dynamic programming equation itself. Convergence is established for the new algorithm. The algorithm has favorable potential applicability when the number of actions is very large or even infinite.     

The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

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.The solution of a great number of problems has shown that efficient problem solving requires automatic adaptation of algorithmic techniques upon problem characteristics. We show when a given technique should be used for a particular problem.The last part of this paper describes 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.All these features have been implemented in the IBM Mathematical Programming System, MPSX/370, and its feature MIP/370. Extensive numerical results and comparisons on real-life problems are provided and commented upon.Presented at the IXth International Symposium on Mathematical Programming in Budapest (1976).     

Parallel distributed-memory simplex for large-scale stochastic LP problems

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.     

A o(n logn) algorithm for LP knapsacks with GUB constraints

A specialization of the dual simplex method is developed for solving the linear programming (LP) knapsack problem subject to generalized upper bound (GUB) constraints. The LP/GUB knapsack problem is of interest both for solving more general LP problems by the dual simplex method, and for applying surrogate constraint strategies to the solution of 0–1 Multiple Choice integer programming problems. We provide computational bounds for our method of o(n logn), wheren is the total number of problem variables. These bounds reduce the previous best estimate of the order of complexity of the LP/GUB knapsack problem (due to Witzgall) and provide connections to computational bounds for the ordinary knapsack problem.We further provide a variant of our method which has only slightly inferior worst case bounds, yet which is ideally suited to solving integer multiple choice problems due to its ability to post-optimize while retaining variables otherwise weeded out by convex dominance criteria.     

A Dual Simplex Implementation of a Constraint Selection Algorithm for Linear Programming

The constraint selection approach to linear programming begins by solving a relaxed version of the problem using only a few of the original constraints. If the solution obtained to this relaxation satisfies the remaining constraints it is optimal for the original LP. Otherwise, additional constraints must be incorporated in a larger relaxation. The procedure successively generates larger subproblems until an optimal solution is obtained which satisfies all of the original constraints. Computational results for a dual simplex implementation of this technique indicate that solving several small subproblems in this manner is more computationally efficient than solving the original LP using the revised simplex method.     

On simplex method with most-obtuse-angle rule and cosine rule

A class of simplex methods for solving linear programming (LP) problems, with cosine pivot rule, have been presented in some recent papers. In this paper we show that the cosine rule used in this class is equivalent to the most-obtuse-angle pivot rule, proposed by Pan (1990) [6]. The relation between the direct method for LP and the most-obtuse-angle rule is discussed.     

Finding all Solutions of Systems of Nonlinear Equations Using the Dual Simplex Method

An efficient algorithm is proposed for finding all solutions of nonlinear equations using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, the system of nonlinear equations is transformed into an LP problem, to which the simplex method is applied. However, although the LP test is very powerful, it requires many pivotings for each region. In this paper, we use the dual simplex method in the LP test, which makes the average number of pivotings per region much smaller (less than one, for example) and makes the algorithm very efficient. By numerical examples, it is shown that the proposed algorithm can find all solutions of systems of 200 nonlinear equations in practical computation time.     

Solving singularly constrained generalized network problems

The singularly constrained generalized network problem represents a large class of capacitated linear programming (LP) problems. This class includes any LP problem whose coefficient matrix, ignoring single upper bound constraints, containsm + 1 rows which may be ordered such that each column has at most two non-zero entries in the firstm rows. The paper describes efficient procedures for solving such problems and presents computational results which indicate that, on large problems, these procedures are at least twenty-five times more efficient than the state of the art LP systemapex-iii.This research was partly supported by ONR Contract N00014-76-C-0383 with Decision Analysis and Research Institute and by Project NR047-021, ONR Contracts N00014-75-C-0616 and N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Implementing pure adaptive search for global optimization using Markov chain sampling

The Pure Adaptive Search (PAS) algorithm for global optimization yields a sequence of points, each of which is uniformly distributed in the level set corresponding to its predecessor. This algorithm has the highly desirable property of solving a large class of global optimization problems using a number of iterations that increases at most linearly in the dimension of the problem. Unfortunately, PAS has remained of mostly theoretical interest due to the difficulty of generating, in each iteration, a point uniformly distributed in the improving feasible region. In this article, we derive a coupling equivalence between generating an approximately uniformly distributed point using Markov chain sampling, and generating an exactly uniformly distributed point with a certain probability. This result is used to characterize the complexity of a PAS-implementation as a function of (a) the number of iterations required by PAS to achieve a certain solution quality guarantee, and (b) the complexity of the sampling algorithm used. As an application, we use this equivalence to show that PAS, using the so-called Random ball walk Markov chain sampling method for generating nearly uniform points in a convex region, can be used to solve most convex programming problems in polynomial time.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

Stopping Rules for the Stochastic Nested Partitions Method

We have recently developed a global optimization methodology for solving combinatorial problems with either deterministic or stochastic performance functions. This method, the Nested Partitions (NP) method has been shown to generate a Markov chain and with probability one to converge to a global optimum. In this paper, we study the rate of convergence of the method through the use of Markov Chain Monte Carlo (MCMC) methods, and use this to derive stopping rules that can be applied during simulation-based optimization. A numerical example serves to illustrate the feasibility of our approach.     

树指标马氏链的等价定义

国内外关于树指标随机过程的研究已经取得了一定的成果.Benjamini和Peres首先给出了树指标马氏链的定义.Berger和叶中行研究了齐次树图上平稳随机场熵率的存在性.杨卫国与刘文研究了树上马氏场的强大数定律与渐近均分性.杨卫国又研究了一般树指标马氏链的强大数定律.为了以后更有效的研究树指标随机过程的一系列相关问题,本文在分析研究前人成果的基础上,给出了树指标马氏链的等价定义,并用数学归纳法证明了其等价性.