Quantitative Convergence Analysis of Iterated Expansive, Set-Valued Mappings

References

• Artacho FA, Geoffroy M (2007) Uniformity and inexact version of a proximal method for metrically regular mappings. J. Math. Anal. Appl. 335(1):168–183.Crossref, Google Scholar
• Artacho FA, Dontchev A, Geoffroy M (2007) Convergence of the proximal point method for metrically regular mappings. ESAIM: Proc. 17:1–8.Crossref, Google Scholar
• Aspelmeier T, Charitha C, Luke DR (2016) Local linear convergence of the ADMM/Douglas-Rachford algorithms without strong convexity and application to statistical imaging. SIAM J. Imaging Sci. 9(2):842–868.Crossref, Google Scholar
• Attouch H, Peypouquet J (2016) The rate of convergence of Nesterov’s accelerated forward-backward method is actually faster than 1/k2. SIAM J. Optim. 26(3):1824–1834.Crossref, Google Scholar
• Aubin J, Center WUMMR (1980) Contingent Derivatives of Set-Valued Maps and Existence of Solutions to Nonlinear Inclusions and Differential Inclusions. Cahiers du CEREMADE (Mathematics Research Center, University of Wisconsin, Madison, WI).Google Scholar
• Aubin JP, Frankowska H (1990) Set-Valued Analysis (Birkhäuser, Boston).Google Scholar
• Azé D (2006) A unified theory for metric regularity of multifunctions. J. Convex Anal. 13(2):225–252.Google Scholar
• 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
• Bauschke HH, Borwein JM (1993) On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1(2):185–212.Crossref, Google Scholar
• Bauschke HH, Combettes PL (2011) Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books Math./Ouvrages Math. SMC (Springer, New York).Crossref, Google Scholar
• Bauschke HH, Moursi WM (2016) The Douglas-Rachford algorithm for two (not necessarily intersecting) affine subspaces. SIAM J. Optim. 26:968–985.Crossref, Google Scholar
• Bauschke HH, Borwein JM, Lewis AS (1997) The method of cyclic projections for closed convex sets in Hilbert space. Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995) (American Mathematical Society, Providence, RI), 1–38.Crossref, Google Scholar
• Bauschke HH, Combettes PL, Luke DR (2002) Phase retrieval, error reduction algorithm and Fienup variants: A view from convex feasibility. J. Optim. Soc. Amer. A. 19(7):1334–1345.Crossref, Google Scholar
• Bauschke HH, Combettes PL, Luke DR (2003) A hybrid projection reflection method for phase retrieval. J. Optim. Soc. Amer. A. 20(6):1025–1034.Crossref, Google Scholar
• Bauschke HH, Combettes PL, Luke DR (2004) Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approximation Theory 127:178–192.Crossref, Google Scholar
• Bauschke HH, Luke DR, Phan HM, Wang X (2013) Restricted normal cones and the method of alternating projections: Applications. Set-Valued Var. Anal. 21:475–501.Crossref, Google Scholar
• Bauschke HH, Luke DR, Phan HM, Wang X (2013) Restricted normal cones and the method of alternating projections: Theory. Set-Valued Variational Anal. 21:431–473.Crossref, Google Scholar
• Bauschke HH, Luke DR, Phan HM, Wang X (2014) Restricted normal cones and sparsity optimization with affine constraints. Foundations Comput. Math. 14:63–83.Crossref, Google Scholar
• Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1):183–202.Crossref, Google Scholar
• Bolte J, Daniilidis A, Ley O, Mazet L (2010) Characterizations of Łojasiewicz inequalities: Subgradient flows, talweg, convexity. Transactions American Math. Soc. 362(6):3319–3363.Crossref, Google Scholar
• Borwein JM, Li G, Tam MK (2017) Convergence rate analysis for averaged fixed point iterations in common fixed point problems. SIAM J. Optim. 27(1):1–33.Crossref, Google Scholar
• Bruck RE, Reich S (1977) Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3(4):459–470.Google Scholar
