Higher-order set-valued Hadamard directional derivatives: calculus rules and sensitivity analysis of equilibrium problems and generalized equations

In this paper, we propose a notion of higher-order directional derivatives in the sense of Hadamard for set-valued maps, which is a natural extension of the classical directional derivatives. Some of the usual calculus rules, for unions, intersections, products, sums, and compositions are given under directional metric subregularity conditions. The Hadamard differentiability of the efficient value mapping and a formula to compute its derivative are also obtained. Then, we apply these derivatives to establish an implicit set-valued map theorem and employ it to higher-order sensitivity analysis of the solution mapping for a parametric vector equilibrium problem. Sensitivity for solutions to a parametric generalized equation is also investigated. Many examples are provided for analyzing and illustrating the obtained results.