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2020 | OriginalPaper | Buchkapitel

Convergence Analysis of Penalty Decomposition Algorithm for Cardinality Constrained Convex Optimization in Hilbert Spaces

verfasst von : Michael Pleshakov, Sergei Sidorov, Kirill Spiridonov

Erschienen in: Mathematical Optimization Theory and Operations Research

Verlag: Springer International Publishing

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Abstract

The paper examines an algorithm for finding approximate sparse solutions of convex cardinality constrained optimization problem in Hilbert spaces. The proposed algorithm uses the penalty decomposition (PD) approach and solves sub-problems on each iteration approximately. We examine the convergence of the algorithm to a stationary point satisfying necessary optimality conditions. Unlike other similar works, this paper discusses the properties of PD algorithms in infinite-dimensional (Hilbert) space. The results showed that the convergence property obtained in previous works for cardinality constrained optimization in Euclidean space also holds for infinite-dimensional (Hilbert) space. Moreover, in this paper we established a similar result for convex optimization problems with cardinality constraint with respect to a dictionary (not necessarily the basis).

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Bollhofer, M., Eftekhari, A., Scheidegger, S., Schenk, O.: Large-scale sparse inverse covariance matrix estimation. SIAM J. Sci. Comput. 41(1), A380–A401 (2019). https://doi.org/10.1137/17M1147615MathSciNetCrossRefMATH

Bubeck, S.: Convex optimization: algorithms and complexity. Found. Trends Mach. Learn. 8(3–4), 231–358 (2015)CrossRef

Chen, Z., Huang, C., Lin, S.: A new sparse representation framework for compressed sensing MRI. Knowl.-Based Syst. 188, 104969 (2020). https://doi.org/10.1016/j.knosys.2019.104969. http://www.sciencedirect.com/science/article/pii/S0950705119303983CrossRef

Dempster, A.P.: Covariance selection. Biometrics 28(1), 157–175 (1972). https://doi.org/10.2307/2528966MathSciNetCrossRef

Deng, Q., et al.: Compressed sensing for image reconstruction via back-off and rectification of greedy algorithm. Sig. Process. 157, 280–287 (2019). https://doi.org/10.1016/j.sigpro.2018.12.007. http://www.sciencedirect.com/science/article/pii/S0165168418303980CrossRef

Dereventsov, A., Temlyakov, V.N.: Biorthogonal greedy algorithms in convex optimization. CoRR abs/2001.05530 (2020). https://arxiv.org/abs/2001.05530

Dong, Z., Zhu, W.: An improvement of the penalty decomposition method for sparse approximation. Sig. Process. 113, 52–60 (2015). https://doi.org/10.1016/j.sigpro.2015.01.012. http://www.sciencedirect.com/science/article/pii/S0165168415000353CrossRef

Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theor. 52(4), 1289–1306 (2006). https://doi.org/10.1109/TIT.2006.871582MathSciNetCrossRefMATH

Dou, H.X., Huang, T.Z., Deng, L.J., Zhao, X.L., Huang, J.: Directional \(l_0\) sparse modeling for image stripe noise removal. Remote Sens. 10(3) (2018). https://doi.org/10.3390/rs10030361. https://www.mdpi.com/2072-4292/10/3/361

10.

Fan, J., Lv, J., Qi, L.: Sparse high-dimensional models in economics. Ann. Rev. Econ. 3(1), 291–317 (2011). https://doi.org/10.1146/annurev-economics-061109-080451CrossRef

11.

Figueiredo, M.A.T., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Sig. Process. 1(4), 586–597 (2007). https://doi.org/10.1109/JSTSP.2007.910281CrossRef

12.

Gajare, S., Sonawani, S.: Improved logistic regression approach in feature selection for EHR. In: Abraham, A., Cherukuri, A.K., Melin, P., Gandhi, N. (eds.) ISDA 2018 2018. AISC, vol. 940, pp. 325–334. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-16657-1_30CrossRef

13.

Gudkov, A.A., Mironov, S.V., Sidorov, S.P., Tyshkevich, S.V.: A dual active set algorithm for optimal sparse convex regression. Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki (J. Samara State Tech. Univ. Ser. Phys. Math. Sci.) 23(1), 113–130 (2019). https://doi.org/10.14498/vsgtu1673CrossRefMATH

14.

