Minty Variational Inequalities, Increase-Along-Rays Property and Optimization1

Set-valued increasing-along-rays maps and well-posed set-valued star-shaped optimization

“…By using increasing-along-rays property, we investigate the stability and well-posedness of set-valued star-shaped optimization. Our results extend the corresponding results in Crespi et al [4,5] and Fang and Huang [8].…”

“…So it is interesting and important to study nonconvex problems. Star-shapedness can be regarded as a generalization of convexity and has been attracting more and more authors' attention (see, e.g., [4,5,8,[18][19][20]). Meanwhile, the wellposedness issue has become one of the most interesting and important subjects of optimization problems.…”

On Bilevel Variational Inequalities

“…This shows that the variational inequalities of differential type studied by Crespi, Ginchev and Rocca [16,17] is a special case of the bifunction variational inequality, which are closely related to nonsmooth optimization problems. We observe that the following relationships among (BVI) and other types of bilevel problems.…”

“…is the lower Dini directional derivative of the function g : R m → R ∪ {+∞} at z in the direction d, then (LSVI) is reduced to the following differential type variational inequality studied by Crespi, Ginchev and Rocca [16,17]: find z * ∈ Ω such that…”

Bifunction hemivariational inequalities

“…In recent years, it has been shown [2][3][4][5]30] that the minimum of directionally differentiable convex function on a convex set can be characterized by a class of variational inequalities, which is called the bifunction variational inequality. For the formulation, applications, numerical methods and other aspects of the bifunction variational inequalities, see [2-5, 7, 30, 32] and the references therein.…”