Journal of Convex Analysis, Vol. 7, No. 1, pp. 47-71 (2000)

Epi-Distance Convergence of Parametrised Sums of Convex Functions in Non-Reflexive Spaces

A. Eberhard and R. Wenczel

Department of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, Australia,

Abstract: A weakened set of conditions is established for the epi-distance convergence of a sum $\{f_v+g_v\}_{v\in W}$ of parametrised closed convex functions $\{f_v\}_{v\in W}$ and $\{g_v\}_{v\in W}$ for $v\to w$, on an arbitrary Banach space. They are as follows: (1) $0\in \operatorname{sqri}(\operatorname{dom} f_w-\operatorname{dom} g_w)$; and (2) $X_w:=\operatorname{cone}(\operatorname{dom} f_w-\operatorname{dom} g_w)$ has closed algebraic complement $Y_w$; and (3) $X_v\cap Y_w=\{0\}$ for all $v$ near $w$, (where $X_v:=\overline{\operatorname{span}}(\operatorname{dom} f_v-\operatorname{dom} g_v)$). These are motivated by similar interiority conditions found in Fenchel duality theory. Our results are then used to investigate saddle-point convergence in Young-Fenchel duality in which both functions vary in a very general fashion.

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