Abstract
We suggest a new greedy strategy for convex optimization in Banach spaces and prove its convergence rates under a suitable behavior of the modulus of uniform smoothness of the objective function. We show that this algorithm is a generalization of the recently discovered Rescaled Pure Greedy Algorithm for approximation in Hilbert spaces.
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References
Borwein, J., Guiro, A., Hajek, P., Vanderwerff, J.: Uniformly convex functions on Banach Spaces. Proc. Am. Math. Soc. 137, 1081–1091 (2009)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2009)
Dereventsov, A.: Convergence and rate of convergence of approximate greedy-type algorithms. Ph.D. dissertation, University of South Carolina (2017)
Dereventsov, A., Temlyakov, V.: A unified way of analyzing some greedy algorithms. arXiv:1801.06198v1
DeVore, R., Temlyakov, V.: Some remarks on greedy algorithms. Adv. Comput. Math. 5, 173–187 (1996)
DeVore, R., Temlyakov, V.: Convex optimization on Banach spaces. Found. Comput. Math. 16(2), 369–394 (2016)
Nemirovski, A.: Optimization II: Numerical Methods for Nonlinear Continuous Optimization. Lecture Notes. Israel Institute of Technology, Haifa (1999)
Nguyen, H., Petrova, G.: Greedy strategies for convex optimization. Calcolo 54(1), 207–224 (2017)
Petrova, G.: Rescaled pure greedy algorithm for Hilbert and Banach spaces. Appl. Comput. Harm. Anal. 41, 852–866 (2016)
Temlyakov, V.: Greedy expansions in convex optimization. Proc. Steklov Inst. Math. 284(1), 244–262 (2014)
Temlyakov, V.: Greedy approximation in convex optimization. Constr. Approx. 41(2), 269–296 (2015)
Temlyakov, V.: Greedy Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2011)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co., Inc., River Edge (2002)
Zhang, T.: Sequential greedy approximation for certain convex optimization problems. IEEE Trans. Inf. Theory 49(3), 682–691 (2003)
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This research was supported by the NSF Grants DMS 1521067, DMS 1817603, and by the DARPA Grant HR0011619523 through Oak Ridge National Laboratory.
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Gao, Z., Petrova, G. Rescaled Pure Greedy Algorithm for convex optimization. Calcolo 56, 15 (2019). https://doi.org/10.1007/s10092-019-0311-x
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DOI: https://doi.org/10.1007/s10092-019-0311-x