Interior projection-like methods for monotone variational inequalities
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.
KeywordsVariational inequalities Convergence analysis Extragradient method Hyperplane projection algorithms Interior projection-like maps Convex-concave saddle point problems Duality and decomposition schemes Conic and semidefinite programming
Unable to display preview. Download preview PDF.
- 1.Auslender, A.: Optimisation: Méthodes Numériques. Masson, Paris, 1976Google Scholar
- 3.Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monographs in Mathematics, Springer-Verlag New-York, 2002Google Scholar
- 4.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, 2003Google Scholar
- 5.Auslender, A., Teboulle, M.: A unified framewok for interior gradient/subgradient and proximal methods in convex optimization. Preprint, February 2003. Submitted for publicationGoogle Scholar
- 10.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, 2003Google Scholar
- 14.Konnov, I.V.: Combined relaxation methods for variational inequalities. Springer Verlag, Berlin, 2001Google Scholar
- 15.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)]Google Scholar
- 16.Nemirovsky, A.: Prox-method with rate of convergence O(1/k) for smooth variational inequalities and saddle point problems. Draft of 30/01/03Google Scholar
- 18.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ, 1970Google Scholar
- 19.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. 1–25Google Scholar
- 21.Rockafellar, R.T., B Wets, R.J.: Variational Analysis. Springer Verlag, New York, 1998Google Scholar
- 27.Weisstein, E.W.: Cubic Equation. From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/CubicEquation.html