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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.

In this paper, the following generalized linear multiplicative programming is considered:

Since a large number of practical applications in various fields can be put into problem (

It is well known that the product of affine functions need not be (quasi) convex, thus the problem can have multiple locally optimal solutions, many of which fail to be globally optimal, that is, problem (

In the last decade, many solution algorithms have been proposed for globally solving special forms of (

The aim of this paper is to provide a new branch-and-reduce algorithm for globally solving problem (

The main features of this algorithm: (1) the problem investigated in this paper has a more general form than those in [

This paper is organized as follows. In Section

Without loss of generality, assume that, for

By using the property of logarithmic function, the equivalent problem (

Thus, for solving problem (

Suppose that

Consider the function

First, we consider the function

By Theorem 1 in [

Second, we consider function

Finally, from (

For all

Let

First, consider

Second, consider

Taken together above, it implies that

From Theorem

Based on the above discussion, the linear relaxation programming problem (

Obviously, the feasible region for the problem (

In this section, we pay our attention on how to form the new reducing technique for eliminate the region in which the global minimum of (

Assume that

For any subrectangle

First, we show that for all

For all

In this section, based on the former results, we present a branch-and-reduce algorithm to solve the problem (

Firstly, based on Section

The second fundamental process iteratively subdivides the rectangle

Let

Let

Let

By this branching rule, the rectangle

The third process is to update the upper and lower bounds of the optimal value of (

The basic steps of the proposed algorithm are summarized as follows. In this algorithm, let

Let the set all active node

Solve the problem (

Select the midpoint

Using the branching rule to partition

If

Set

In this subsection, we give the global convergence properties of the above algorithm.

The above algorithm either terminates finitely with a globally

When the algorithm is finite, by the algorithm, it terminates at some step

When the algorithm is infinite, by [

A bounding operation is called consistent if at every step any unfathomed partition can be further refined, and if any infinitely decreasing sequence of successively refined partition elements satisfies

Since the employed subdivision process is rectangle bisection, the process is exhaustive. Consequently, from Theorem

A selection operation is called bound improving if at least one partition element where the actual lower bound is attained is selected for further partition after a finite number of refinements. Clearly, the employed selection operation is bound improving because the partition element where the actual lower bound is attained is selected for further partition in the immediately following iteration.

From the above discussion, and Theorem IV.3 in [

In this section, some numerical experiments are reported to verify the performance of the proposed algorithm. The algorithm is coded in Matlab 7.1. The simplex method is applied to solve the linear relaxation programming problems. The test problems are implemented on a Pentium IV (3.06 GHZ) microcomputer, and the convergence tolerance is set at

The results of problems (

Computational results of test problems (

Example | Methods | Optimal solution | Optimal value | Iter | Time |
---|---|---|---|---|---|

1 | [ | (1.0, 1.0) | 997.661265160 | 49 | 0 |

[ | (1.0, 1.0) | 997.6613 | 5 | 0.0984 | |

ours | (1.0, 1.0) | 997.6613 | 1 | 0.0160 | |

2 | [ | (1.0, 2.0, 1.0) | 3.7127 | 10 | 0.2717 |

ours | (1.0, 2.0, 1.0) | 3.7127 | 1 | 0.0150 | |

3 | [ | (1.0, 1.0, 1.0) | 60.0 | 64 | 0 |

[ | (1.0, 1.0, 1.0) | 60.0 | 1 | 0.0126 | |

ours | (1.0, 1.0, 1.0) | 60.0 | 1 | 0.0148 | |

4 | [ | (0.0, 0.0) | 0.533333333 | 3 | 0 |

[ | (0.0, 0.0) | 0.533333 | 16 | 0.05 | |

ours | (0.0, 0.0) | 0.5333 | 2 | 0.0221 | |

5 | [ | (1.0, 1.0) | 275.074284 | 1 | 0 |

[ | (1.0, 1.0) | 275.0743 | 1 | 0.0105 | |

ours | (1.0, 1.0) | 275.0743 | 1 | 0.0102 |

The results in Table

The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have greatly improved the earlier version of this paper. This paper is supported by the National Natural Science Foundation of China (60974082) and the Fundamental Research Funds for the Central Universities (K50510700004).