• Burke JV, Luke DR (2003) Variational analysis applied to the problem of optical phase retrieval. SIAM J. Control. Optim. 42(2):576–595.Crossref, Google Scholar
• Candès E, Eldar Y, Strohmer T, Voroninski V (2013) Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6(1):199–225.Crossref, Google Scholar
• Chambolle A, Dossal C (2014) On the convergence of the iterates of FISTA, https://hal.inria.fr/hal-0106013, hAL Id: hal-01060130.Google Scholar
• Combettes PL, Pennanen T (2004) Proximal methods for cohypomonotone operators. SIAM J. Control. Optim. 43(2):731–742.Crossref, Google Scholar
• Daniilidis A, Georgiev P (2004) Approximate convexity and submonotonicity. J. Math. Anal. Appl. 291:292–301.Crossref, Google Scholar
• Daubechies I, Defrise M, Mol CD (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57:1413–1457.Crossref, Google Scholar
• Dontchev AL, Rockafellar RT (2014) Implicit Functions and Solution Mapppings, 2nd ed. (Springer-Verlag, Dordrecht, Netherlands).Google Scholar
• Drusvyatskiy D, Ioffe AD, Lewis AS (2015) Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15(6):1637–1651.Crossref, Google Scholar
• Edelstein M (1966) A remark on a theorem of M. A. Krasnoselski. Amer. Math. Monthly 73(5):509–510.Crossref, Google Scholar
• Gubin L, Polyak B, Raik E (1967) The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7(6):1–24.Crossref, Google Scholar
• Hesse R, Luke DR (2013) Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. SIAM J. Optim. 23(4):2397–2419.Crossref, Google Scholar
• Hesse R, Luke DR, Neumann P (2014) Alternating projections and Douglas-Rachford for sparse affine feasibility. IEEE Transactions Signal. Processing 62(18):4868–4881.Crossref, Google Scholar
• Hesse R, Luke DR, Sabach S, Tam M (2015) The proximal heterogeneous block implicit-explicit method and application to blind ptychographic imaging. SIAM J. Imaging Sci. 8(1):426–457.Crossref, Google Scholar
• Ioffe AD (2011) Regularity on a fixed set. SIAM J. Optim. 21(4):1345–1370.Crossref, Google Scholar
• Ioffe AD (2013) Nonlinear regularity models. Math. Program. 139(1–2):223–242.Crossref, Google Scholar
• Iusem A, Pennanen T, Svaiter B (2003) Inexact versions of the proximal point algorithm without monotonicity. SIAM J. Optim. 13:1080–1097.Crossref, Google Scholar
• Klatte D, Kummer B (2009) Optimization methods and stability of inclusions in Banach spaces. Math. Program. 117(1–2):305–330.Crossref, Google Scholar
• Kohlenbach U, López-Acedo G, Nicolae A (2017) Quantitative asymptotic regularity results for the composition of two mappings. Optimization 66(8):1291–1299.Crossref, Google Scholar
• Krasnoselski MA (1955) Two remarks on the method of successive approximations. Math. Nauk. (N.S.) 63(1):123–127, (Russian).Google Scholar
• Kruger AY (2006) About regularity of collections of sets. Set-Valued Anal. 14:187–206.Crossref, Google Scholar
• Kruger AY, Thao NH (2015) Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164:41–67.Crossref, Google Scholar
• Kruger AY, Luke DR, Thao NH (2016) Set regularities and feasibility problems. Math. Programming 168(1–2):279–311.Crossref, Google Scholar
• Lewis AS, Malick J (2008) Alternating projections on manifolds. Math. Oper. Res. 33:216–234.Link, Google Scholar
• Lewis AS, Luke DR, Malick J (2009) Local linear convergence of alternating and averaged projections. Found. Comput. Math. 9(4):485–513.Crossref, Google Scholar
• Luke DR (2005) Relaxed averaged alternating reflections for diffraction imaging. Inverse Problems 21:37–50.Crossref, Google Scholar
• Luke DR (2008) Finding best approximation pairs relative to a convex and a prox-regular set in Hilbert space. SIAM J. Optim. 19(2):714–739.Crossref, Google Scholar
