scholarly journals On validation of solutions to linear programming problems on cluster computing systems

2021 ◽  
pp. 252-261
Author(s):  
Л.Б. Соколинский ◽  
И.М. Соколинская
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Large Scale ◽  
Cluster Computing ◽  
Parallel Implementation ◽  
Main Idea ◽  
Small Radius ◽  
Feasible Region ◽  
Computing Systems ◽  
Scalable Algorithm

В статье представлен параллельный алгоритм валидации решений задач линейного программирования. Идея метода состоит в том, чтобы генерировать регулярный набор точек на гиперсфере малого радиуса, центрированной в точке тестируемого решения. Целевая функция вычисляется для каждой точки валидационного множества, принадлежащей допустимой области. Если все полученные значения меньше или равны значению целевой функции в точке, проверяемой как решение, то эта точка считается корректным решением. Параллельная реализация алгоритма VaLiPro выполнена на языке C++ с использованием параллельного BSF-каркаса, инкапсулирующего в проблемно-независимой части своего кода все аспекты, связанные с распараллеливанием программы на базе библиотеки MPI. Приводятся результаты масштабных вычислительных экспериментов на кластерной вычислительной системе, подтверждающие эффективность предложенного подхода. The paper presents and evaluates a scalable algorithm for validating solutions to linear programming (LP) problems on cluster computing systems. The main idea of the method is to generate a regular set of points (validation set) on a small-radius hypersphere centered at the solution point submitted to validation. The objective function is computed at each point of the validation that belongs to the feasible region. If all the values are less than or equal to the value of the objective function at the point that is to be validated, then this point is the correct solution. The parallel implementation of the VaLiPro algorithm is written in C++ through the parallel BSF-skeleton, which encapsulates all aspects related to the MPI-based parallelization of the program. We provide the results of large-scale computational experiments on a cluster computing system to study the scalability of the VaLiPro algorithm.

scholarly journals On an iterative method for solving linear programming problems on cluster computing systems

2020 ◽  
pp. 329-340
Author(s):  
И.М. Соколинская ◽  
Л.Б. Соколинский
Keyword(s):  
Linear Programming ◽  
Large Scale ◽  
Cluster Computing ◽  
Parallel Implementation ◽  
Computing System ◽  
Feasible Region ◽  
Computing Systems ◽  
C Language ◽  
Linear Programming Problems ◽  
Predictor Corrector

Статья посвящена исследованию нового метода решения сверхбольших задач линейного программирования. Указанный метод получил название "апекс-метод". Апекс-метод работает по схеме предиктор-корректор. На фазе предиктор находится точка, лежащая на границе <em>n</em>-мерного многогранника, задающего допустимую область задачи линейного программирования. На фазе корректор организуется итерационный процесс, в результате которого строится последовательность точек, сходящаяся к точному решению задачи линейного программирования. В статье дается формальное описание апекс-метода и приводятся сведения о его параллельной реализации на языке C++ с использованием библиотеки MPI. Приводятся результаты масштабных вычислительных экспериментов на кластерной вычислительной системе по исследованию масштабируемости апекс-метода. The paper is devoted to a new method for solving large-scale linear programming (LP) problems. This method is called the apex-method. The apex-method uses the predictor–corrector framework. Thepredictor step calculates a point belonging to the feasible region of the LP problem. The corrector step calculates a sequence of points converging to the exact solution of the LP problem. The paper gives a formal description of the apex-method and provides information about its parallel implementation in C++ language using the MPI library. The results of large-scale computational experiments on a cluster computing system to study the scalability of the apex method are discussed.

scholarly journals A scalable algorithm for solving non-stationary linear programming problems

2018 ◽  
pp. 540-550
Author(s):  
И.М. Соколинская ◽  
Л.Б. Соколинский
Keyword(s):  
Linear Programming ◽  
High Dimension ◽  
Upper Bound ◽  
Cluster Computing ◽  
Computation Model ◽  
Computing Systems ◽  
Scalable Algorithm ◽  
Linear Programming Problems ◽  
Algorithm Implementation ◽  
Algorithm Scalability

