Relations between (proper) Pareto optimality of solutions of multicriteria optimization problems and solutions of the minimization problems obtained by replacing the multiple criteria with ${L}_{p}$-norm related functions (depending on the criteria, goals, and scaling factors) are investigated.

For an aggregation function $A$ we know that it is bounded by ${A}^{*}$ and ${A}_{*}$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if ${A}^{*}$ is directionally convex, then $A={A}^{*}$ and ${A}_{*}$ is linear; similarly, if ${A}_{*}$ is directionally concave, then $A={A}_{*}$ and ${A}^{*}$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively.

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