• Luke DR (2012) Local linear convergence of approximate projections onto regularized sets. Nonlinear Anal. 75:1531–1546.Crossref, Google Scholar
• Luke DR (2017) Phase retrieval, what’s new? SIAG/OPT Views and News 25(1):1–5.Google Scholar
• Luke DR, Shefi R (2017) A globally linearly convergent method for pointwise quadratically supportable convex-concave saddle point problems. J. Math. Anal. Appl. 457(2):1568–1590.Crossref, Google Scholar
• Luke DR, Burke JV, Lyon RG (2002) Optical wavefront reconstruction: Theory and numerical methods. SIAM Rev. 44(2):169–224.Crossref, Google Scholar
• Luo ZQ, Tseng P (1993) Error bounds and convergence analysis of feasible descent methods: A general approach. Ann. Oper. Res. 46(1):157–178 Crossref, Google Scholar
• Mann WR (1953) Mean value methods in iterations. Proc. Amer. Math. Soc. 4:506–510.Crossref, Google Scholar
• Minty GJ (1962) Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29(3):341–346.Crossref, Google Scholar
• Mordukhovich B (2006) Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications. Grundlehren der Mathematischen Wissenschaften (Springer, New York).Google Scholar
• Moreau JJ (1962) Fonctions convexes duales et points proximaux dans un espace Hilbertien. Comptes Rendus de l’Académie des Sciences de Paris 255:2897–2899.Google Scholar
• Moreau JJ (1965) Proximité et dualité dans un espace Hilbertian. Bull. de la Soc. Math. de France 93(3):273–299.Crossref, Google Scholar
• Nesterov Y (2007) Gradient methods for minimizing composite objective function. Technical report, CORE Discussion Papers, Catholic University of Louvain, Belgium.Google Scholar
• Ngai HV, Théra M (2004) Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization. Set-Valued Anal. 12(1–2):195–223.Crossref, Google Scholar
• Ngai HV, Théra M (2008) Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19(1):1–20.Crossref, Google Scholar
• Ngai HV, Luc DT, Théra M (2000) Approximate convex functions. J. Nonlinear Convex Anal. 1:155–176.Google Scholar
• Noll D, Rondepierre A (2016) On local convergence of the method of alternating projections. Foundations Comput. Math. 16(2):425–455.Crossref, Google Scholar
• Nussbaum RD (1972) Degree theory for local condensing maps. J. Math. Anal. Appl. 37:741–766.Crossref, Google Scholar
• Opial Z (1967) Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73(4):591–597.Crossref, Google Scholar
• Pennanen T (2002) Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27(1):170–191.Link, Google Scholar
• Penot JP (1989) Metric regularity, openness and lipschitzian behavior multifunctions. Nonlinear Anal. Theor. Meth. Appl. 13(6):629–643.Crossref, Google Scholar
• Phan H (2016) Linear convergence of the Douglas-Rachford method for two closed sets. Optimization 65:369–385.Crossref, Google Scholar
• Poliquin RA, Rockafellar RT, Thibault L (2000) Local differentiability of distance functions. Trans. Amer. Math. Soc. 352(11):5231–5249.Crossref, Google Scholar
• Reich S (1973) Fixed points of condensing functions. J. Math. Anal. Appl. 41(2):460–467.Crossref, Google Scholar
• Reich S (1977) Extension problems for accretive sets in Banach spaces. J. Functional Anal. 26(4):378–395.Crossref, Google Scholar
• Rockafellar RT, Wets RJ (1998) Variational Analysis (Springer, Berlin).Crossref, Google Scholar
• Rouhani BD (1990) Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space. J. Math. Anal. Appl. 151(1):226–235.Crossref, Google Scholar
• Schaefer H (1957) Über die Methode sukzessiver Approximationen. Jahresber. Dtsch. Math.-Ver. 59:131–140.Google Scholar
• Spingarn J (1981) Submonotone subdifferentials of Lipschitz functions. Trans. Amer. Math. Soc. 264:77–89.Crossref, Google Scholar