# zbMATH — the first resource for mathematics

A projection-proximal point algorithm for solving generalized variational inequalities. (English) Zbl 1242.90267
The study of variational inequality problems provides a nice challenge in Mathematics. This paper deals with the folllowing generalized variational inequality problem:
Let $$X$$ be a nonempty, closed and convex subset of a Hilbert space $$H$$ and $$T:\rightrightarrows H$$ be a set-valued mapping. We consider the following generalized variational inequality: find $$x^*\in X$$ and $$w^*\in T(x^*)$$ such that $\langle w^*,y- x^*\rangle\leq 0,\quad\forall y\in X.$ In order to solve it, the authors consider a projection-proximal point method, and investigate a general iterative algorithm, which consists of an inexact proximal point step followed by a suitable orthogonal projection onto a hyperplane. When $$T$$ is pseudo-monotone (in the sense of Karamardian) with weakly upper semicontinuity and weakly compact and convex values, the convergence of such an algorithm is proved. In addition, the convergence rate of the iterative sequence under suitable conditions is also analyzed.

##### MSC:
 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
Full Text:
##### References:
 [1] Burachik, R.S., Scheimberg, S.: A proximal point method for the variational inequality problem in Banach spaces. SIAM J. Control Optim. 39, 1633–1649 (2001) · Zbl 0988.90045 [2] Konnov, I.V.: A combined relaxation method for variational inequalities with nonlinear constraints. Math. Program. 80, 239–252 (1998) · Zbl 0894.90145 [3] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053 [4] Yao, J.C.: A proximal method for pseudomonotone type variational-like inequalities. Taiwan. J. Math. 2, 497–514 (2006) · Zbl 1116.49004 [5] Yao, J.C.: Strong convergence theorems for strictly pseudocontractive mappings of Browder–Petryshyn type. Taiwan. J. Math. 3, 837–850 (2006) · Zbl 1159.47054 [6] Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90002 [7] He, Y.R.: A new double projection algorithm for variational inequalities. J. Comput. Appl. Math. 185, 166–173 (2006) · Zbl 1081.65066 [8] Marcotte, P.: Application of Khobotov’s algorithm to variational inequalities and network equilibrium. Inf. Syst. Oper. Res. 29, 258–270 (1991) · Zbl 0781.90086 [9] Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999) · Zbl 0959.49007 [10] Solodov, M.V., Svaiter, B.F.: A unified framework for some inexact proximal point algorithms. Numer. Funct. Anal. Optim. 22, 1013–1035 (2001) · Zbl 1052.49013 [11] Solodov, M.V., Svaiter, B.F.: Error bounds for proximal point subproblems and associated inexact proximal point algorithms. Math. Program. 88, 371–389 (2000) · Zbl 0963.90064 [12] Krasnoselskii, M.A.: Two observations about the method of successive approximations. Usp. Mat. Nauk 10, 123–127 (1955) [13] Martinet, B.: Régularisation d’inéquations variationelles par approximations successives. Rev. Fr. Inform. Rech. Oper. 4, 154–158 (1970) · Zbl 0215.21103 [14] Moreau, J.: Proximité et dualité dans un espace hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965) · Zbl 0136.12101 [15] Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976) · Zbl 0402.90076 [16] Burke, J.V., Qian, M.: A variable metric proximal point algorithm for monotone operators. SIAM J. Control Optim. 37, 353–375 (1998) · Zbl 0918.90112 [17] Cominetti, R.: Coupling the proximal point algorithm with approximation methods. J. Optim. Theory Appl. 95, 581–600 (1997) · Zbl 0902.90129 [18] Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6, 59–70 (1999) · Zbl 0961.90128 [19] Polyak, B.T.: Introduction to Optimization. Optimization Software Inc. Publications Division, New York (1987) · Zbl 0708.90083 [20] Ding, X.P., Tan, K.K.: A minimax inequality with application to existence of equilibrium point and fixed point theorems. Colloq. Math. 63, 233–247 (1992) · Zbl 0833.49009 [21] Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) · Zbl 0641.47066 [22] Takahashi, W.: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000) · Zbl 0997.47002 [23] Solodov, M.V.: Convergence rate analysis of iterative algorithms for solving variational inequality problems. Math. Program. 96, 513–528 (2003) · Zbl 1042.49008 [24] Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997) · Zbl 0887.90165 [25] Nadler, S.B.: Multi-valued contraction mappings. Pac. J. Math. 30, 475–488 (1969) · Zbl 0187.45002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.