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An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. (English) Zbl 0919.90123
Summary: We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operator $F$, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures.

##### MSC:
 90C25 Convex programming
##### References:
 [1] D. Bertsekas and J.N. Tsitsiklis,Parallel and Distributed Computation: Numerical Methods (Prentice-Hall, Englewood Cliffs, NJ, 1989). [2] L.M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,USSR Computational Mathematics and Mathematical Physics 7 (1967) 200–217. · doi:10.1016/0041-5553(67)90040-7 [3] R. Burachik, L.M. Graña Drummond, A.N. Iusem and B.F. Svaiter, On full convergence of the steepest descent method with inexact line searchesk, Optimization 32 (1995) 137–146. · Zbl 0821.90089 · doi:10.1080/02331939508844042 [4] Y. Censor, A.R. De Pierro, T. Elfving, G.T. Herman and A.N. Iusem, On iterative methods for linearly constrained entropy maximization, in: A. Wakulicz, ed.,Numerical Analysis and Mathematical Modelling, Banach Center Publications, Vol. 24 (PWN-Polish Scientific Publishers, Warsaw, Poland, 1990) 145–163. [5] Y. Censor and A. Lent, An iterative row-action method for interval convex programming,Journal of Optimization Theory and Applications 34 (1981) 321–353. · Zbl 0431.49042 · doi:10.1007/BF00934676 [6] Y. Censor and S. Zenios, The proximal minimization algorithm withD-functions,Journal of Optimization Theory and Applications 73 (1992) 451–464; Y. Censor and S.A. Zenios,Parallel Optimization: Theory, Algorithms and Applications (Oxford Univ. Press, New York, NY, 1997). · Zbl 0794.90058 · doi:10.1007/BF00940051 [7] G. Chen and M. Teboulle, Convergence analysis of a proximal-like optimization algorithm using Bregman functions,SIAM Journal on Optimization 3 (1993) 538–543. · Zbl 0808.90103 · doi:10.1137/0803026 [8] J.E. Dennis and R.B. Schnabel,Numerical Methods for Unconstrained Minimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983). [9] A.R. De Pierro and A.N. Iusem, A relaxed version of Bregman’s method for convex programming, Journal of Optimization Theory and Applications 51 (1986) 421–440. · Zbl 0581.90069 · doi:10.1007/BF00940283 [10] J. Eckstein, Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming,Mathematics of Operations Research 18 (1993) 202–226. · Zbl 0807.47036 · doi:10.1287/moor.18.1.202 [11] A.N. Iusem, On dual convergence and the rate of primal convergence of Bregman’s convex programming method,SIAM Journal on Optimization 1 (1991) 401–423. · Zbl 0753.90051 · doi:10.1137/0801025 [12] A.N. Iusem, An iterative algorithm for the variational inequality problem,Computational and Applied Mathematics 13 (1994) 103–114. [13] A.N. Iusem and B.F. Svaiter, A proximal regularization of the steepest descent method,RAIRO, Recherche Opérationelle 29 (1995) 123–130. [14] G.M. Korpolevich, The extragradient method for finding saddle points and other problems,Ekonomika i Matematcheskie Metody 12 (1976) 747–756. [15] A. Lent, A convergent algorithm for maximum entropy image restoration with a medical X-ray application, in: R. Shaw, ed.,Image Analysis and Evaluation (Society of Photographic Scientists and Engineers, Washington, DC, 1977) 249–257. [16] A. Lent and Y. Censor, The primal-dual algorithm as a constraint-set-manipulation device,Mathematical Programming 50 (1991) 343–357. · Zbl 0734.90066 · doi:10.1007/BF01594943 [17] F. Liese and I. Vajda,Convex Statistical Distances (Teubner, Leipzig, 1987). [18] R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).