Stability of the geometric Ekeland variational principle: Convex case (Q1331102)

The authors study the stability of cone maximal points for convex functions defined on an arbitrary (generally non reflexive) Banach space \(X\). The main stability result is the Theorem: Let \(f\in \Gamma(X)\), \(\lambda>0\), and suppose that \(g_ \lambda(x)= f(x)+ \lambda\| x-x_ 0\|\) has a strong minimum at \(x= x_ 0\). Then, whenever \(\langle f_ n\rangle\) is a sequence in \(\Gamma(X)\) convergent to \(f\) in the slice topology, we have \((x_ 0,f(x_ 0))\in \text{Li}(\lambda-\text{ext }f_ n)\), where \(\Gamma(X)\) is the set of proper lower semicontinuous convex functions defined on \(X\), \(\text{Li}(\lambda-\text{ext } f_ n)\) is the lower limit of the sequence \(\{\lambda-\text{ext }f_ n\}\) of the cone maximal points for the set \(\text{epi }f_ n\) with respect to the cone order induced by \(K_ \lambda\) the convex cone \(K_ \lambda= \{(x,\alpha)\in X\times \mathbb{R}\): \(\lambda\| x\|\leq- \alpha\}\).