Abstract
As well known, the Moreau-Rockafellar-Robinson internal point qualification condition is sufficient to ensure that the infimal convolution of the conjugates of two extended-real-valued convex lower semi-continuous functions defined on a locally convex space is exact, and that the subdifferential of the sum of these functions is the sum of their subdifferentials. This note is devoted to proving that this condition is, in a certain sense, also necessary, provided the underlying space is a Banach space. Our result is based upon the existence of a non-supporting weak*-closed hyperplane to any weak*-closed and convex unbounded linearly bounded subset of the topological dual of a Banach space.
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Dedicated to Stephen Robinson in honor of his 65 th birthday.
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Ernst, E., Théra, M. On the necessity of the Moreau-Rockafellar-Robinson qualification condition in Banach spaces. Math. Program. 117, 149–161 (2009). https://doi.org/10.1007/s10107-007-0162-0
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DOI: https://doi.org/10.1007/s10107-007-0162-0