Existence, Pareto reducibility, and connectedness in multi-objective optimization: Elementary proofs and extension

TL;DR

This paper provides simplified proofs for key results in multi-objective optimization, extends Pareto reducibility to proper efficiency, and explores the relationships between various properties of nondominated solution sets, including connectedness.

Contribution

It offers elementary proofs for fundamental results, extends Pareto reducibility to proper efficiency, and clarifies the conditions for connectedness in solution sets.

Findings

Elementary proofs for important multi-objective optimization results.

Extension of Pareto reducibility from efficiency to proper efficiency.

Redundancy of certain assumptions in connectedness results.

Abstract

This paper is divided to two parts. In the first part, we provide elementary proofs for some important results in multi-objective optimization. The given proofs are so simple and short in compared to the existing ones. Also, a Pareto reducibility result is extended from efficiency to proper efficiency. The second part is devoted to the connections between nonemptiness, $R^p_{\geqq}$-(semi)compactness, external stability and connectedness of the set of nondominated solutions in multi-objective optimization. Furthermore, it is shown that some assumption in an important result, concerning connectedness, is redundant and should be removed.