On Robustness of Set-valued Maps and Marginal Value Functions

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7, 26], and further developed by Shi, Zheng, Zhuang [18, 19, 20], Phú, Hoffmann and Hichert [8, 9, 10, 17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the robust analysis of Zheng et al. to that of robustness of set-valued maps with given structures and marginal value functions. We are also strongly convinced that the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required.