A critical view on invexity

The aim of this note is to emphasize the fact that in many papers on invexity published in prestigious journals there are not clear definitions, trivial or not clear statements and wrong proofs. We also point out the unprofessional way of answering readers' questions by some authors. We think that this is caused mainly by the lack of criticism of the invexity communityComment: The paper was submitted to JOTA in December 2007 and practically accepted by the AE handling it in March 2008. Being a critical paper, the EiC asked the authors of the criticised articles to say their opinion. With the change of the EiC's, apparently the paper was not transmitted to the new Ei

New classes of higher order variational-like inequalities

In this paper, we prove that the optimality conditions of the higher order convex functions are characterized by a class of variational inequalities, which is called the higher order variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving higher order variational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results

NEW ASPECTS OF STRONGLY Log-PREINVEX FUNCTIONS

In this paper, we consider some new classes of log-preinvex functions. Several properties of the log-preinvex functions are studied. We also discuss their relations with convex functions. Several interesting results characterizing the log-convex functions are obtained. Optimality conditions of differentiable strongly $\log$-preinvex are characterized by a class of variational-like inequalities.  Results obtained in this paper can be viewed as significant improvement of previously known results

On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

By using Clarke’s generalized gradients we consider a nonsmooth vector optimization problem with cone constraints and introduce some generalized cone-invex functions called K-α-generalized invex, K-α-nonsmooth invex, and other related functions. Several sufficient optimality conditions and Mond-Weir type weak and converse duality results are obtained for this problem under the assumptions of the generalized cone invexity. The results presented in this paper generalize and extend the previously known results in this area

On the Subdifferentials of Quasiinvex and Pseudoinvex Functions and Cyclic Inmomocity

Abstract: The subdifferential characteristic of quasiinvex and pseudoinvex functions are vital role in convex optimization literature. Inmonicity and cyclic inmonicity properties are equally on the same line. In this paper, we studied the relation among subdifferential characteristics, inmonicity, and cyclic inmonicity of quasiinvex and pseudoinvex functions

Generalized convexities and generalized gradients based on algebraic operations

AbstractIn this paper, we investigate properties of generalized convexities based on algebraic operations introduced by Ben Tal [A. Ben Tal, On generalized means and generalized convex functions, J. Optim. Theory Appl. 21 (1977) 1–13] and relations between these generalized convexities and generalized monotonicities. We also discuss the (h,φ)-generalized directional derivative and gradient, and explore the relation between this gradient and the Clarke generalized gradient. Definitions of some generalized averages of the values of a generalized convex function at n equally spaced points based on the algebraic operations are also presented and corresponding results are obtained. Finally, the (φ,γ)-convexity is defined and some properties of (φ,γ)-convex functions are derived