Journal of Convex Analysis 07 (2000), No. 1, 047--072
Copyright Heldermann Verlag 2000
Epi-Distance Convergence of Parametrised Sums of Convex Functions in Non-Reflexive Spaces
A. Eberhard
Dept. of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, Australia
R. Wenczel
Dept. of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, Australia
A weak set of conditions ensuring epi-distance convergence of the sum of two epi-distance convergent families of closed convex functions, are established. These conditions may be viewed as containing two parts.
The first is that zero is in the strong quasi-relative interior of difference of the domains of the epi-distance limits of these families, a condition which has been used elsewhere in Young-Fenchel duality.
The second part implies (and is essentially equivalent to) the epi-distance convergence of the subspaces generated by the closure of the span of the differences of domains of the corresponding pairs of functions taken from these families. Convergence of saddle points in Young-Fenchel duality is investigated. Although both functions may vary in a very general way it is shown one can always extract a convergent subsequence of dual optimal solutions when both sequences of convex functions epi-converge and satisfy the conditions outlined above.
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