Iterative schemes for solving mixed variational-like inequalities. (English) Zbl 0999.49008

Let \(H\) be a Hilbert space whose inner product and norm are denoted by \(\langle,\rangle\) and \(\|.\|\), respectively. Let \(K\) be any subset of \(H\) and \(\eta(.,.): K\times K\to H\) be a mapping. Let \(f: K\to R\) be a real valued function. For a given nonlinear operator \(T: K\to H\), consider the problem of finding \(u\in K\) such that \[ \langle Tu,\eta(v,u)\rangle+ f(v)- f(u)\geq 0,\quad\forall v\in K,\tag{1} \] which is called the variational-like inequality problem, which has been studied extensively in recent years.
It is well known [T. Weir and V. Jeyakumar, Bull. Aust. Math. Soc. 38, No. 2, 177-189 (1988; Zbl 0639.90082)] that a set \(K\) is called a preinvex set, if for all \(u,v\in K\), \(t\in [0,1]\) there exists a mapping \(\eta(.,.): K\times K\to H\) such that \(u+ t\eta(v,u)\in K\) and a function \(F: K\to H\) is called a preinvex function if and only if \[ F(u+ t\eta(v,u))\leq (1-t)F(u)+ tF(v),\quad \forall u,v\in K,\;t\in [0,1]. \] It is also known that preinvex sets and preinvex functions may not be convex sets and convex functions. Consider the function \(I[v]\) defined by \[ I[v]= F(v)+ f(v),\tag{2} \] where \(F\) and \(f\) are preinvex functions. If the preinvex function \(F\) is differentiable, then it has been shown [M. A. Noor, Optimization 30, No. 4, 323-330 (1994; Zbl 0816.49005)] that the minimum of the functional \(I[v]\) defined by (2) can be characterized by the inequality of type: \[ \langle F'(u), \eta(v,u)\rangle+ f(v)- f(u)\geq 0,\quad \forall v\in K,\tag{3} \] where \(K\) is a preinvex set in \(H\). This clearly shows that variational-like inequalities are associated with the concept of preinvex functions and are only defined in the setting of preinvexity. It is very important to note that the variational-like inequalities are well defined only on the invex sets. If the underlying set \(K\) is not an invex set, then one can’t characterize the minimum of the convex functions by variational-like inequalities.
In spite of these known facts, the present authors have studied the variational-like inequalities in the convexity settings. Due to this reasons, all the results obtained in this paper are misleading and incorrect. One can’t apply the KKM result, since it does not hold for preinvex functions. This implies that the main result of this paper (Theorem 3.1) is wrong. Even in the setting of convexity, all the results are simple modifications of the results of D. L. Zhu and P. Marcotte [SIAM J. Optimization 6, No. 3, 714-726 (1996; Zbl 0855.47043)]. It is worth mentioning that variational-like inequalities have the same relationship with preinvex functions as variational inequalities with convex functions. This is the main motivation of the papers [M. A. Noor, J. Optimization Theory Appl. 87, No. 3, 615-630 (1995; Zbl 0840.90107); (loc. cit.)] and [X. Q. Yang and G.-Y. Chen, J. Math. Anal. Appl. 169, No. 2, 359-373 (1992; Zbl 0779.90067)].


49J40 Variational inequalities
65J15 Numerical solutions to equations with nonlinear operators
65K10 Numerical optimization and variational techniques
Full Text: DOI


[1] Ding, X. P., Random Mixed Variational-Like Inequalities in Topological Vector Spaces, Journal of the Sichuan Normal University, Vol. 20, pp. 1–13, 1997.
[2] Noor, M. A., Nonconvex Functions and Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 87, pp. 615–630, 1995. · Zbl 0840.90107
[3] Dien, N. H., Some Remarks on Variational-Like and Quasi-Variational-Like Inequalities, Bulletin of the Australian Mathematical Society, Vol. 46, pp. 335–342, 1992. · Zbl 0773.90071
[4] Noor, M. A., Variational-Like Inequalities, Optimization, Vol. 30, pp. 323–330, 1994. · Zbl 0816.49005
[5] Ansari, Q. H., and Yao, J. C., Prevariational Inequalities in Banach Spaces, Optimization: Techniques and Applications, Edited by L. Cacetta et al., Curtin University of Technology, Perth, Australia, Vol. 2, pp. 1165–1172, 1998.
[6] Ansari, Q. H., and Yao, J. C., Nonlinear Variational Inequalities for Pseudomonotone Operators with Applications, Advances in Nonlinear Variational Inequalities, Vol. 3, pp. 61–69, 2000. · Zbl 1014.47035
[7] Parida, J., Sahoo, M., and Kumar, A., A Variational-Like Inequality Problem, Bulletin of the Australian Mathematical Society, Vol. 39, pp. 225–231, 1989. · Zbl 0649.49007
[8] Siddiqi, A. H., Khaliq, A., and Ansari, Q. H., On Variational-Like Inequalities, Annales des Sciences Mathématiques du Québec, Vol. 18, pp. 39–48, 1994. · Zbl 0807.47054
[9] Yang, X. Q., and Chen, G. Y., A Class of Nonconvex Functions and Prevariational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 169, pp. 359–373, 1992. · Zbl 0779.90067
[10] Cohen, G., Optimization by Decomposition and Coordination: A Unified Approach, IEEE Transactions on Automatic Control, Vol. 23, pp. 222–232, 1978. · Zbl 0391.90074
[11] Cohen, G., Auxiliary Problem Principle and Decomposition of Optimization Problems, Journal of Optimization Theory and Applications, Vol. 32, pp. 277–305, 1980. · Zbl 0417.49046
[12] Cohen, G., and Zhu, D. L., Decomposition Coordination Methods in Large-Scale Optimization Problems: The Nondifferentiable Case and the Use of Augmented Lagrangians, Advances in Large-Scale Systems: Theory and Applications, Edited by J. B. Cruz, JAI Press, Greenwich, Connecticut, Vol. 1, pp. 203–266, 1984.
[13] Cohen, G., Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 49, pp. 325–333, 1988. · Zbl 0628.90066
[14] Glowinski, R., Lions, J. L., and TrÉmoliÈres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, Holland, 1981.
[15] Zhu, D. L., and Marcotte, P., Cocoercivity and Its Role in the Convergence of Iterative Schemes for Solving Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714–726, 1996. · Zbl 0855.47043
[16] Tseng, P., Further Applications of a Splitting Algorithm to Decomposition in Variational Inequalities and Convex Programming, Mathematical Programming, Vol. 48, pp. 249–263, 1990. · Zbl 0725.90079
[17] Karamardian, S., The Nonlinear Complementarity Problem with Applications, Part 2, Journal of Optimization Theory and Applications, Vol. 4, pp. 167–181, 1969. · Zbl 0169.51302
[18] Hanson, M. A., On Sufficiency of the Kuhn–Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981. · Zbl 0463.90080
[19] KÖthe, G., Topological Vector Spaces, I, Springer Verlag, Berlin, Germany, 1983.
[20] Fan, K., A Generalization of Tychonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701
[21] Deimling, K., Nonlinear Functional Analysis, Springer Verlag, Berlin, Germany, 1985. · Zbl 0559.47040
[22] Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994. · Zbl 0812.47055
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.