Isotonicity of the proximity operator and stochastic optimization problems in Hilbert quasi-lattices endowed with Lorentz cones

In this paper, we discuss the isotonicity of the proximity operator in Hilbert quasi-lattices endowed with different Lorentz cones. The extended Lorentz cone is first defined by the Minkowski functionals of some subsets. We then establish some sufficient conditions for the isotonicity of the proximity operator concerning one order and two mutually dual orders induced by Lorentz cones, respectively. Similarly, the cases of the extended Lorentz cones and other ordered inequality properties of the proximity operator are analysed. By adopting these characterizations, some solvability and iterative algorithm theorems for the stochastic optimization problem are established by different order approaches. For solvability, the gradient of the mappings does not need to be continuous, and the solutions are optimal with respect to the orders. In the stochastic proximal algorithms, the mappings satisfy inequality conditions just for comparable elements, but the convergence direction and convergence rate are more optimal.