Set-valued robust mappings and approximable mappings
The second author recently proposed the concept of robustness of sets and functions for weakening the requirement of continuity of the objective functions and constraints in global optimization problems. In this paper the authors propose the concept of lower robustness and of lower approximation for a set-valued mapping and show that any lower approximating set-valued mapping is lower robust. This result extends a result of the same authors for robust mappings and approximating mappings. The authors also introduce the concept of in-lower approximable and bi-lower approximable for set-valued functions and establish their relationship. They demonstrate an interesting property of bi-robust single-valued mappings. They also present some in-lower-robust mapping theorems which are the counterparts of open or inverse (set-valued) mapping theorems. For the upper robustness and upper approximability of set-valued mapping they obtain similar results as in the lower robust and lower approximate cases.
scientific article; zbMATH DE number 599930