Abstract
We study subgradient projection type methods for solving non-differentiable convex minimization problems and monotone variational inequalities. The methods can be viewed as a natural extension of subgradient projection type algorithms, and are based on using non-Euclidean projection-like maps, which generate interior trajectories. The resulting algorithms are easy to implement and rely on a single projection per iteration. We prove several convergence results and establish rate of convergence estimates under various and mild assumptions on the problem’s data and the corresponding step-sizes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslender A. (1973). Résolution numérique d’inégalites variationnelles. RAIRO 2: 67–72
Auslender A. (1976). Optimisation, methodes numeriques. Masson, Paris
Auslender A., Teboulle M. (2006). Interior gradient and proximal methods for convex and conic optimization. SIAM J. Optim. 16: 697–725
Auslender A., Teboulle M. (2004). Interior gradient and Epsilon-subgradient methods for constrained convex minimization. Math. Oper. Res. 29: 1–26
Auslender A., Teboulle M. (2005). Interior projection-like methods for monotone variational inequalities. Math. Program. Ser. A 104: 39–68
Auslender A., Guillet A., Gourgand M. (1974). Résolution numérique d’inégalites variationnelles. C.R.A.S serie A t.279: 341–344
Beck A., Teboulle M. (2003). Mirror descent and nonlinear projected subgradient methods for convexoptimization. Oper. Res. Lett. 31: 167–175
Ben-Tal A., Margalit T., Nemirovsky A. (2001). The ordered subsets mirror descent optimization method with applications to tomography. SIAM J. Optim. 12: 79–108
Bruck R.E. (1977). On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space. J. Math. Anal. Appl. 61: 159–164
Ermoliev Y.M., Ermoliev M. (1966). Methods of solution of nonlinear extremal problems. Cybernetics 2: 1–16
Konnov I.V. (2001). Combined relaxation methods for variational inequalities. Springer, Berlin
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomie. i Mathe. Metody 12, 746–756 (1976) [english translation: Matecon 13 35–49 (1977)]
Nemirovsky A. (1981). Effective iterative methods for solving equations with monotone operators. MATECON 17: 344–359
Nemirovsky, A.: Prox-method with rate of convergence O(1/k) for smooth variational inequalities and saddle point problems. Draft of 30/01/03
Nemirovsky A., Yudin D. (1983). Problem complexity and method efficiency in optimization. Wiley, New York
Nurminski, E.A.: Bibliography on nondifferentiable optimization. In: Progress in nondifferentiable optimization, IIASA Collaborative Proceedings Series, CP-82-98 (1982)
Polyak B.T. (1969). Minimization of nonsmooth functionals. U.S.S.R. Comput. Math. Math. Phys. 18: 14–29
Polyak B.T. (1987). Introduction to optimization. Optimization Software Inc., New York
Rockafellar R.T. (1970). Convex analysis. Princeton University Press, Princeton
Shor N.S. (1969). The generalized gradient descent. Trudy I Zimnei Skoly po Mat. Programmirovanyi 3: 578–585
Author information
Authors and Affiliations
Corresponding author
Additional information
We dedicate this paper to Boris Polyak on the occasion of his 70th birthday.
Rights and permissions
About this article
Cite this article
Auslender, A., Teboulle, M. Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities. Math. Program. 120, 27–48 (2009). https://doi.org/10.1007/s10107-007-0147-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-007-0147-z
Keywords
- Non-differentiable convex optimization
- Variational inequalities
- Ergodic convergence
- Subgradient methods
- Interior projection-like maps