Статья посвящена исследованию алгоритма NSLP для решения нестационарных задач линейного программирования сверхбольшой размерности, ориентированного на кластерные вычислительные системы. В основе анализа лежит модель параллельных вычислений BSF, основанная на моделях BSP и SPMD. Даются краткие описания алгоритма NSLP и модели BSF. Рассматривается реализация алгоритма NSLP в виде BSF-программы. На базе стоимостной метрики модели BSF выводится верхняя граница масштабируемости алгоритма NSLP и оценивается эффективность его параллелизации. Описывается реализация алгоритма NSLP на основе программного каркаса BSF на языке Си и приводятся результаты экспериментов, исследующих масштабируемость указанной реализации на модельной задаче линейного программирования. Делается сравнение результатов, полученных аналитическим и экспериментальным путем. This paper is devoted to the scalability study of an NSLP algorithm for solving non-stationary high-dimension linear programming problems on cluster computing systems. The analysis is based on the BSF model of parallel computations. The BSF model is a new parallel computation model designed on the basis of BSP and SPMD models. The brief descriptions of the NSLP algorithm and the BSF model are given. The NSLP algorithm implementation in the form of a BSF program is considered. On the basis of the BSF cost metric, the upper bound of the NSLP algorithm scalability is derived and its parallel efficiency is estimated. The NSLP algorithm implementation using BSF skeleton is described. The scalability estimates obtained analytically and experimentally are compared.

scholarly journals Didáctica de la programación lineal con el método gráfico

2007 ◽  
pp. 17
Author(s):  
Fernando León Parada
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Model ◽  
Geometric Analysis ◽  
Linear Programming Model ◽  
Feasible Region ◽  
Optimal Point ◽  
Decision Variables ◽  
Programación Lineal

Every Linear Programming model has a Objective Function determined by the coefficients of the decision variables. The Objective Function can be represented as a line evaluated on an optimal point which slope depends from the variation of the coefficients. The segments adjacent to the optimal point in the Feasible Region can also determine the variation of this slope. Finally a geometric analysis allows to establishing the constraining of the boundary for the Feasible Region keeping as a domain for the Objective Function.

Balancing the Degree of Exploration and Exploitation of Swarm Intelligence Using Parallel Computing

2019 ◽  
Vol 28 (03) ◽  
pp. 1950014
Author(s):  
Surafel Luleseged Tilahun
Keyword(s):  
Parallel Computing ◽  
Objective Function ◽  
Swarm Intelligence ◽  
Parallel Implementation ◽  
Metaheuristic Algorithms ◽  
Feasible Region ◽  
Central Research ◽  
Exploration And Exploitation ◽  
Specific Distribution ◽  
Search Behaviors

The two basic search behaviors used in metaheuristic algorithms in general and swarm intelligence in particular are exploration and exploitation. Exploration refers to searching the unexplored area of the feasible region while exploitation refers to the search of the neighborhood of a promising region. The success of these algorithms highly depends on how these two search behaviors are balanced. Increasing the number of initial solutions indeed increase the performance of the algorithm over the expense of the runtime. Different exploration and exploitation adjustments are used in different metaheuristic algorithms. To well balance the degree of exploration and exploitation, which ultimately leads to finding good solutions, needs a careful implementation and sufficient runtime. However, the runtime is another issue which has been a bottleneck for different application, especially for high dimensional, highly constrained and expensive objective function cases. Using parallel computing to overcome this limitation has been a central research issue in the study of metaheuristic algorithms. The three basic parallelization approaches used are the parallel move model which is a master slave model where solutions will be distributed to different processors for updating, parallel multi start model which uses several updating methods for the solutions in parallel and move acceleration method, which refers to a parallel implementation with parallelizing the objective function. There is no specific parallelization used towards balancing of exploration and exploitation is mentioned. Hence, in this paper a parallel implementation of swarm intelligence in general and two algorithms, namely particle swarm optimization and prey predator algorithm, in particular will be discussed. The parallelization proposed is aimed to merge the parallel move model and the parallel multi start model with the aim of adjusting the degree of exploration and exploitation. Experimental study based on ten benchmark problems from constrained as well as unconstrained problems will be used for simulation. It will be shown that the parallelization with specific distribution of exploration and exploitation runs better and efficiently compared to the serial and also the parallel version without exporation/exploitation adjustment.

