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Abstract

Over the past decade, numerous approaches have been put forward to address multi-objective optimization problems-focusing on approximating proper efficient solutions instead of the efficient solutions. These methods are important because, according to Geoffrion’s conjecture, proper non-dominated solutions constitute a dense subset of the non-dominated solution set. In the current study, first, using the definition of Kuhn-Tucker’s proper efficiency, we present a linear subproblem that investigates the proper efficiency of a randomly selected feasible point. Additionally, the relationship between the different types of solutions to this sub-problem and the proper efficient solutions of the main problem has been analyzed. Next, we presented another sub-problem that upgrades the first sub-problem to a scalarization method to verify the efficiency of an arbitrary feasible point. Finally, we conducted an in-depth analysis of the interaction between the solutions of this upgraded sub-problem and the critical points of the main problem.We also include examples to showcase how our results can be applied in practice.

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