Generalized Monotonicity and the Proximal Point Algorithm

Published Online:https://doi.org/10.1287/moor.2025.0863

References

  • [1] Ariza-Ruiz D, Leuştean L, López-Acedo G (2014) Firmly nonexpansive mappings in classes of geodesic spaces. Trans. Amer. Math. Soc. 366(8):4299–4322.CrossrefGoogle Scholar
  • [2] Artacho FA, Dontchev A, Geoffroy M (2007) Convergence of the proximal point method for metrically regular mappings. ESAIM Proc. 17:1–8.CrossrefGoogle Scholar
  • [3] Baillon JB, Bruck RE, Reich S (1978) On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houston J. Math. 4(1):1–9.Google Scholar
  • [4] Bauschke HH, Combettes PL (2017) Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd ed. (Springer, Cham, Switzerland).CrossrefGoogle Scholar
  • [5] Bërdëllima A, Lauster F, Luke DR (2022) α-Firmly nonexpansive operators on metric spaces. J. Fixed Point Theory Appl. 24:14.CrossrefGoogle Scholar
  • [6] Bruck RE, Reich S (1977) Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3(4):459–470.Google Scholar
  • [7] Burachik RS, Iusem AN (2008) Set-Valued Mappings and Enlargements of Monotone Operators, Springer Optimization and Its Applications, vol. 8 (Springer, New York).Google Scholar
  • [8] Cohen E, Luke DR, Pinta T, Sabach S, Teboulle M (2024) A semi-Bregman proximal alternating method for a class of nonconvex problems: Local and global convergence analysis. J. Global Optim. 89(1):33–55.CrossrefGoogle Scholar
  • [9] Combettes PL, Pennanen T (2004) Proximal methods for cohypomonotone operators. SIAM J. Control Optim. 43(2):731–742.CrossrefGoogle Scholar
  • [10] Daniilidis A, Georgiev P (2004) Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291(1):292–301.CrossrefGoogle Scholar
  • [11] Dontchev AL, Rockafellar RT (2014) Implicit Functions and Solution Mappings, 2nd ed. (Springer-Verlag, Dordrecht, Netherlands).CrossrefGoogle Scholar
  • [12] Edelstein M (1966) A remark on a theorem of M. A. Krasnoselski. Amer. Math. Monthly 73(5):509–510.CrossrefGoogle Scholar
  • [13] Esser E, Lou Y, Xin J (2013) A method for finding structured sparse solutions to nonnegative least squares problems with applications. SIAM J. Imaging Sci. 6(4):2010–2046.CrossrefGoogle Scholar
  • [14] Hermer N, Luke DR, Sturm A (2019) Random function iterations for consistent stochastic feasibility. Numer. Functional Anal. Optim. 40(4):386–420.CrossrefGoogle Scholar
  • [15] Hermer N, Luke DR, Sturm A (2023) Rates of convergence for chains of expansive Markov operators. Trans. Math. Its Appl. 7(1):tnad001.Google Scholar
  • [16] Ioffe AD (2011) Regularity on a fixed set. SIAM J. Optim. 21(4):1345–1370.CrossrefGoogle Scholar
  • [17] Ioffe AD (2013) Nonlinear regularity models. Math. Programming 139(1–2):223–242.CrossrefGoogle Scholar
  • [18] Iusem AN, Pennanen T, Svaiter BF (2003) Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13(4):1080–1097.CrossrefGoogle Scholar
  • [19] Krasnoselski MA (1955) Two remarks on the method of successive approximations. Uspekhi Matematicheskikh Nauk 63(1):123–127.Google Scholar
  • [20] Kruger AY, Luke DR, Thao NH (2018) Set regularities and feasibility problems. Math. Programming 168(1–2):279–311.CrossrefGoogle Scholar
  • [21] Lasry JM, Lions PL (1986) A remark on regularization in Hilbert spaces. Israel J. Math. 55:257–266.CrossrefGoogle Scholar
  • [22] Lauster F, Luke DR (2021) Convergence of proximal splitting algorithms in CAT(κ) spaces and beyond. Fixed Point Theory Algorithms Sci. Engrg. 2021:13.CrossrefGoogle Scholar
  • [23] Leventhal D (2009) Metric subregularity and the proximal point method. J. Math. Anal. Appl. 360(2):681–688.CrossrefGoogle Scholar
  • [24] Luke DR, Tam MK (2023) Generalized monotonicity and the proximal point algorithm. Proc. 35th RAMP Sympos. (Research Association of Mathematical Programming, Tokyo), 39–48. Google Scholar
  • [25] Luke DR, Teboulle M, Thao NH (2020) Necessary conditions for linear convergence of iterated expansive, set-valued mappings. Math. Programming 180:1–31.CrossrefGoogle Scholar
  • [26] Luke DR, Thao NH, Tam MK (2018) Quantitative convergence analysis of iterated expansive, set-valued mappings. Math. Oper. Res. 43(4):1143–1176.LinkGoogle Scholar
  • [27] Mann WR (1953) Mean value methods in iterations. Proc. Amer. Math. Soc. 4(3):506–510.CrossrefGoogle Scholar
  • [28] Martinet B (1970) Régularisation d’inéquations variationnelles par approximations successives. Revue Française D’automatique Informatique Recherche Opérationnelle 3:154–158.Google Scholar
  • [29] Minty GJ (1962) Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29(3):341–346.CrossrefGoogle Scholar
  • [30] Ngai HV, Luc DT, Théra M (2000) Approximate convex functions. J. Nonlinear Convex Anal. 1(2):155–176.Google Scholar
  • [31] Pennanen T (2002) Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27(1):170–191.LinkGoogle Scholar
  • [32] Phan H (2016) Linear convergence of the Douglas–Rachford method for two closed sets. Optimization 65(2):369–385.CrossrefGoogle Scholar
  • [33] Poliquin RA, Rockafellar RT (1996) Prox-regular functions in variational analysis. Trans. Amer. Math. Soc. 348(5):1805–1838.CrossrefGoogle Scholar
  • [34] Poliquin RA, Rockafellar RT, Thibault L (2000) Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11):5231–5249.CrossrefGoogle Scholar
  • [35] Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5):877–898.CrossrefGoogle Scholar
  • [36] Rockafellar RT (2023) Generic linear convergence through metric subregularity in a variable-metric extension of the proximal point algorithm. Comput. Optim. Appl. 86(3):1327–1346.CrossrefGoogle Scholar
  • [37] Rolewicz S (1999) On α(·)-monotone multifunctions and differentiability of γ-paraconvex functions. Stud. Math. 133(1):29–37.CrossrefGoogle Scholar
  • [38] Spingarn J (1981) Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264(1):77–89.CrossrefGoogle Scholar
  • [39] Vial J-P (1983) Strong and weak convexity of sets and functions. Math. Oper. Res. 8(2):231–259.LinkGoogle Scholar
  • [40] Yin P, Lou Y, He Q, Xin J (2015) Minimization of ℓ1−2 for compressed sensing. SIAM J. Sci. Comput. 37(1):A536–A563.CrossrefGoogle Scholar
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