Integral-Type Representations for the Subdifferential of Suprema
本文给出了任意一族凸函数上确界的ε-次微分的积分型表示，通过数据函数的离散和与奇异测度来刻画，并在自反Banach空间或可分赋范空间中用涉及数据函数的极限替代这些测度。
Abstract An integral-like representation is provided for the $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> -subdifferential of the supremum of an arbitrary family of convex functions. Our characterizations are expressed through appropriate discrete sums performed on the data functions $$f_{t}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> ’s together with specific singular measures operating on them. Moreover, as long as the underlying space is a reflexive Banach space or a separable normed space, we substitute these additional measures with related limits that involve the data functions. All the objects involved in our characterizations rely intrinsically on the data functions that are (almost) active at the reference point.