Convergent Nested Alternating Minimization Algorithms for Nonconvex Optimization Problems

Published Online:https://doi.org/10.1287/moor.2022.1256

References

  • [1] Abatzoglou TJ, Mendel JM, Harada GA (1991) The constrained total least squares technique and its applications to harmonic superresolution. IEEE Trans. Signal Processing 39(5):1070–1087.CrossrefGoogle Scholar
  • [2] Attouch H, Bolte J (2009) On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Programming 116(1-2):5–16.CrossrefGoogle Scholar
  • [3] Attouch H, Bolte J, Svaiter BF (2013) Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods. Math. Programming 137(1-2):91–129.CrossrefGoogle Scholar
  • [4] Attouch H, Bolte J, Redont P, Soubeyran A (2010) Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality. Math. Oper. Res. 35(2):438–457.LinkGoogle Scholar
  • [5] Beck A (2017) First-Order Methods in Optimization (SIAM, Philadelphia).CrossrefGoogle Scholar
  • [6] Beck A, Ben-Tal A (2006) On the solution of the Tikhonov regularization of the total least squares problem. SIAM J. Optim. 17(1):98–118.CrossrefGoogle Scholar
  • [7] Beck A, Teboulle M (2009) Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Processing 18(11):2419–2434.CrossrefGoogle Scholar
  •  [8] Beck A, Sabach S, Teboulle M (2016) An alternating semiproximal method for nonconvex regularized structured total least squares problems. SIAM J. Matrix Anal. Appl. 37(3):1129–1150.CrossrefGoogle Scholar
  •  [9] Bolte J, Daniilidis A, Lewis A (2007) The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4):1205–1223.CrossrefGoogle Scholar
  • [10] Bolte J, Sabach S, Teboulle M (2014) Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Programming 146(1-2):459–494.CrossrefGoogle Scholar
  • [11] Bolte J, Sabach S, Teboulle M, Vaisbourd Y (2018) First order methods beyond convexity and lipschitz gradient continuity with applications to quadratic inverse problems. SIAM J. Optim. 28(3):2131–2151.CrossrefGoogle Scholar
  • [12] Bonettini S, Prato M, Rebegoldi S (2018) A block coordinate variable metric linesearch based proximal gradient method. Comput. Optim. Appl. 71(1):5–52.CrossrefGoogle Scholar
  • [13] Chouzenoux E, Pesquet JC, Repetti A (2016) A block coordinate variable metric forward–backward algorithm. J. Global Optim. 66(3):457–485.CrossrefGoogle Scholar
  • [14] Frankel P, Garrigos G, Peypouquet J (2015) Splitting methods with variable metric for Kurdyka–łojasiewicz functions and general convergence rates. J. Optim. Theory Appl. 165(3):874–900.CrossrefGoogle Scholar
  • [15] Golub GH, Van Loan CF (1980) An analysis of the total least squares problem. SIAM J. Numer. Anal. 17(6):883–893.CrossrefGoogle Scholar
  • [16] Golub GH, Hansen PC, O’Leary DP (1999) Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21(1):185–194.CrossrefGoogle Scholar
  • [17] Groenen PJ, van de Velden M (2016) Multidimensional scaling by majorization: A review. J. Statist. Software 73(8):1–26.CrossrefGoogle Scholar
  • [18] Gutjahr WJ, Pichler A (2016) Stochastic multi-objective optimization: A survey on non-scalarizing methods. Ann. Oper. Res. 236(2):475–499.CrossrefGoogle Scholar
  • [19] Hansen PC, Nagy JG, O’Leary DP (2006) Deblurring Images: Matrices, Spectra, and Filtering (SIAM, Philadelphia).CrossrefGoogle Scholar
  • [20] Hesse R, Luke DR, Sabach S, Tam MK (2015) Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging. SIAM J. Imaging Sci. 8(1):426–457.CrossrefGoogle Scholar
  • [21] Jain P, Kar P (2017) Non-convex optimization for machine learning. Foundations Trends Machine Learning 10(3–4):142–336.CrossrefGoogle Scholar
  • [22] Kurdyka K (1998) On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48(3):769–783.CrossrefGoogle Scholar
  • [23] Łojasiewicz S (1963) Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles (Paris, 1962) (Éditions du Centre National de la Recherche Scientifique, Paris), 87–89.Google Scholar
  • [24] Mohammadi FG, Amini MH, Arabnia HR (2020) Evolutionary computation, optimization, and learning algorithms for data science. Amini MH, ed. Optimization, Learning, and Control for Interdependent Complex Networks (Springer, Cham, Switzerland), 37–65.CrossrefGoogle Scholar
  • [25] Mordukhovich BS (2006) Variational Analysis and Generalized Differentiation I: Basic Theory (Springer, Berlin).CrossrefGoogle Scholar
  • [26] Moreau JJ (1965) Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. France 93:273–299.CrossrefGoogle Scholar
  • [27] Nesterov YE (1983) A method for solving the convex programming problem with convergence rate O(1/k2). Dokl. Akad. Nauk SSSR 269(3):543–547.Google Scholar
  • [28] Ochs P (2019) Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano. SIAM J. Optim. 29(1):541–570.CrossrefGoogle Scholar
  • [29] Ochs P, Chen Y, Brox T, Pock T (2014) iPiano: Inertial proximal algorithm for nonconvex optimization. SIAM J. Imaging Sci. 7(2):1388–1419.CrossrefGoogle Scholar
  • [30] Piovesan N, Erseghe T (2018) Cooperative localization in WSNs: A hybrid convex/nonconvex solution. IEEE Trans. Signal Inform. Processing Networks 4(1):162–172.CrossrefGoogle Scholar
  • [31] Pock T, Sabach S (2016) Inertial proximal alternating linearized minimization (iPALM) for nonconvex and nonsmooth problems. SIAM J. Imaging Sci. 9(4):1756–1787.CrossrefGoogle Scholar
  • [32] Rockafellar RT, Wets RJB (2009) Variational Analysis (Springer, Berlin).Google Scholar
  • [33] Sabach S, Teboulle M, Voldman S (2018) A smoothing alternating minimization-based algorithm for clustering with sum-min of Euclidean norms. Pure Appl. Funct. Anal. 3:653–679.Google Scholar
  • [34] Wen F, Chu L, Liu P, Qiu RC (2018) A survey on nonconvex regularization-based sparse and low-rank recovery in signal processing, statistics, and machine learning. IEEE Access 6:69883–69906.CrossrefGoogle Scholar
  • [35] Xu Y, Yin W (2017) A globally convergent algorithm for nonconvex optimization based on block coordinate update. J. Sci. Comput. 72(2):700–734.CrossrefGoogle Scholar
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