Directionally Variational Analysis and Second-Order Optimality Conditions for Mathematical Programs with Switching Constraints

The switching constraint refers to a constraint in which the product of two functions is equal to zero, which exists extensively in the control and optimization community. This paper is devoted to investigate directionally variational analysis and the second-order optimality conditions for mathematical programs with switching constraints (MPSC) originated from optimal control theory. Developing the variational analysis associated with switching constraints, we present explicit formulas of radial cone, (directional) tangent cones, (directional) normal cones, outer (inner) second-order tangent set as well as geometrically derivability and parabolically derivability of the cross set. The relations among the (directional) tangent cones and normal cones, and second-order tangent set are established. The decompositions of normal cones, tangent cones and second-order tangent sets associated with Cartesian product of the cross set and a nonempty closed set are also obtained. We derive the characterizations, such as the non-emptiness, uniqueness, convexity and compactness, of the multiplier sets and directional multiplier sets associated with the stationary points and directional stationary points of MPSC. Then the second-order necessary conditions and the second-order growth conditions of MPSC are established by the second-order tangent sets. Besides, the second-order optimality conditions of MPSC are also established by using the Fréchet second-order subdifferentials, where the involved functions are lack of the twice-differentiability.