We study vector optimization problems with solid non-polyhedral convex ordering cones, without assuming any convexity or quasiconvexity assumption. We state a Weierstrass-type theorem and existence results for weak efficient solutions for coercive and noncoercive problems. Our approach is based on a new coercivity notion for vector-valued functions, two realizations of the Gerstewitz scalarization function, asymptotic analysis and a regularization of the objective function. We define new boundedness and lower semicontinuity properties for vector-valued functions and study their properties. These new tools rely heavily on the solidness of the ordering cone through the notion of colevel and level sets. As a consequence of this approach, we improve various existence results from the literature, since weaker assumptions are required.

中文翻译：

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非凸向量优化问题弱有效解的存在性

我们研究具有实体非多面体凸排序锥的向量优化问题，不假设任何凸性或拟凸性假设。我们陈述了一个 Weierstrass 型定理和强制和非强制问题的弱有效解决方案的存在结果。我们的方法基于向量值函数的新矫顽力概念、Gerstewitz 标量函数的两种实现、渐近分析和目标函数的正则化。我们为向量值函数定义了新的有界性和下半连续性，并研究了它们的性质。这些新工具通过 colevel 和 level 集的概念在很大程度上依赖于排序锥的坚固性。由于这种方法，我们改进了文献中的各种存在结果，因为需要更弱的假设。