Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions

References

• Aronszajn N (1950) Theory of reproducing kernels. Trans. Amer. Math. Soc. 68(3):337–404.Crossref, Google Scholar
• Baillon J-B, Haddad G (1977) Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J. Math. 26(2):137–150.Crossref, 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, Borwein JM (1994) Dykstra’s alternating projection algorithm for two sets. J. Approximation Theory 79(3):418–443.Crossref, Google Scholar
• Bauschke HH, Borwein JM (1996) On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3):367–426.Crossref, Google Scholar
• Bauschke HH, Combettes PL (2011) Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics (Springer, New York).Crossref, Google Scholar
• Bauschke HH, Borwein JM, Li W (1999) Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Programming 86(1):135–160.Crossref, Google Scholar
• Bauschke HH, Noll D, Phan HM (2015) Linear and strong convergence of algorithms involving averaged nonexpansive operators. J. Math. Anal. Appl. 421(1):1–20.Crossref, Google Scholar
• Bauschke HH, Phan HM, Wang X (2014) The method of alternating relaxed projections for two nonconvex sets. Vietnam J. Math. 42(4):421–450.Crossref, Google Scholar
• Bauschke HH, Bello Cruz JY, Nghia TTA, Phan HM, Wang X (2014) The rate of linear convergence of the Douglas-Rachford algorithm for subspaces is the cosine of the friedrichs angle. J. Approximation Theory 185:63–79.Crossref, Google Scholar
• Bertsekas DP (2010) Incremental gradient, subgradient, and proximal methods for convex optimization: A survey. Technical Report LIDS-P-2848, MIT.Google Scholar
• Bertsekas DP, Tsitsiklis JN (1989) Parallel and Distributed Computation: Numerical Methods (Prentice-Hall, Englewood Cliffs, NJ).Google Scholar
• Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations Trends Machine Learn. 3(1):1–122.Crossref, Google Scholar
• Combettes PL, Pesquet J-C (2011) Proximal Splitting Methods in Signal Processing. Fixed-Point Algorithms for Inverse Problems in Science and Engineering (Springer, New York), 185–212.Crossref, Google Scholar
• Davis D, Yin W (2016) Convergence rate analysis of several splitting schemes. Glowinski R, Osher S, Yin W, eds. Splitting Methods in Communication and Imaging, Science and Engineering (Springer, New York).Crossref, Google Scholar
• Deng W, Yin W (2016) On the global and linear convergence of the generalized alternating direction method of multipliers. J. Scientific Comput. 66(3):889–916.Crossref, Google Scholar
• Deutsch F, Hundal H (2008) The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets. J. Approximation Theory 155(2):155–184.Crossref, Google Scholar
• Gabay D (1983) Applications of the method of multipliers to variational inequalities. Fortin M, Glowinski R, eds. Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Vol. 15. Studies in Mathematics and Its Applications, Chap. IX (Elsevier, Amsterdam), 299–331.Crossref, Google Scholar
• Gabay D, Mercier B (1976) A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1):17–40.Crossref, Google Scholar
• Glowinski R, Marrocco A (1975) Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problèmes de Dirichlet nonlinéaires. ESAIM: Mathematical Modelling and Numerical Analysis—Modélisation Mathématique et Analyse Numérique 9(R2):41–76.Google Scholar
• Han D, Yuan X (2012) Convergence analysis of the Peaceman-Rachford splitting method for nonsmooth convex optimization. Technical report, Nanjing Normal University, Nanjing, China.Google Scholar
• Kayalar S, Weinert HL (1988) Error bounds for the method of alternating projections. Math. Control Signals Systems 1(1):43–59.Crossref, Google Scholar
• Lions PL, Mercier B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numerical Anal. 16(6):964–979.Crossref, Google Scholar
• Luke DR (2008) Finding best approximation pairs relative to a convex and prox-regular set in a hilbert space. SIAM J. Optim. 19(2): 714–739.Crossref, Google Scholar
• Phan HM (2016) Linear convergence of the Douglas-Rachford method for two closed sets. Optimization 65(2):369–385.Crossref, Google Scholar