On the closedness of the mapping defined by the generalized gradient of the support function of a Lipschitz set-valued map (Q923357)

The paper considers the set-valued mapping \((y^*,x)\to \partial \sigma_ F(y^*,x)\), where \(\sigma_ F(y^*,x)\) is the support function of a locally Lipschitz set-valued map F from a Hilbert space into another one and \(\partial \sigma_ F(y^*,x)\) denotes the generalized gradient (in Clarke's sense) of \(\sigma_ F(y^*,.)\) at x. The closedness of this mapping is important for many purposes such as: characterization of the solution set of an inclusion system, establishing optimality conditions for optimization problems involving inclusions constraints,... The paper gives some sufficient conditions for the closedness of the mentioned mapping, after showing by examples that this property is not automatically satisfied, in general.