scholarly journals An Effective Branch and Bound Algorithm for Minimax Linear Fractional Programming

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Hong-Wei Jiao ◽  
Feng-Hui Wang ◽  
Yong-Qiang Chen
Keyword(s):  
Branch And Bound ◽  
Fractional Programming ◽  
Optimal Solution ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Feasible Region ◽  
Test Problems ◽  
Linear Fractional Programming ◽  
Successive Refinement ◽  
Global Optimal

An effective branch and bound algorithm is proposed for globally solving minimax linear fractional programming problem (MLFP). In this algorithm, the lower bounds are computed during the branch and bound search by solving a sequence of linear relaxation programming problems (LRP) of the problem (MLFP), which can be derived by using a new linear relaxation bounding technique, and which can be effectively solved by the simplex method. The proposed branch and bound algorithm is convergent to the global optimal solution of the problem (MLFP) through the successive refinement of the feasible region and solutions of a series of the LRP. Numerical results for several test problems are reported to show the feasibility and effectiveness of the proposed algorithm.

scholarly journals An Efficient Branch-and-Bound Algorithm for Globally Solving Minimax Linear Fractional Programming Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Pujun Jia ◽  
Hongwei Jiao ◽  
Dongwei Shi ◽  
Jingben Yin
Keyword(s):  
Programming Problem ◽  
Branch And Bound ◽  
Fractional Programming ◽  
Outer Space ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Linear Fractional Programming ◽  
Fractional Programming Problem ◽  
Wide Range ◽  
Linear Fractional Programming Problem

This paper presents an efficient outer space branch-and-bound algorithm for globally solving a minimax linear fractional programming problem (MLFP), which has a wide range of applications in data envelopment analysis, engineering optimization, management optimization, and so on. In this algorithm, by introducing auxiliary variables, we first equivalently transform the problem (MLFP) into the problem (EP). By using a new linear relaxation technique, the problem (EP) is reduced to a sequence of linear relaxation problems over the outer space rectangle, which provides the valid lower bound for the optimal value of the problem (EP). Based on the outer space branch-and-bound search and the linear relaxation problem, an outer space branch-and-bound algorithm is constructed for globally solving the problem (MLFP). In addition, the convergence and complexity of the presented algorithm are given. Finally, numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.

scholarly journals A New Linearizing Method for Sum of Linear Ratios Problem with Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Hongwei Jiao ◽  
Yongqiang Chen
Keyword(s):  
Lower Bound ◽  
Optimal Solution ◽  
Global Optimum ◽  
Initial Problem ◽  
Linear Relaxation ◽  
Feasible Region ◽  
Global Optimal Solution ◽  
Optimal Value ◽  
Global Optimal ◽  
Methods Numerical

A new linearizing method is presented for globally solving sum of linear ratios problem with coefficients. By using the linearizing method, linear relaxation programming (LRP) of the sum of linear ratios problem with coefficients is established, which can provide the reliable lower bound of the optimal value of the initial problem. Thus, a branch and bound algorithm for solving the sum of linear ratios problem with coefficients is put forward. By successively partitioning the linear relaxation of the feasible region and solving a series of the LRP, the proposed algorithm is convergent to the global optimal solution of the initial problem. Compared with the known methods, numerical experimental results show that the proposed method has the higher computational efficiency in finding the global optimum of the sum of linear ratios problem with coefficients.

scholarly journals An Efficient Algorithm for Solving a Class of Fractional Programming Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yongjian Qiu ◽  
Yuming Zhu ◽  
Jingben Yin
Keyword(s):  
Computational Complexity ◽  
Global Convergence ◽  
Programming Problem ◽  
Branch And Bound ◽  
Fractional Programming ◽  
Financial Engineering ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Optimal Solutions ◽  
Communication Engineering

This paper presents an efficient branch-and-bound algorithm for globally solving a class of fractional programming problems, which are widely used in communication engineering, financial engineering, portfolio optimization, and other fields. Since the kind of fractional programming problems is nonconvex, in which multiple locally optimal solutions generally exist that are not globally optimal, so there are some vital theoretical and computational difficulties. In this paper, first of all, for constructing this algorithm, we propose a novel linearizing method so that the initial fractional programming problem can be converted into a linear relaxation programming problem by utilizing the linearizing method. Secondly, based on the linear relaxation programming problem, a novel branch-and-bound algorithm is designed for the kind of fractional programming problems, the global convergence of the algorithm is proved, and the computational complexity of the algorithm is analysed. Finally, numerical results are reported to indicate the feasibility and effectiveness of the algorithm.

scholarly journals An Interior Point Method for Solving Semidefinite Programs Using Cutting Planes and Weighted Analytic Centers

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
John Machacek ◽  
Shafiu Jibrin
Keyword(s):  
Newton’S Method ◽  
Interior Point ◽  
Newton's Method ◽  
Interior Point Method ◽  
Optimal Solution ◽  
Point Method ◽  
Feasible Region ◽  
Test Problems ◽  
Analytic Centers ◽  
Semidefinite Programs

We investigate solving semidefinite programs (SDPs) with an interior point method called SDP-CUT, which utilizes weighted analytic centers and cutting plane constraints. SDP-CUT iteratively refines the feasible region to achieve the optimal solution. The algorithm uses Newton’s method to compute the weighted analytic center. We investigate different stepsize determining techniques. We found that using Newton's method with exact line search is generally the best implementation of the algorithm. We have also compared our algorithm to the SDPT3 method and found that SDP-CUT initially gets into the neighborhood of the optimal solution in less iterations on all our test problems. SDP-CUT also took less iterations to reach optimality on many of the problems. However, SDPT3 required less iterations on most of the test problems and less time on all the problems. Some theoretical properties of the convergence of SDP-CUT are also discussed.

scholarly journals Ranking Function Application for Optimal Solution of Fractional programming problem

2020 ◽  
Vol 25 (1) ◽  
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.

scholarly journals Global Optimization for Solving Linear Multiplicative Programming Based on a New Linearization Method

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Chun-Feng Wang ◽  
Yan-Qin Bai
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Optimization Algorithm ◽  
Linearization Method ◽  
Branch And Bound Algorithm ◽  
Linear Relaxation ◽  
Global Optimization Algorithm ◽  
Multiplicative Programming ◽  
Linear Multiplicative Programming ◽  
Pruning Techniques

This paper presents a new global optimization algorithm for solving a class of linear multiplicative programming (LMP) problem. First, a new linear relaxation technique is proposed. Then, to improve the convergence speed of our algorithm, two pruning techniques are presented. Finally, a branch and bound algorithm is developed for solving the LMP problem. The convergence of this algorithm is proved, and some experiments are reported to illustrate the feasibility and efficiency of this algorithm.

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