Motivated by the recent works on proximity operators and isotone projection cones, in this paper, we discuss the isotonicity of the proximity operator in quasi-lattices, endowed with general cones. First, we show that Hilbert spaces, endowed with general cones, are quasi-lattices, in which the isotonicity of the proximity operator with respect to one order and two mutually dual orders is then, respectively, studied. Various sufficient conditions and examples are introduced. Moreover, we compare the proximity operator with the identity operator with respect to the orders. As applications, we study the solvability and approximation results for the nonconvex nonsmooth optimization problem by the order approaches. By establishing the increasing sequences, we, respectively, discuss the region of the solutions and the convergence rate, which vary with combinations of the mappings, and hence, one can choose the proper combination of the mappings under specific conditions. Compared to other approaches, the optimal solutions are obtained and inequality conditions hold only for comparable elements with respect to the orders.

中文翻译：

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一般拟格中近似算子的等渗性和优化问题

受最近关于邻近算子和等音投影锥的研究的启发，在本文中，我们讨论了具有一般锥体的准晶格中邻近算子的等渗性。首先，我们证明了具有一般锥体的希尔伯特空间是准格，其中分别研究了邻近算子相对于一个阶和两个相互对偶阶的等渗性。介绍了各种充分条件和例子。此外，我们在订单方面比较了邻近算子和身份算子。作为应用，我们通过阶数方法研究了非凸非光滑优化问题的可解性和近似结果。通过建立递增序列，我们分别讨论解的区域和收敛速度，随映射的组合而变化，因此，可以在特定条件下选择适当的映射组合。与其他方法相比，获得了最优解，并且不等式条件仅适用于关于阶数的可比元素。