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

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.

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

Journal of Applied Mathematics ◽

10.1155/2013/984168 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 3

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.

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A new linearization method for generalized linear multiplicative programming

Computers & Operations Research ◽

10.1016/j.cor.2010.10.016 ◽

2011 ◽

Vol 38 (7) ◽

pp. 1008-1013 ◽

Cited By ~ 10

Author(s):

Chun-Feng Wang ◽

San-Yang Liu

Keyword(s):

Linearization Method ◽

Multiplicative Programming ◽

Linear Multiplicative Programming

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

Scientific Programming ◽

10.1155/2016/3204368 ◽

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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A New Global Optimization Algorithm for Solving a Class of Nonconvex Programming Problems

Journal of Applied Mathematics ◽

10.1155/2014/697321 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Xue-Gang Zhou ◽

Bing-Yuan Cao

Keyword(s):

Linear Programming ◽

Nonconvex Programming ◽

Optimal Solution ◽

Linearization Method ◽

Global Optimal Solution ◽

Natural Logarithm ◽

Lower Bounding ◽

Linear Programming Problems ◽

Global Optimal ◽

Logarithm Function

A new two-part parametric linearization technique is proposed globally to a class of nonconvex programming problems (NPP). Firstly, a two-part parametric linearization method is adopted to construct the underestimator of objective and constraint functions, by utilizing a transformation and a parametric linear upper bounding function (LUBF) and a linear lower bounding function (LLBF) of a natural logarithm function and an exponential function witheas the base, respectively. Then, a sequence of relaxation lower linear programming problems, which are embedded in a branch-and-bound algorithm, are derived in an initial nonconvex programming problem. The proposed algorithm is converged to global optimal solution by means of a subsequent solution to a series of linear programming problems. Finally, some examples are given to illustrate the feasibility of the presented algorithm.

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