A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.

[1]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[2]  B. Martinet.  Brève communication. Régularisation d'inéquations variationnelles par approximations successives , 1970 .

[3]  Ronald E. Bruck.  An iterative solution of a variational inequality for certain monotone operators in Hilbert space , 1975 .

[4]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[5]  R. Tyrrell Rockafellar,et al.  Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming , 1976, Math. Oper. Res..

[6]  Z.-Q. Luo,et al.  Error bounds and convergence analysis of feasible descent methods: a general approach , 1993, Ann. Oper. Res..

[7]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[8]  Mounir Haddou,et al.  An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities , 1995, Math. Program..

[9]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[10]  Marc Teboulle,et al.  Convergence of Proximal-Like Algorithms , 1997, SIAM J. Optim..

[11]  Alfredo N. Iusem,et al.  An interior point method with Bregman functions for the variational inequality problem with paramonotone operators , 1998, Math. Program..

[12]  Jonathan Eckstein,et al.  Approximate iterations in Bregman-function-based proximal algorithms , 1998, Math. Program..

[13]  Alfredo N. Iusem,et al.  A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space , 1998, SIAM J. Optim..

[14]  A. Nemirovski.  On self-concordant convex–concave functions , 1999 .

[15]  Marc Teboulle,et al.  Interior Proximal and Multiplier Methods Based on Second Order Homogeneous Kernels , 1999, Math. Oper. Res..

[16]  G. Duclos.  New York 1987 , 2000 .