Holmberg, K.: Creative modeling: variable and constraint duplicationin primal - dual decomposition methods. Ann. Oper. Res. 82, 355–390 (1998). https://doi.org/10.1023/A:1018927123151MathSciNetCrossRefMATH

15.

Kampa, K., Mehta, S., Chou, C.A., Chaovalitwongse, W.A., Grabowski, T.J.: Sparse optimization in feature selection: application in neuroimaging. J. Global Optim. 59(2), 439–457 (2014). https://doi.org/10.1007/s10898-013-0134-2MathSciNetCrossRefMATH

16.

Lu, Z., Li, X.: Sparse recovery via partial regularization: models, theory, and algorithms. Math. Oper. Res. 43(4), 1290–1316 (2018). https://doi.org/10.1287/moor.2017.0905MathSciNetCrossRefMATH

17.

Lu, Z., Zhang, Y.: Sparse approximation via penalty decomposition methods. SIAM J. Optim. 23(4), 2448–2478 (2013). https://doi.org/10.1137/100808071MathSciNetCrossRefMATH

18.

Luo, X., Chang, X., Ban, X.: Regression and classification using extreme learning machine based on l1-norm and l2-norm. Neurocomputing 174, 179–186 (2016). https://doi.org/10.1016/j.neucom.2015.03.112. http://www.sciencedirect.com/science/article/pii/S092523121501139XCrossRef

19.

Pan, L.L., Xiu, N.H., Fan, J.: Optimality conditions for sparse nonlinear programming. Sci. China Math. 60(5), 759–776 (2017). https://doi.org/10.1007/s11425-016-9010-xMathSciNetCrossRefMATH

20.

Pun, C.S., Wong, H.Y.: A linear programming model for selection of sparse high-dimensional multiperiod portfolios. Eur. J. Oper. Res. 273(2), 754–771 (2019). https://doi.org/10.1016/j.ejor.2018.08.025. http://www.sciencedirect.com/science/article/pii/S0377221718307203MathSciNetCrossRefMATH

21.

Sidorov, S.P., Faizliev, A.R., Khomchenko, A.A.: Algorithms for \(l_1\)-norm minimisation of index tracking error and their performance. Int. J. Math. Oper. Res. 11(4), 497–519 (2017). https://ideas.repec.org/a/ids/ijmore/v11y2017i4p497-519.htmlMathSciNetCrossRef

22.

Temlyakov, V.N.: Greedy approximation in convex optimization. Constr. Approx. 41(2), 269–296 (2015). https://doi.org/10.1007/s00365-014-9272-0MathSciNetCrossRefMATH

23.

Teng, Y., Yang, L., Yu, B., Song, X.: A penalty palm method for sparse portfolio selection problems. Optim. Methods Softw. 32(1), 126–147 (2017). https://doi.org/10.1080/10556788.2016.1204299MathSciNetCrossRefMATH

24.

Wipf, D.P., Rao, B.D.: Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process. 52(8), 2153–2164 (2004). https://doi.org/10.1109/TSP.2004.831016MathSciNetCrossRefMATH

25.

Xu, F., Deng, R.: Fast algorithms for sparse inverse covariance estimation. Int. J. Comput. Math. 96(8), 1668–1686 (2019). https://doi.org/10.1080/00207160.2018.1506108MathSciNetCrossRef

26.

Zhu, W., Dong, Z., Yu, Y., Chen, J.: Lagrange dual method for sparsity constrained optimization. IEEE Access 6, 28404–28416 (2018). https://doi.org/10.1109/ACCESS.2018.2836925CrossRef

Titel: Convergence Analysis of Penalty Decomposition Algorithm for Cardinality Constrained Convex Optimization in Hilbert Spaces
verfasst von: Michael Pleshakov
Sergei Sidorov
Kirill Spiridonov
Verlag: Springer International Publishing
Buch: Mathematical Optimization Theory and Operations Research
Print ISBN: 978-3-030-49987-7

Electronic ISBN: 978-3-030-49988-4

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-49988-4_10

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