# Proper efficiency, scalarization and transformation in multi-objective optimization: Unified approaches

## TL;DR

This paper explores the connections between proper efficiency and scalarization solutions in multi-objective optimization, providing conditions for their equivalence and analyzing transformations that preserve proper efficiency.

## Contribution

It offers a unified theoretical framework linking proper efficiency, scalarization, and transformations in multi-objective optimization, extending existing results.

## Key findings

- Scalarization solutions are properly efficient under certain conditions.
- Unbounded scalar problems imply no properly efficient solutions exist.
- Transformations can preserve proper efficiency in multi-objective problems.

## Abstract

In this paper, we investigate the relationships between proper efficiency and the solutions of a general scalarization problem in multi-objective optimization. We provide some conditions under which the solutions of the dealt with scalar program are properly efficient and vice versa. We also show that, under some conditions, if the considered general scalar problem is unbounded, then the original multi-objective problem does not have any properly efficient solution. In another part of the work, we investigate a general transformation of the objective functions which preserves proper efficiency. We show that several important results existing in the literature are direct consequences of the results of the present paper.