The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint sets. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, $M$-tameness, and the Palais--Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems.

中文翻译：

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约束向量优化问题的有效和适当有效的解决方案的存在

本文致力于在约束集上没有有界假设的约束向量优化的一般类非光滑问题的全局最优解的存在性。主要关注向量问题中最优性的两个主要概念：帕累托效率和 Geoffrion 意义上的适当效率。使用足够的变分分析和广义微分工具，我们首先建立适当性概念、$M$-tameness 和 Palais-Smale 条件之间的关系，这些条件是为约束集上的向量成本映射的限制而制定的。这些结果有助于为向量优化中帕累托有效解的存在推导出可验证的充分必要条件。此外，