Linear-quadratic programming and its application to data correction of improper linear programming problems

AbstractThe problem of maximizing a linear function with linear and quadratic constraints is considered. The solution of the problem is obtained in a constructive form using the Lagrange function and the optimality conditions. Many optimization problems can be reduced to the problem of this type. In this paper, as an application, we consider an improper linear programming problem formalized in the form of maximization of the initial linear criterion with a restriction to the Euclidean norm of the correction vector of the right-hand side of the constraints or the Frobenius norm of the correction matrix of both sides of the constraints.

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A Global Optimization Algorithm for Generalized Quadratic Programming

Journal of Applied Mathematics ◽

10.1155/2013/215312 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 4

Author(s):

Hongwei Jiao ◽

Yongqiang Chen

Keyword(s):

Linear Programming ◽

Global Optimization ◽

Quadratic Programming ◽

Optimization Algorithm ◽

Global Minimum ◽

Nonconvex Programming ◽

Nonconvex Quadratic Programming ◽

Global Optimization Algorithm ◽

Quadratic Constraints ◽

Linear Programming Problems

We present a global optimization algorithm for solving generalized quadratic programming (GQP), that is, nonconvex quadratic programming with nonconvex quadratic constraints. By utilizing a new linearizing technique, the initial nonconvex programming problem (GQP) is reduced to a sequence of relaxation linear programming problems. To improve the computational efficiency of the algorithm, a range reduction technique is employed in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (GQP) by means of the subsequent solutions of a series of relaxation linear programming problems. Finally, numerical results show the robustness and effectiveness of the proposed algorithm.

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A METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS WITH FUZZY PARAMETERS BASED ON MULTIOBJECTIVE LINEAR PROGRAMMING TECHNIQUE

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595907001395 ◽

2007 ◽

Vol 24 (04) ◽

pp. 557-573 ◽

Cited By ~ 3

Author(s):

M. ZANGIABADI ◽

H. R. MALEKI

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Linear Programming Problem ◽

Fuzzy Numbers ◽

Optimization Problems ◽

Multiobjective Linear Programming ◽

Programming Technique ◽

Linear Programming Problems ◽

Linear Programming Technique ◽

Fuzzy Parameters

In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, we propose several theorems which are used to obtain optimal solutions of linear programming problems with fuzzy parameters. Finally some examples are given for illustrating the proposed method of solving linear programming problem with fuzzy parameters.

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On Linear and Quadratic Two-Stage Transportation Problem

Cybernetics and Computer Technologies ◽

10.34229/2707-451x.20.4.1 ◽

2020 ◽

pp. 5-14

Author(s):

P. Stetsyuk ◽

O. Lykhovyd ◽

A. Suprun

Keyword(s):

Linear Programming ◽

Mathematical Programming ◽

Quadratic Programming ◽

Programming Problem ◽

Linear Programming Problem ◽

Transportation Problem ◽

Quadratic Programming Problem ◽

Modeling Language ◽

Two Stage ◽

Linear Programming Problems

Introduction. When formulating the classical two-stage transportation problem, it is assumed that the product is transported from suppliers to consumers through intermediate points. Intermediary firms and various kinds of storage facilities (warehouses) can act as intermediate points. The article discusses two mathematical models for two-stage transportation problem (linear programming problem and quadratic programming problem) and a fairly universal way to solve them using modern software. It uses the description of the problem in the modeling language AMPL (A Mathematical Programming Language) and depends on which of the known programs is chosen to solve the problem of linear or quadratic programming. The purpose of the article is to propose the use of AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems, to formulate a mathematical model of a quadratic programming two-stage transportation problem and to investigate its properties. Results. The properties of two variants of a two-stage transportation problem are described: a linear programming problem and a quadratic programming problem. An AMPL code for solving a linear programming two-stage transportation problem using modern software for linear programming problems is given. The results of the calculation using Gurobi program for a linear programming two-stage transportation problem, which has many solutions, are presented and analyzed. A quadratic programming two-stage transportation problem was formulated and conditions were found under which it has unique solution. Conclusions. The developed AMPL-code for a linear programming two-stage transportation problem and its modification for a quadratic programming two-stage transportation problem can be used to solve various logistics transportation problems using modern software for solving mathematical programming problems. The developed AMPL code can be easily adapted to take into account the lower and upper bounds for the quantity of products transported from suppliers to intermediate points and from intermediate points to consumers. Keywords: transportation problem, linear programming problem, AMPL modeling language, Gurobi program, quadratic programming problem.

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A Fast Approach to Solve Matrix Games with Payoffs of Trapezoidal Fuzzy Numbers

Asia Pacific Journal of Operational Research ◽

10.1142/s0217595916500470 ◽

2016 ◽

Vol 33 (06) ◽

pp. 1650047 ◽

Cited By ~ 2

Author(s):

Sanjiv Kumar ◽

Ritika Chopra ◽

Ratnesh R. Saxena

Keyword(s):

Linear Programming ◽

Linear Programming Problem ◽

Fuzzy Numbers ◽

Optimization Problems ◽

Fuzzy Linear Programming ◽

Matrix Game ◽

Matrix Games ◽

Trapezoidal Fuzzy Numbers ◽

Fast Approach ◽

The Matrix

The aim of this paper is to develop an effective method for solving matrix game with payoffs of trapezoidal fuzzy numbers (TrFNs). The method always assures that players’ gain-floor and loss-ceiling have a common TrFN-type fuzzy value and hereby any matrix game with payoffs of TrFNs has a TrFN-type fuzzy value. The matrix game is first converted to a fuzzy linear programming problem, which is converted to three different optimization problems, which are then solved to get the optimum value of the game. The proposed method has an edge over other method as this focuses only on matrix games with payoff element as symmetric trapezoidal fuzzy number, which might not always be the case. A numerical example is given to illustrate the method.

