Abstract
We consider the problem of optimizing the ratio of two convex functions over a closed and convex set in the space of matrices. This problem appears in several applications and can be classified as a double-convex fractional programming problem. In general, the objective function is nonconvex but, nevertheless, the problem has some special features. Taking advantage of these features, a conditional gradient method is proposed and analyzed, which is suitable for matrix problems. The proposed scheme is applied to two different specific problems, including the well-known trace ratio optimization problem which arises in many engineering and data processing applications. Preliminary numerical experiments are presented to illustrate the properties of the proposed scheme.
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Acknowledgements
We are grateful to the referees for their helpful comments and suggestions. The fifth author would like to thank Abderrahman Bouhamidi, Khalide Jbilou, and Hassane Sadok for their hospitality during a one-month visit to the Laboratory LMPA at Université du Littoral Côte d’Opale, Calais, France, in June 2017.
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Dedicated to the memory of Mohammed Bellalij.
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Bouhamidi, A., Bellalij, M., Enkhbat, R. et al. Conditional Gradient Method for Double-Convex Fractional Programming Matrix Problems. J Optim Theory Appl 176, 163–177 (2018). https://doi.org/10.1007/s10957-017-1203-3
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DOI: https://doi.org/10.1007/s10957-017-1203-3