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For a set-valued mapping M defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x,y)∈E×F, we study the minimal time function associated with the images of M and a bounded set Ω⊂F defined by M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of M,Ω at those points of (x,y)∈E×F (both cases: points in the graph gph M and points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph of M at points inside gph M and to the graph of the enlargement set-valued mapping at points outside gph M. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function M,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

journal of function spaces

Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces