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A Partially Inertial Customized Douglas–Rachford Splitting Method for a Class of Structured Optimization Problems

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Abstract

In this paper, we are concerned with a class of structured optimization problems frequently arising from image processing and statistical learning, where the objective function is the sum of a quadratic term and a nonsmooth part, and the constraint set consists of a linear equality constraint and two simple convex sets in the sense that projections onto simple sets are easy to compute. To fully exploit the quadratic and separable structure of the problem under consideration, we accordingly propose a partially inertial Douglas–Rachford splitting method. It is noteworthy that our algorithm enjoys easy subproblems for the case where the underlying two simple convex sets are not the whole spaces. Theoretically, we establish the global convergence of the proposed algorithm under some mild conditions. A series of computational results on the constrained Lasso and constrained total-variation (TV) based image restoration demonstrate that our proposed method is competitive with some state-of-the-art first-order solvers.

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References

  1. Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projection-proximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773–782 (2004)

    MathSciNet  Google Scholar 

  2. Alves, M.M., Eckstein, J., Geremia, M., Melo, J.G.: Relative-error inertial-relaxed inexact versions of Douglas–Rachford and ADMM splitting algorithms. Comput. Optim. Appl. 75(2), 389–422 (2020)

    MathSciNet  Google Scholar 

  3. Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward-backward algorithm for convex minimization. SIAM J. Optim. 24(1), 232–256 (2014)

    MathSciNet  Google Scholar 

  4. Barzilai, J., Borwein, J.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    MathSciNet  Google Scholar 

  5. Bennett, J., Lanning, S.: The Netflix prize. In Proceedings of KDD Cup and Workshop, vol. 2007, p. 35. New York (2007)

  6. Bertsekas, D., Tsitsiklis, J.: Parallel and Distributed Computation. Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)

    Google Scholar 

  7. Bot, R.I., Csetnek, E.R.: An inertial alternating direction method of multipliers. Minimax Theory Appl. 1, 29–49 (2016)

    MathSciNet  Google Scholar 

  8. Boţ, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256, 472–487 (2015)

    MathSciNet  Google Scholar 

  9. Boţ, R.I., Csetnek, E.R., László, S.C.: An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions. EURO J. Comput. Optim. 4(1), 3–25 (2016)

    MathSciNet  Google Scholar 

  10. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3, 1–122 (2010)

    Google Scholar 

  11. Briceno-Arias, L.M., Combettes, P.L.: A monotone+ skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230–1250 (2011)

    MathSciNet  Google Scholar 

  12. Cai, X., Guo, K., Jiang, F., Wang, K., Wu, Z., Han, D.: The developments of proximal point algorithms. J. Oper. Res. Soc. China 10(2), 197–239 (2022)

    MathSciNet  Google Scholar 

  13. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)

    MathSciNet  Google Scholar 

  14. Chambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal-dual algorithm. Math. Program. 159(1–2), 253–287 (2016)

    MathSciNet  Google Scholar 

  15. Chen, C., Chan, R.H., Ma, S., Yang, J.: Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8(4), 2239–2267 (2015)

    MathSciNet  Google Scholar 

  16. Chen, C., He, B., Yuan, X.: Matrix completion via alternating direction method. IMA J. Numer. Anal. 32, 227–245 (2012)

    MathSciNet  Google Scholar 

  17. Chen, Y., Nasrabadi, N.M., Tran, T.D.: Hyperspectral image classification using dictionary-based sparse representation. IEEE Trans. Geosci. Remote sens. 49(10), 3973–3985 (2011)

    Google Scholar 

  18. Cheng, H., Liu, Z., Yang, L., Chen, X.: Sparse representation and learning in visual recognition: theory and applications. Signal Process. 93(6), 1408–1425 (2013)

    Google Scholar 

  19. Dai, Y., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21–47 (2005)

    MathSciNet  Google Scholar 

  20. Deng, Z., Liu, S.: Inertial generalized proximal Peaceman–Rachford splitting method for separable convex programming. Calcolo 58(1), 1–30 (2021)

    MathSciNet  Google Scholar 

  21. Dou, M., Li, H., Liu, X.: An inertial proximal Peaceman–Rachford splitting method. Sci. Sin. Math. 47(2), 333–348 (2016)

    Google Scholar 

  22. Eaves, B.: On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971)

    MathSciNet  Google Scholar 

  23. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)

    Google Scholar 

  24. Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximations. Comput. Math. Appl. 2, 16–40 (1976)

    Google Scholar 

  25. Gaines, B.R., Kim, J., Zhou, H.: Algorithms for fitting the constrained Lasso. J. Comput. Graph. Stat. 27(4), 861–871 (2018)

    MathSciNet  Google Scholar 

  26. Glowinski, R., Marrocco, A.: Approximation par éléments finis d’ordre un et résolution par pénalisation-dualité d’une classe de problèmes non linéaires. R.A.I.R.O. R2, 41–76 (1975)

    Google Scholar 

  27. Goldberg, K., Roeder, T., Gupta, D., Perkins, C.: Eigentaste: a constant time collaborative filtering algorithm. Inf. Retr. J. 4(2), 133–151 (2001)

    Google Scholar 

  28. Han, D.: A survey on some recent developments of alternating direction method of multipliers. J. Oper. Res. Soc. China 10, 1–52 (2022)

    MathSciNet  Google Scholar 

  29. Han, D., He, H., Yang, H., Yuan, X.: A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167–200 (2014)

    MathSciNet  Google Scholar 

  30. Han, D., Xu, W., Yang, H.: An operator splitting method for variational inequalities with partially unknown mappings. Numer. Math. 111, 207–237 (2008)

    MathSciNet  Google Scholar 

  31. Hansen, P.C., Nagy, J.G., O’leary, D.P.: Deblurring images: matrices, spectra, and filtering. SIAM (2006)

  32. He, B., Liao, L., Wang, S.: Self-adaptive operator splitting methods for monotone variational inequalities. Numer. Math. 94, 715–737 (2003)

    MathSciNet  Google Scholar 

  33. He, H., Cai, X., Han, D.: A fast splitting method tailored for Dantzig selectors. Comput. Optim. Appl. 62, 347–372 (2015)

    MathSciNet  Google Scholar 

  34. He, H., Xu, H.K.: Splitting methods for split feasibility problems with application to dantzig selectors. Inverse Probl. 33(5), 055003 (2017)

    MathSciNet  Google Scholar 

  35. Hu, L., Zhang, W., Cai, X., Han, D.: A parallel operator splitting algorithm for solving constrained total-variation retinex. Inverse Probl. Imaging 14(6), 1135 (2020)

  36. Iordache, M.D., Bioucas-Dias, J.M., Plaza, A.: Sparse unmixing of hyperspectral data. IEEE Trans. Geosci. Remote. Sens. 49(6), 2014–2039 (2011)

    Google Scholar 

  37. Liu, Y.J., Sun, D., Toh, K.C.: An implementable proximal point algorithmic framework for nuclear norm minimization. Math. Program. 133(1), 399–436 (2012)

    MathSciNet  Google Scholar 

  38. Lorenz, D.A., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51(2), 311–325 (2015)

    MathSciNet  Google Scholar 

  39. Morini, S., Porcelli, M., Chan, R.: A reduced Newton method for constrained linear least squares problems. J. Comput. Appl. Math. 233, 2200–2212 (2010)

    MathSciNet  Google Scholar 

  40. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)

    Google Scholar 

  41. Rockafellar, R.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209–216 (1970)

    MathSciNet  Google Scholar 

  42. Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 227–238 (1992)

    MathSciNet  Google Scholar 

  43. Tibshirani, R.: Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1996)

    MathSciNet  Google Scholar 

  44. Toh, K., Yun, S.: An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. Pac. J. Optim. 6, 615–640 (2010)

    MathSciNet  Google Scholar 

  45. Varga, R.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1966)

    Google Scholar 

  46. Wang, X., Yuan, X.: The linearized alternating direction method of multipliers for Dantzig selector. SIAM J. Sci. Comput. 34, A2792–A2811 (2012)

    MathSciNet  Google Scholar 

  47. Wu, L., Yang, Y., Liu, H.: Nonnegative-lasso and application in index tracking. Comput. Stat. Data Anal. 70, 116–126 (2014)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous referees for their valuable comments, which helped us improve the quality of this paper. H. He was supported in part by the National Natural Science Foundation of China (NSFC) No. 12371303 and Ningbo Natural Science Foundation (No. 2023J014). D. Han was supported by NSFC Nos. 12126603 and 12131004 and Ministry of Science and Technology of China (Grant No. 2021YFA1003600).

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  1. LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Yunfei Qu & Deren Han

  2. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

    Hongjin He

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  1. Yunfei Qu
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Qu, Y., He, H. & Han, D. A Partially Inertial Customized Douglas–Rachford Splitting Method for a Class of Structured Optimization Problems. J Sci Comput 98, 9 (2024). https://doi.org/10.1007/s10915-023-02397-x

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