# Some remarks on weak generalizations of minima and quasi efficiency

## TL;DR

This paper critically examines weak generalizations of minima and quasi efficiency, revealing many are not true generalizations, and provides criteria based on the Ekeland Variational Principle to distinguish genuine extensions.

## Contribution

It offers the first rigorous criteria for weaker minima generalizations and demonstrates that quasi efficiency is not a true generalization, clarifying inconsistencies in prior results.

## Key findings

- Many weak minima generalizations are invalid.
- Quasi efficiency is not a genuine generalization.
- Inconsistencies in duality results are identified.

## Abstract

In this note, we remark, with sufficient mathematical rigor, that many weak generalizations of the usual minimum available in the literature are not true generalizations. Motivated by the Ekeland Variational Principle, we provide, first time, the criteria for weaker generalizations of the usual minimum. Further, we show that the quasi efficiency, recently used in Bhatia et al. (Optim. Lett. 7, 127-135 (2013)) and introduced in Gupta et al. ( Bull. Aust. Math. Soc. 74, 207-218 (2006)) is not a true generalization of the usual efficiency. Since the former paper relies heavily on the results of later one, so we discuss the later paper. We show that the necessary optimality condition is a consequence of the local Lipschitzness and sufficiency result is trivial in the later paper. Consequently, the duality results of the same paper are also inconsistent.