scholarly journals A Novel Method for Solving the Fully Fuzzy Bilevel Linear Programming Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Aihong Ren
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Level Sets ◽  
Fuzzy Numbers ◽  
Main Idea ◽  
Bilevel Linear Programming ◽  
Decision Variables ◽  
Upper Level

We address a fully fuzzy bilevel linear programming problem in which all the coefficients and variables of both objective functions and constraints are expressed as fuzzy numbers. This paper is to develop a new method to deal with the fully fuzzy bilevel linear programming problem by applying interval programming method. To this end, we first discretize membership grade of fuzzy coefficients and fuzzy decision variables of the problem into a finite number ofα-level sets. By usingα-level sets of fuzzy numbers, the fully fuzzy bilevel linear programming problem is transformed into an interval bilevel linear programming problem for eachα-level set. The main idea to solve the obtained interval bilevel linear programming problem is to convert the problem into two deterministic subproblems which correspond to the lower and upper bounds of the upper level objective function. Based on theKth-best algorithm, the two subproblems can be solved sequentially. Based on a series ofα-level sets, we introduce a linear piecewise trapezoidal fuzzy number to approximate the optimal value of the upper level objective function of the fully fuzzy bilevel linear programming problem. Finally, a numerical example is provided to demonstrate the feasibility of the proposed approach.

Grey-fuzzy solution for multi-objective linear programming with interval coefficients

2018 ◽  
Vol 8 (3) ◽  
pp. 312-327 ◽  
Author(s):  
Amin Mahmoudi ◽  
Mohammad Reza Feylizadeh ◽  
Davood Darvishi ◽  
Sifeng Liu
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Fuzzy Programming ◽  
Feasible Region ◽  
Grey Theory ◽  
Acceptable Solution ◽  
Content Type ◽  
Multi Objective ◽  
Interval Coefficients ◽  
Interactive Fuzzy Programming

Purpose The purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and interactive fuzzy programming approaches. Design/methodology/approach In the proposed algorithm, first, lower and upper bounds of each objective function in its feasible region will be determined. Afterwards using fuzzy approach, considering a membership function for each objective function and finally using grey linear programming, the solution for this problem will be obtained. Findings According to the presented example, in this paper, the proposed method is both simple in use and suitable for solving different problems. In the numerical example mentioned in this paper, the proposed method provides an acceptable solution for such problems. Practical implications As in most real-world situations, the coefficients of decision models are not known and exact. In this paper, the authors consider the model of MOLP with interval data, since one of the solutions to cover uncertainty is using interval theory. Originality/value Based on using grey theory and interactive fuzzy programming approaches, an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions.

scholarly journals Linear programming and pediatric dietetics

1990 ◽  
Vol 64 (2) ◽  
pp. 307-317 ◽  
Author(s):  
Ciro Colavita ◽  
Renato D'orsi
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Operations Research ◽  
Analytical Methods ◽  
Simple Problem ◽  
Feasible Region ◽  
Recommended Dietary Allowances ◽  
Geometrical Representation

The composition of 500 foods has been stored in a computer in order to analyse a child's diet. The methodology of operations research is applied to a very simple problem: a diet with only two foods. The geometrical representation of the ‘feasible region’ and of the ‘objective function’ is illustrated. One of the analytical methods employable with many variables (foods) is considered. This method was used in trying to find diets allowing for the preferential use of selected foods while respecting recommended dietary allowances, the tastes of the child and other constraints. The theoretical difficulty of transferring this methodology to pediatric dietetics was examined. We solved a simple case utilizing this procedure.

Sign in / Sign up

Export Citation Format

Share Document