Solving the sum-of-ratios problem by a stochastic search algorithm

• Published in: Journal of global optimization Vol. 42; no. 1; pp. 91 - 109
• Main Authors: Wu, Wei-Ying, Sheu, Ruey-Lin, Birbil, Ş. İlker
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.09.2008; Springer Nature B.V
• Subjects: Algorithms; Bond portfolios; Computer Science; Electromagnetism; Linear programming; Mathematics; Mathematics and Statistics; Methods; Operations Research/Decision Theory; Optimization; Random variables; Ratios; Real Functions; Studies; Stochastic search method; Branch-and-bound method; Sum-of-ratios problems; Dinkelbach-type method; Min-max problems
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-008-9285-y

In spite of the recent progress in fractional programming, the sum-of-ratios problem remains untoward. Freund and Jarre proved that this is an NP-complete problem. Most methods overcome the difficulty using the deterministic type of algorithms, particularly, the branch-and-bound method. In this paper, we propose a new approach by applying the stochastic search algorithm introduced by Birbil, Fang and Sheu to a transformed image space. The algorithm then computes and moves sample particles in the q  − 1 dimensional image space according to randomly controlled interacting electromagnetic forces. Numerical experiments on problems up to sum of eight linear ratios with a thousand variables are reported. The results also show that solving the sum-of-ratios problem in the image space as proposed is, in general, preferable to solving it directly in the primal domain.