128
Views
0
CrossRef citations to date
0
Altmetric
Research Articles

On Proximal Algorithms with Inertial Effects Beyond Monotonicity

&
Pages 1583-1601 | Received 23 Apr 2023, Accepted 29 Sep 2023, Published online: 13 Oct 2023

References

  • Martinet, B. (1972). Algorithmes pour la résolution de problèmes d’optimisation et de minimax, Université Joseph-Fourier, Grenoble.
  • Martinet, B. (1970). Régularisation d’inéquations variationnelles par approximations successives. Revue Française de Informatique 4:154–159. DOI: 10.1051/m2an/197004R301541.
  • Rockafellar, R. T. (1973). A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Program. 5(1):354–373. DOI: 10.1007/BF01580138.
  • Rockafellar, R. T. (1976). Monotone operators and proximal point algorithms. SIAM J. Control Optim. 14:877–898. DOI: 10.1137/031405.
  • Parikh, N., Boyd, S. (2014). Proximal algorithms. Found. Trends Optim. 1(3):127–239. DOI: 10.1561/2400000003.
  • Bauschke, H. H., Combettes, P. L. (2017). Convex Analysis and Monotone Operators Theory in Hilbert Spaces. CMS Books in Mathematics, 2nd ed. New York: Springer-Verlag.
  • Burachik, R. S., Svaiter, B. F. (2001). A relative error tolerance for a family of generalized proximal point methods. Math. Oper. Res. 26:816–831. DOI: 10.1287/moor.26.4.816.10011.
  • Alves, M. M., Eckstein, J., Geremia, M., Melo, J. G. (2020). Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms. Comput. Optim. Appl. 75:389–422. DOI: 10.1007/s10589-019-00165-y.
  • Eckstein, J., Silva, P. J. S. (2013). A practical relative error criterion for augmented Lagrangians. Math. Program. 141(1):319–348. DOI: 10.1007/s10107-0.
  • Combettes, P. L., Pennanen, T. (2004). Proximal methods for cohypomonotone operators. SIAM J. Control Optim. 43:731–742. DOI: 10.1137/S0363012903427336.
  • Iusem, A., Pennanen, T., Svaiter, B.F. (2003). Inexact variants of the proximal point algorithm without monotonicity. SIAM J. Optim. 13:1080–1097. DOI: 10.1137/S1052623401399587.
  • Gárciga Otero, R., Iusem, A. (2007). Proximal methods in reflexive Banach spaces without monotonicity. J. Math. Anal. Appl. 330:433–450. DOI: 10.1016/j.jmaa.2006.07.076.
  • Kohlenbach, U. (2022). On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space. Optim. Lett. 16:611–621. DOI: 10.1007/s11590-021-01738-9.
  • Moudafi, A. (2020). About proximal-type methods for a class of nonmonotone operators. Bull. Iran. Math. Soc. 46:115–125. DOI: 10.1007/s41980-019-00244-0.
  • Pennanen, T. (2002). Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27:170–191. DOI: 10.1287/moor.27.1.170.331.
  • Alvarez, A. (2000). On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM J. Control Optim. 38:1102–1119. DOI: 10.1137/S0363012998335802.
  • Antipin, A. S. (1994). Minimization of convex functions on convex sets by means of differential equations. Differ. Equ. 30:1365–1375.
  • Attouch, H., Chbani, Z., Fadili, J., Riahi, H. (2022). Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics. J. Optim. Theory Appl. 193:704–736. DOI: 10.1007/s10957-021-01859-2.
  • Attouch, H., Goudou, X., Redont, P. (2000). The heavy ball with friction I. The continuous dynamical system. Commun. Contemp. Math. 2(1):1–34. DOI: 10.1142/S0219199700000025.
  • Polyak, B. T. (1964). Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Phys. 4(5):1–17. DOI: 10.1016/0041-5553(64)90137-5.
  • Alvarez, F., Attouch, H. (2001). An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Var. Anal. 9:3–11. DOI: 10.1023/A:1011253113155.
