The Avriel-Ben-Tal algebraic operations approach for a short version proof of the Karush-Kuhn-Tucker optimality conditions

In this paper, by using (h, ϕ)-generalized directional derivative and (h, ϕ)-generalized gradient, the authors directly derives the Karush- Kuhn-Tucker conditions by applying a corollary of Farkas lemma under the Mangasarian-Fromovitz constraint qualification.Furthermore, the boundedness of Lagrange multipliers is showed.

Nonsmooth quasi-variational-like inequalities with applications to nonsmooth vector quasi-optimization

We consider nonsmooth vector quasi-variational-like inequalities and nonsmooth vector quasioptimization problems. We utilize the method of scalarization to define nonsmooth quasi-variational-like inequalities by means of Clarke generalized directional derivative. We then study their relations with the problem of vector quasi-optimization and its scalarized version. Under the assumption of pseudomonotonicity and then densely pseudomonotonicity, we present some existence results for solutions to nonsmooth quasi-variational-like inequalities. To the best of our knowledge, the results we obtained are new in the sense of utilizing the scalarization method.

Existence results for general inequality problems with constraints

This paper is concerned with existence results for inequality problems of typeF0(u;v)+Ψ′(u;v)≥0, for allv∈X, whereXis a Banach space,F:X→ℝis locally Lipschitz, andΨ:X→(−∞+∞]is proper, convex, and lower semicontinuous. HereF0stands for the generalized directional derivative ofFandΨ′denotes the directional derivative ofΨ. The applications we consider focus on the variational-hemivariational inequalities involving thep-Laplacian operator.