scholarly journals A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xue-Gang Zhou ◽  
Bing-Yuan Cao
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Duality Theory ◽  
Global Minimum ◽  
Auxiliary Variables ◽  
Multiplicative Programming ◽  
Branch And Bound Search ◽  
Linear Multiplicative Programming ◽  
Linear Programming Problems ◽  
Present Algorithm

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP). We firstly convert the problem (LMP) into an equivalent programming one by introducingpauxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.

scholarly journals A Branch-and-Reduce Approach for Solving Generalized Linear Multiplicative Programming

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Chun-Feng Wang ◽  
San-Yang Liu ◽  
Geng-Zhong Zheng
Keyword(s):  
Linear Programming ◽  
Linearization Method ◽  
Multiplicative Programming ◽  
Nonconvex Problem ◽  
Linear Multiplicative Programming ◽  
Linear Programming Problems ◽  
Overall Performance ◽  
Approximate Linearization

We consider a branch-and-reduce approach for solving generalized linear multiplicative programming. First, a new lower approximate linearization method is proposed; then, by using this linearization method, the initial nonconvex problem is reduced to a sequence of linear programming problems. Some techniques at improving the overall performance of this algorithm are presented. The proposed algorithm is proved to be convergent, and some experiments are provided to show the feasibility and efficiency of this algorithm.

scholarly journals Global Optimization for Generalized Linear Multiplicative Programming Using Convex Relaxation

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yingfeng Zhao ◽  
Ting Zhao
Keyword(s):  
Global Optimization ◽  
Branch And Bound ◽  
Recent Literature ◽  
Convex Relaxation ◽  
Relaxation Method ◽  
Engineering Practice ◽  
Multiplicative Programming ◽  
Wide Range ◽  
Linear Multiplicative Programming ◽  
Convergence And Optimality

Applications of generalized linear multiplicative programming problems (LMP) can be frequently found in various areas of engineering practice and management science. In this paper, we present a simple global optimization algorithm for solving linear multiplicative programming problem (LMP). The algorithm is developed by a fusion of a new convex relaxation method and the branch and bound scheme with some accelerating techniques. Global convergence and optimality of the algorithm are also presented and extensive computational results are reported on a wide range of problems from recent literature and GLOBALLib. Numerical experiments show that the proposed algorithm with a new convex relaxation method is more efficient than usual branch and bound algorithm that used linear relaxation for solving the LMP.

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.

scholarly journals A Global Optimization Algorithm for Generalized Quadratic Programming

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
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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