scholarly journals A new branch and bound algorithm for minimax ratios problems

2017 ◽  
Vol 15 (1) ◽  
pp. 840-851 ◽  
Author(s):  
Yingfeng Zhao ◽  
Sanyang Liu ◽  
Hongwei Jiao
Keyword(s):  
Programming Problem ◽  
Branch And Bound ◽  
Convex Relaxation ◽  
Numerical Test ◽  
Auxiliary Variable ◽  
Branch And Bound Algorithm ◽  
Computational Performance ◽  
Minimax Fractional Programming ◽  
Convexity And Concavity ◽  
Algorithm Scheme

Abstract This study presents an efficient branch and bound algorithm for globally solving the minimax fractional programming problem (MFP). By introducing an auxiliary variable, an equivalent problem is firstly constructed and the convex relaxation programming problem is then established by utilizing convexity and concavity of functions in the problem. Other than usual branch and bound algorithm, an adapted partition skill and a practical reduction technique performed only in an unidimensional interval are incorporated into the algorithm scheme to significantly improve the computational performance. The global convergence is proved. Finally, some comparative experiments and a randomized numerical test are carried out to demonstrate the efficiency and robustness of the proposed algorithm.

scholarly journals A Deterministic Method for Solving the Sum of Linear Ratios Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhenping Wang ◽  
Yonghong Zhang ◽  
Paolo Manfredi
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Branch And Bound ◽  
Nonconvex Programming ◽  
Real Life ◽  
Branch And Bound Algorithm ◽  
Separation Technique ◽  
Linear Programming Relaxation ◽  
Deterministic Method ◽  
Linearization Technique

Since the sum of linear ratios problem (SLRP) has many applications in real life, for globally solving it, an efficient branch and bound algorithm is presented in this paper. By utilizing the characteristic of the problem (SLRP), we propose a convex separation technique and a two-part linearization technique, which can be used to generate a sequence of linear programming relaxation of the initial nonconvex programming problem. For improving the convergence speed of this algorithm, a deleting rule is presented. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency 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 reduced space branch and bound algorithm for a class of sum of ratios problems

2018 ◽  
Vol 16 (1) ◽  
pp. 539-552
Author(s):  
Yingfeng Zhao ◽  
Ting Zhao
Keyword(s):  
Branch And Bound ◽  
Convex Relaxation ◽  
Branch And Bound Algorithm ◽  
Engineering Practice ◽  
Auxiliary Variables ◽  
Equivalent Problem ◽  
Sum Of Ratios ◽  
Relaxation Problem ◽  
Reduction Techniques ◽  
Reduced Space

AbstractSum of ratios problem occurs frequently in various areas of engineering practice and management science, but most solution methods for this kind of problem are often designed for determining local solutions . In this paper, we develop a reduced space branch and bound algorithm for globally solving sum of convex-concave ratios problem. By introducing some auxiliary variables, the initial problem is converted into an equivalent problem where the objective function is linear. Then the convex relaxation problem of the equivalent problem is established by relaxing auxiliary variables only in the outcome space. By integrating some acceleration and reduction techniques into branch and bound scheme, the presented global optimization algorithm is developed for solving these kind of problems. Convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and MINLPLib are carried out to validate the performance of the proposed algorithm.

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.

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