Sensitivity analysis of variational inequalities by normal-map techniques. (English) Zbl 0861.49009

Giannessi, F. (ed.) et al., Variational inequalities and network equilibrium problems. Proceedings of a conference, Erice, Italy, June 19-25, 1994. New York, NY: Plenum, 257-269 (1995).
Let \(K\) be a nonempty closed convex set in a real Hilbert space \(H\). Let \(T:K\to H\) be a nonlinear operator. Consider the problem of finding \(u\in K\) such that \[ \langle Tu,\nu-u\rangle\geq 0, \qquad\text{for all }\nu\in K,\tag{1} \] which is called the variational inequality problem. Let \(P_K\) be the projection of \(H\) onto the closed convex set \(K\). Consider the problem of finding \(z\in H\) such that \[ TP_Kz+ z-P_Kz=0, \tag{2} \] which is called the normal map induced by \(T\). It can be shown that these both problems are equivalent. This equivalence has been used to study the sensitivity analysis of variational inequalities.
Remark: It is worth mentioning that P. Shi [Proc. Am. Math. Soc. 111, No. 2, 339-346 (1991)] was the first who established the equivalence between the variational inequalities (1) and the problem (2), which he called the Wiener-Hopf equations. Later on, M. A. Noor [Panam. Math. J. 2, No. 2, 17-26 (1992; Zbl 0842.49012), J. Optimization Theory Appl. 79, No. 1, 197-206 (1993; Zbl 0799.49010)] has established the equivalence between various classes of variational inequalities and different classes of Wiener-Hopf equations. This equivalence has been exploited to suggest a number of iterative algorithms for variational inequalities. M. A. Noor [“Some recent advances in variational inequalities”, N.Z. J. Math. (to appear) and preprint, Mathematics Department, King Saud University, Riyadh 11451, Saudi Arabia (1996)] has also used this equivalence to study the sensitivity analysis of various classes of variational inequalities.
For the entire collection see [Zbl 0834.00044].
Reviewer: M.A.Noor (Riyadh)


49J40 Variational inequalities
49Q12 Sensitivity analysis for optimization problems on manifolds
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)