Iterative Schemes for Nonconvex Variational Inequalities

Existence and convergence theorems for the split quasi variational inequality problems on proximally smooth sets

“…Also, in 1995, Clarke et al [16] introduced a new class of nonconvex sets, which is called proximally smooth sets, and it has played an important part in many nonconvex applications such as optimization, dynamic systems and differential inclusions. Subsequently, in recent years, Bounkhel et al [4], Cho et al [12], Noor [25,26], Petrot [28] and J. Suwannait and N. Petrot [26,29,32] have considered both variational inequalities and equilibrium problems in the context of proximally smooth sets. They suggested and analyzed some projection type iterative algorithms by using the prox-regular technique and auxiliary principle technique.…”

“…However, in fact, the convexity assumption may not be required, because the algorithm may be well defined even if the considered set is nonconvexs (e.g., when the considered set is a closed subset of a finite dimensional space or a compact subset of Hilbert space, etc.) see [1,4,20,25,26,27]. While, it may be from the practical point of view, one may see that the nonconvex problems are more useful and general than convex case, subsequently, now many researchers are convinced and paid attention to many nonconvex cases.…”

Projection methods for nonconvex variational inequalities

“…See also [2,13,18,21,25] for the variant forms of nonconvex variational inequalities. We note that, if K r ≡ K , the convex set in H, then problem (2.1) is equivalent to…”

Generalized System of Variational Inequalities in Banach Spaces

“…Variational inequality and complementarity problems are of fundamental importance in a wide range of mathematical and applied sciences problems, such as mathematical programming, traffic engineering, economics and equilibrium problems, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The ideas and techniques of the variational inequalities are being applied in a variety of diverse areas of sciences and proved to be productive and innovative.…”