Mathematical Programming

, Volume 104, Issue 1, pp 39–68 | Cite as

Interior projection-like methods for monotone variational inequalities

Article

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.

Keywords

Variational 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 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Departement de MathematiquesUniversity Lyon ILyonFrance
  2. 2.School of Mathematical SciencesTel-Aviv UniversityRamat-AvivIsrael

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