Continuousization of the family of point-to-set maps and its applications

AbstractThis paper gives necessary and sufficient conditions for extending a family of point-to-set maps to a family of point-to-set maps which can be applied to abstract algorithms with global convergence, and also gives the convergence theorems of abstract algorithms with global convergence. The condition that the family of point-to-set maps is ϱ-decreasing or ϱ-increasing is not essential in the existing algorithm theory of the family of point-to-set maps [4, 5, 8]. Corresponding to three abstract algorithms, three different concrete examples are provided, one of which (i.e., the variable metric method) can be obtained by neither Denel's method of ϱ-decreasing family of point-to-set maps nor M. Yue's method of ϱ-increasing family of point-to-set maps