  • Boţ, R. I., Csetnek, E. R., Hendrich, C. (2015). Inertial Douglas-Rachford splitting for monotone inclusion problems. Appl. Math. Comput. 256:472–487. DOI: 10.1016/j.amc.2015.01.017.
  • Chen, C., Chan, R. H., Ma, S., Yang, J. (2015). Inertial proximal ADMM for linearly constrained separable convex optimization. SIAM J. Imaging Sci. 8(4):2239–2267. DOI: 10.1137/15100463X.
  • Lorenz, D. A., Pock, T. (2015). An inertial forward-backward algorithm for monotone inclusions. Math. Imaging Vision 51(2):311–325. DOI: 10.1007/s10851-014-0523-2.
  • Alves, M. M., Marcavillaca, R. T. (2020). On inexact relative-error hybrid proximal extragradient, forward-backward and Tseng’s modified forward-backward methods with inertial effects. Set-Valued Var. Anal. 28:301–325. DOI: 10.1007/s11228-019-00510-7.
  • Boţ, R. I., Csetnek, E. R. (2014). A hybrid proximal-extragradient algorithm with inertial effects. Numer. Funct. Anal. Optim. 36(8):951–963. DOI: 10.1080/01630563.2015.1042113.
  • Grad, S. G., Lara, F., Marcavillaca, R. T. (2023). Relaxed-inertial proximal point type algorithms for quasiconvex minimization. J. Glob. Optim. 85:615–635. DOI: 10.1007/s10898-022-01226-z.
  • Ochs, P., Chen, Y., Brox T., Pock, T. (2014). iPiano: Inertial proximal algorithm for non-convex optimization. SIAM J. Imaging Sci. 7(2):1388–1419. DOI: 10.1137/130942954.
  • Boţ, R. I., Dao, M. N., Li, G. (2022). Extrapolated proximal subgradient algorithms for nonconvex and nonsmooth fractional programs. Appl. Math. Comput. 47(3):2415–2443. DOI: 10.1287/moor.2021.1214.
  • Wu, Z., Li, C., Li, M., Lim, A. (2021). Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems. J. Glob. Optim. 79:617–644. DOI: 10.1007/s10898-020-00943-7.
  • Bauschke, H. H., Mouri, W. M., Wang, X. (2021). Generalized monotone operators and their averaged resolvents. Math. Program. 189(1-2, Ser. B):55–74. DOI: 10.1007/s10107-020-01500-6.
  • Rockafellar, R. T., Wets, R. J.-B. (1998). Variational Analysis. volume 317 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag.
  • Davis, D., Drusvyatskiy, D. (2019). Stochastic model-based minimization of weakly convex functions. SIAM J. Optim. 29(1):207–239. DOI: 10.1137/18M1178244.
  • Davis, D., Drusvyatskiy, D. (2022). Proximal methods avoid active strict saddles of weakly convex functions. Found. Comput. Math. 22:561–606. DOI: 10.1007/s10208-021-09516-w.
  • Mollenhoff, T., Strekalovskiy, E., Moeller, M., Cremers, D. (2015). The primal-dual hybrid gradient method for semiconvex splittings. SIAM J. Imaging Sci. 8(2):827–857. DOI: 10.1137/140976601.
  • Selesnick, I., Lanza, A., Morigi, S., Sgallari, F. (2020). Non-convex total variation regularization for convex denoising of signals. J. Math. Imaging Vision 62(6):825–841. DOI: 10.1007/s10851-019-00937-5.
  • Rafique, H., Liu, M., Lin, Q., Yang, T. (2022). Weakly-convex-concave min-max optimization: provable algorithms and applications in machine learning. Optim. Methods Softw. 37(3):1087–1121. DOI: 10.1080/10556788.2021.1895152.
  • Guo, K., Han, D. R., Yuan, M. X. (2017). Convergence analysis of Douglas-Rachford splitting method for “strongly + weakly” convex programming. SIAM J. Numer. Anal. 55(4):1549–1577. DOI: 10.1137/16M1078604.
  • Opial, Z. (1967). Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73:591–597. DOI: 10.1016/0022-247X(73)90087-5.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.