##
**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)].

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)].

Reviewer: Muhammad Aslam Noor (Halifax)

### MSC:

49J40 | Variational inequalities |

65J15 | Numerical solutions to equations with nonlinear operators |

65K10 | Numerical optimization and variational techniques |

### Keywords:

preinvex sets; preinvex functions; convex sets; convex functions; variational-like inequalities; invex sets
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\textit{Q. H. Ansari} and \textit{J. C. Yao}, J. Optim. Theory Appl. 108, No. 3, 527--541 (2001; Zbl 0999.49008)

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### References:

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