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

Xue-Gang Zhou,Bing-Yuan Cao

doi:10.1155/2013/984168

Abstract

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.

Highlights

Since problem (QC) is a special case of the linear multiplicative programming (LMP), it follows that all of the numerous applications of linear zero-one programming are embodied among the applications of the LMP
A simplicial branch and bound dualitybounds algorithm is presented to the problem (LMP) by solving a sequence of linear programming one over partitioned subsets
We show how to determine an upper bound of the global optimal value for (LMP(S))

Summary

Introduction

Linear multiplicative programming problems are given by: p (LMP) V = min h (x) = ∑ (ciTx + ci0) (diTx + di0) , i=1. Since problem (QC) is a special case of the LMP, it follows that all of the numerous applications of linear zero-one programming are embodied among the applications of the LMP. A simplicial branch and bound dualitybounds algorithm is presented to the problem (LMP) by solving a sequence of linear programming one over partitioned subsets. The branch and bound search takes place in a space of only dimension p, where p is the number of terms in the objective function of problem (LMP), rather than in the decision space Rn. Secondly, the subproblems that must be solved during the search are all linear programming problems that can be solved very efficiently, for example, by a simplex method.

Preliminaries

Global Optimizing Algorithm

Convergence of the Algorithm

Numerical Examples

Conclusion