Primal and dual objective space algorithms for solving linear multiplicative programmes

SHAO Li-zhen; Ehrgott Matthias

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Primal and dual objective space algorithms for solving linear multiplicative programmes

SHAO Li-zhen 1 * #,Ehrgott Matthias 2

1.School of Automation, University of Science and Technology Beijing, Beijing 100083

2.Department of Management Science, Lanacaster University Management School, Bailrigg, Lancaster LA1 4YX

*Correspondence author

#Submitted by

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Funding: Specialized Research Fund for the Doctoral Program of Higher Education (No. 20100006120016) and the National Natural Science Foundation of（No.No. 81000650）

Opened online:22 October 2013

Accepted by: none

Citation: SHAO Li-zhen,Ehrgott Matthias.Primal and dual objective space algorithms for solving linear multiplicative programmes[OL]. [22 October 2013] http://en.paper.edu.cn/en_releasepaper/content/4564266

%It is known that for most of gene expression data for cancer classification, the number of samples is quite small compared to the number of genes. Therefore, feature selection is an essential pre-processing step and a challenging problem to remove the irrelevant or redundant genes before classification.Multiplicative programming (MPP) problems are global optimisation problems known to be NP-hard.In this paper, we focus on linear MPP problems. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear program at each iteration step instead of two as the previous version indicates. We call this algorithm ``primal objective space algorithm". Then based on the dual variant of Benson's algorithm, we propose a ``dual objective space algorithm" for solving linear MPP problems. Again the dual algorithm requires solving only one scalar linear program in each iteration step. We prove the correctness of the dual algorithm and use computational experiments to demonstrate the superiority of the new algorithms compared to existing algorithms from the literature.

Keywords:Operations research; linear multiplicative programming; multi-objective optimisation; approximation algorithm; nondominated point.

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