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Ranking Function Application for Optimal Solution of Fractional programming problem

Al-Qadisiyah Journal Of Pure Science ◽

10.29350/jops.2020.25.1.1048 ◽

2020 ◽

Vol 25 (1) ◽

Cited By ~ 2

Author(s):

Rasha Jalal

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Fractional Programming ◽

Linear Programming Problem ◽

Fuzzy Numbers ◽

Optimal Solution ◽

Ranking Function ◽

Linear Fractional Programming ◽

Fractional Programming Problem ◽

Linear Programming Problems

The aim of this paper is to suggest a solution procedure to fractional programming problem based on new ranking function (RF) with triangular fuzzy number (TFN) based on alpha cuts sets of fuzzy numbers. In the present procedure the linear fractional programming (LFP) problems is converted into linear programming problems. We concentrate on linear programming problem problems in which the coefficients of objective function are fuzzy numbers, the right- hand side are fuzzy numbers too, then solving these linear programming problems by using a new ranking function. The obtained linear programming problem can be solved using win QSB program (simplex method) which yields an optimal solution of the linear fractional programming problem. Illustrated examples and comparisons with previous approaches are included to evince the feasibility of the proposed approach.

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A Comparative Study of Redundant Constraints Identification Methods in Linear Programming Problems

Mathematical Problems in Engineering ◽

10.1155/2010/723402 ◽

2010 ◽

Vol 2010 ◽

pp. 1-16 ◽

Cited By ~ 6

Author(s):

Paulraj S. ◽

Sumathi P.

Keyword(s):

Linear Programming ◽

Large Scale ◽

Optimization Problems ◽

Inequality Constraints ◽

Computational Effort ◽

Linear Functions ◽

Redundant Constraints ◽

Computer Scientists ◽

Constraints Solving ◽

Linear Programming Problems

The objective function and the constraints can be formulated as linear functions of independent variables in most of the real-world optimization problems. Linear Programming (LP) is the process of optimizing a linear function subject to a finite number of linear equality and inequality constraints. Solving linear programming problems efficiently has always been a fascinating pursuit for computer scientists and mathematicians. The computational complexity of any linear programming problem depends on the number of constraints and variables of the LP problem. Quite often large-scale LP problems may contain many constraints which are redundant or cause infeasibility on account of inefficient formulation or some errors in data input. The presence of redundant constraints does not alter the optimal solutions(s). Nevertheless, they may consume extra computational effort. Many researchers have proposed different approaches for identifying the redundant constraints in linear programming problems. This paper compares five of such methods and discusses the efficiency of each method by solving various size LP problems and netlib problems. The algorithms of each method are coded by using a computer programming language C. The computational results are presented and analyzed in this paper.

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Convergence Analysis on an Accelerated Proximal Point Algorithm for Linearly Constrained Optimization Problems

Mathematical Problems in Engineering ◽

10.1155/2020/8873507 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Sha Lu ◽

Zengxin Wei

Keyword(s):

Machine Learning ◽

Linear Programming ◽

Proximal Point Algorithm ◽

Original Problem ◽

Optimization Problems ◽

Linear Constraints ◽

Proximal Point ◽

Constrained Optimization Problems ◽

Linearly Constrained ◽

Linear Programming Problems

Proximal point algorithm is a type of method widely used in solving optimization problems and some practical problems such as machine learning in recent years. In this paper, a framework of accelerated proximal point algorithm is presented for convex minimization with linear constraints. The algorithm can be seen as an extension to G u ¨ ler’s methods for unconstrained optimization and linear programming problems. We prove that the sequence generated by the algorithm converges to a KKT solution of the original problem under appropriate conditions with the convergence rate of O 1 / k 2 .

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A dual simplex method for grey linear programming problems based on duality results

Grey Systems Theory and Application ◽

10.1108/gs-10-2019-0044 ◽

2020 ◽

Vol 10 (2) ◽

pp. 145-157

Author(s):

Davood Darvishi Salookolaei ◽

Seyed Hadi Nasseri

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Linear Programming Problem ◽

Simplex Method ◽

Dual Problem ◽

Complementary Slackness ◽

Dual Simplex Method ◽

Content Type ◽

Dual Simplex ◽

Linear Programming Problems

PurposeFor extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.Design/methodology/approachThe authors discuss the solution concepts of primal and dual of grey linear programming problems without converting them to classical linear programming problems. A numerical example is provided to illustrate the theory developed.FindingsBy using arithmetic operations between interval grey numbers, the authors prove the complementary slackness theorem for grey linear programming problem and the associated dual problem.Originality/valueComplementary slackness theorem for grey linear programming is first presented and proven. After that, a dual simplex method in grey environment is introduced and then some useful concepts are presented.

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Possibilistic Linear Programming Problems involving Normal Random Variables

International Journal of Fuzzy System Applications ◽

10.4018/ijfsa.2016070101 ◽

2016 ◽

Vol 5 (3) ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Suresh Kumar Barik ◽

M. P. Biswal

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Random Variables ◽

Fuzzy Programming ◽

Solution Procedure ◽

Mathematical Programming Problem ◽

Possibility Distribution ◽

Linear Programming Problems ◽

The Right ◽

Possibilistic Linear Programming

A new solution procedure of possibilistic linear programming problem is developed involving the right hand side parameters of the constraints as normal random variables with known means and variances and the objective function coefficients are considered as triangular possibility distribution. In order to solve the proposed problem, convert the problem into a crisp equivalent deterministic multi-objective mathematical programming problem and then solved by using fuzzy programming method. A numerical example is presented to illustrate the solution procedure and developed methodology.

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