Abstract
We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.
Similar content being viewed by others
References
Auslender, A.: Optimisation: Méthodes Numériques. Masson, Paris, 1976
Auslender, A., Teboulle, M., Ben Tiba, S.: Interior Proximal and Multiplier Methods based on Second Order Homogeneous Kernels. Mathematics of Operations Research 24, 645–668 (1999)
Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monographs in Mathematics, Springer-Verlag New-York, 2002
Auslender, A., Teboulle, M.: The Log-Quadratic proximal methodology in convex optimization algorithms and variational inequalities. In: P. Daniel, F. Gianessi, A. Maugeri (eds.), Equilibrium problems and variational models, Vol. 68, Nonconvex optimization and its applications, Kluwer Academic Press, 2003
Auslender, A., Teboulle, M.: A unified framewok for interior gradient/subgradient and proximal methods in convex optimization. Preprint, February 2003. Submitted for publication
Beck, A., Teboulle, M.: Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett. 31, 167–175 (2003)
Bruck, R.D.: An iterative solution of a variational inequality for certain monotone operators in Hilbert space. Bulletin of The American Math. Soc. 81, 890–892 (1975)
Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3, 538–543 (1993)
Doljanski, M., Teboulle, M.: An interior proximal algorithm and the exponential multiplier method for semidefinite programming. SIAM J. Optim. 9, 1–13 (1998)
Facchinei, F., Pang, J.S.: Finite-dimensional variational inequalities and complementarity problems. Vol. I and II. Springer Series in Operations Research. Springer-Verlag, New York, 2003
He, B.S., Liao, L.Z.: Improvements of some projection methods for monotone nonlinear variational inequalities. J. Optim. Theory and Appl. 112, 111–128 (2002)
Khobotov, E.N.: A modification of the extragradient method for the solution of variational inequalities and some optimization problems. USSR Comput. Math. and Math. Phys. 27, 120–127 (1987)
Konnov, I.V.: Combined relaxation methods for finding equilbrium points and solving related problems. Russian Mathematics 37, 46–53 (1993)
Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer Verlag, Berlin, 2001
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomie. i Mathematik Metody 12, 746–756 (1976). [english translation: Matecon 13, 35–49 (1977)]
Nemirovsky, A.: Prox-method with rate of convergence O(1/k) for smooth variational inequalities and saddle point problems. Draft of 30/01/03
Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ, 1970
Rockafellar, R.T.: Monotone operators and augmented Lagrangians in nonlinear programming. In: O. L. Mangasarian, et al. (eds.), “Nonlinear Programming 3”, Academic Press, New York, 1978, pp. 1–25
Rockafellar, R.T.: Linear-Quadratic Programming and Optimal Control. SIAM J. Control and Optim. 25, 781–814 (1987)
Rockafellar, R.T., B Wets, R.J.: Variational Analysis. Springer Verlag, New York, 1998
Rockafellar, R.T., B Wets, R.J.: A Lagrangian finite generatioon technique for solving linear-quadratic problems in stochastic programming. Math. Programming Studies 28, 63–93 (1986)
Sibony, M.: Methodes itératives pour les equations et inequations aux dérivees partielles non linéaires de type monotone. Calcolo 7, 65–183 (1970)
Teboulle, M.: Convergence of Proximal-like Algorithms. SIAM J. Optim. 7, 1069–1083 (1997)
Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951–965 (1997)
Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38 (2), 431–446 (2000)
Weisstein, E.W.: Cubic Equation. From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/CubicEquation.html
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010.
Rights and permissions
About this article
Cite this article
Auslender, A., Teboulle, M. Interior projection-like methods for monotone variational inequalities. Math. Program. 104, 39–68 (2005). https://doi.org/10.1007/s10107-004-0568-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-004-0568-x