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doi: 10.3934/naco.2021028
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## A dual Bregman proximal gradient method for relatively-strongly convex optimization

 1 School of Science, Hebei University of Technology, Tianjin, China 2 Institute of Mathematics, Hebei University of Technology, Tianjin, China

* Corresponding author: Xin-Wei Liu

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: The research is partially supported by NSFC grant 11671116

We consider a convex composite minimization problem, whose objective is the sum of a relatively-strongly convex function and a closed proper convex function. A dual Bregman proximal gradient method is proposed for solving this problem and is shown that the convergence rate of the primal sequence is $O(\frac{1}{k})$. Moreover, based on the acceleration scheme, we prove that the convergence rate of the primal sequence is $O(\frac{1}{k^{\gamma}})$, where $\gamma\in[1,2]$ is determined by the triangle scaling property of the Bregman distance.

Citation: Jin-Zan Liu, Xin-Wei Liu. A dual Bregman proximal gradient method for relatively-strongly convex optimization. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021028
##### References:
 [1] H. H. Bauschke, J. Bolte and M. Teboulle, A descent lemma beyond Lipschitz gradient continuity: first-order method revisited and applications, Mathematics of Operations Research, 42 (2017), 330-348.  doi: 10.1287/moor.2016.0817.  Google Scholar [2] H. H. Bauschke and J. M. Borwein, Joint and separate convexity of the Bregman distance, in Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (eds. D. Butnariu, Y. Censor, and S. Reich), Elsevier, (2001), 23–26. doi: 10.1016/S1570-579X(01)80004-5.  Google Scholar [3] A. Beck, First-Order Methods in Optimization, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974997.  Google Scholar [4] A. Beck and M. Teboulle, Gradient-based algorithms with applications to signal recovery, Convex Optimization in Signal Processing and Communications, (2009), 42–88. doi: 10.1017/CBO9780511804458.003.  Google Scholar [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Science, 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar [6] A. Beck and M. Teboulle, A fast dual proximal gradient algorithm for convex minimization and applications, Operations Research Letters, 42 (2014), 1-6.  doi: 10.1016/j.orl.2013.10.007.  Google Scholar [7] M. Bertero, P. Boccacci, G. Desider and G. Vicidomini, Image deblurring with Poisson data: from cells to galaxies, Inverse Problems, 25 (2009), 1-26.  doi: 10.1088/0266-5611/25/12/123006.  Google Scholar [8] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.  doi: 10.1016/0041-5553(67)90040-7.  Google Scholar [9] R. Bruck, On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, Journal of Mathematical Analysis and Applications, 61 (1977), 159-164.  doi: 10.1016/0022-247X(77)90152-4.  Google Scholar [10] G. Chen and M. Teboulle, Convergence analysis of a proximal-like minimization algorithm using Bregman functions, SIAM Journal on Optimization, 3 (1993), 538-543.  doi: 10.1137/0803026.  Google Scholar [11] M. Fukushima and H. Milne, A generalized proximal point algorithm for certain nonconvex minimization problems, International Journal of Systems Science, 12 (1981), 989-1000.  doi: 10.1080/00207728108963798.  Google Scholar [12] D. H. Gutman and J. F. Pena, Perturbed Fenchel duality and first-order methods, preprint, arXiv: 1812.10198. Google Scholar [13] F. Hanzely, P. Richtarik and L. Xiao, Accelerated Bregman proximal gradient methods for relatively smooth convex optimization, preprint, arXiv: 1808.03045. doi: 10.1007/s10589-021-00273-8.  Google Scholar [14] J. Kiefer and J. Wolfowitz, Optimal design in regression problems, The Annals of Mathematical Statistics, 30 (1959), 271-294.  doi: 10.1214/aoms/1177706252.  Google Scholar [15] Fast dual proximal gradient algorithms with rate $O(1/k^{1.5} )$ for convex minimization, preprint, arXiv: 1609.09441., Google Scholar [16] E. Klintberg and S. Gros, Approximate inverses in preconditioned fast dual gradient methods for MPC, IFAC-PapersOnLine, 50 (2017), 5901-5906.  doi: 10.1016/j.ifacol.2017.08.1321.  Google Scholar [17] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar [18] H. Lu, M. Freund and Y. Nesterov, Relatively smooth convex optimization by first-order methods and applications, SIAM Journal on Optimization, 28 (2018), 333-354.  doi: 10.1137/16M1099546.  Google Scholar [19] J. Li, G. Chen and Z. Dong, A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints, Computational Optimization and Applications, 64 (2016), 671-697.  doi: 10.1007/s10589-016-9826-0.  Google Scholar [20] D. P. Palomar and Y. C. Eldar, Convex Optimization in Signal Processing and Communications, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511804458.  Google Scholar [21] S. Sra, S. Nowozin and S. J. Wright, Optimization for Machine Learning, MIT Press, Massachusetts, 2011.  doi: 10.7551/mitpress/8996.001.0001.  Google Scholar [22] M. Teboulle, A simplified view of first order methods for optimization, Mathematical Programming, 170 (2018), 67-96.  doi: 10.1007/s10107-018-1284-2.  Google Scholar [23] T. Treskatis, M. A. Moyers-Gonzalez and C. J. Price, An accelerated dual gradient method and applications in viscoplasticity, Journal of Non-Newtonian Fluid Mechanics, 238 (2016), 115-130.  doi: 10.1016/j.jnnfm.2016.09.004.  Google Scholar [24] H. Zhang, Y. H. Dai and L. Guo, Linear convergence of random dual coordinate incremental aggregated gradient methods, preprint, arXiv: 2008.13080. Google Scholar

show all references

##### References:
 [1] H. H. Bauschke, J. Bolte and M. Teboulle, A descent lemma beyond Lipschitz gradient continuity: first-order method revisited and applications, Mathematics of Operations Research, 42 (2017), 330-348.  doi: 10.1287/moor.2016.0817.  Google Scholar [2] H. H. Bauschke and J. M. Borwein, Joint and separate convexity of the Bregman distance, in Inherently Parallel Algorithms in Feasibility and Optimization and their Applications (eds. D. Butnariu, Y. Censor, and S. Reich), Elsevier, (2001), 23–26. doi: 10.1016/S1570-579X(01)80004-5.  Google Scholar [3] A. Beck, First-Order Methods in Optimization, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974997.  Google Scholar [4] A. Beck and M. Teboulle, Gradient-based algorithms with applications to signal recovery, Convex Optimization in Signal Processing and Communications, (2009), 42–88. doi: 10.1017/CBO9780511804458.003.  Google Scholar [5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Science, 2 (2009), 183-202.  doi: 10.1137/080716542.  Google Scholar [6] A. Beck and M. Teboulle, A fast dual proximal gradient algorithm for convex minimization and applications, Operations Research Letters, 42 (2014), 1-6.  doi: 10.1016/j.orl.2013.10.007.  Google Scholar [7] M. Bertero, P. Boccacci, G. Desider and G. Vicidomini, Image deblurring with Poisson data: from cells to galaxies, Inverse Problems, 25 (2009), 1-26.  doi: 10.1088/0266-5611/25/12/123006.  Google Scholar [8] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Computational Mathematics and Mathematical Physics, 7 (1967), 200-217.  doi: 10.1016/0041-5553(67)90040-7.  Google Scholar [9] R. Bruck, On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, Journal of Mathematical Analysis and Applications, 61 (1977), 159-164.  doi: 10.1016/0022-247X(77)90152-4.  Google Scholar [10] G. Chen and M. Teboulle, Convergence analysis of a proximal-like minimization algorithm using Bregman functions, SIAM Journal on Optimization, 3 (1993), 538-543.  doi: 10.1137/0803026.  Google Scholar [11] M. Fukushima and H. Milne, A generalized proximal point algorithm for certain nonconvex minimization problems, International Journal of Systems Science, 12 (1981), 989-1000.  doi: 10.1080/00207728108963798.  Google Scholar [12] D. H. Gutman and J. F. Pena, Perturbed Fenchel duality and first-order methods, preprint, arXiv: 1812.10198. Google Scholar [13] F. Hanzely, P. Richtarik and L. Xiao, Accelerated Bregman proximal gradient methods for relatively smooth convex optimization, preprint, arXiv: 1808.03045. doi: 10.1007/s10589-021-00273-8.  Google Scholar [14] J. Kiefer and J. Wolfowitz, Optimal design in regression problems, The Annals of Mathematical Statistics, 30 (1959), 271-294.  doi: 10.1214/aoms/1177706252.  Google Scholar [15] Fast dual proximal gradient algorithms with rate $O(1/k^{1.5} )$ for convex minimization, preprint, arXiv: 1609.09441., Google Scholar [16] E. Klintberg and S. Gros, Approximate inverses in preconditioned fast dual gradient methods for MPC, IFAC-PapersOnLine, 50 (2017), 5901-5906.  doi: 10.1016/j.ifacol.2017.08.1321.  Google Scholar [17] P. L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.  doi: 10.1137/0716071.  Google Scholar [18] H. Lu, M. Freund and Y. Nesterov, Relatively smooth convex optimization by first-order methods and applications, SIAM Journal on Optimization, 28 (2018), 333-354.  doi: 10.1137/16M1099546.  Google Scholar [19] J. Li, G. Chen and Z. Dong, A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints, Computational Optimization and Applications, 64 (2016), 671-697.  doi: 10.1007/s10589-016-9826-0.  Google Scholar [20] D. P. Palomar and Y. C. Eldar, Convex Optimization in Signal Processing and Communications, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511804458.  Google Scholar [21] S. Sra, S. Nowozin and S. J. Wright, Optimization for Machine Learning, MIT Press, Massachusetts, 2011.  doi: 10.7551/mitpress/8996.001.0001.  Google Scholar [22] M. Teboulle, A simplified view of first order methods for optimization, Mathematical Programming, 170 (2018), 67-96.  doi: 10.1007/s10107-018-1284-2.  Google Scholar [23] T. Treskatis, M. A. Moyers-Gonzalez and C. J. Price, An accelerated dual gradient method and applications in viscoplasticity, Journal of Non-Newtonian Fluid Mechanics, 238 (2016), 115-130.  doi: 10.1016/j.jnnfm.2016.09.004.  Google Scholar [24] H. Zhang, Y. H. Dai and L. Guo, Linear convergence of random dual coordinate incremental aggregated gradient methods, preprint, arXiv: 2008.13080. Google Scholar
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