Conditional Gradient Method for Double-Convex Fractional Programming Matrix Problems
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.
KeywordsFractional programming Conditional gradient method Trace ratio problem
Mathematics Subject Classification65F10 15A18 90C32
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.
- 12.Barysbek, J.B., Tungalag, N., Enkhbat, R.: Application of conditional gradient method to a corporate financing problem. J. Pure Appl. Math. 107(1), 179–186 (2016)Google Scholar
- 15.Jaggi, M.: Revisiting Frank–Wolfe: projection-free sparse convex optimization. In: Proceedings of the 30th International Conference on Machine Learning, Atlanta, Georgia, USA, JMLR: W&CP 28 (2013)Google Scholar
- 21.Wang, H., Yan, S., Xu, D., Tang, X., Huang, T.: Trace ratio vs. ratio trace for dimensionality reduction. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR-07 (2007)Google Scholar
- 22.Yan, S., Tang, X.: Trace quotient problems revisited. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) Proceedings of the European Conference on Computer Vision, vol. 2, Number 3952, pp. 232–244. Lecture Notes in Computer Science. Springer, Berlin (2006)Google Scholar
- 25.Han, I., Malioutov, D., Avron, H., Shin, J.: Approximating the spectral sums of large-scale matrices using Chebyshev approximations. arXiv preprint arXiv:1606.00942v2 (2017)