Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {Kn},{ln} ⊂ [1,∞), limn→∞kn = 1, limn→∞ln = 1, F(T1)∩ F(T2) = {x ∈ K : T1x = T2x = x} =∅, respectively. Suppose that {xn} is a sequence in...

The supremum over all knot sequences of the max-norm of the orthogonal spline projector is studied with respect to the order k of the splines and their smoothness. It is first bounded from below in terms of the max-norm of the orthogonal projector onto a space of incomplete polynomials. Then, for continuous and for differentiable splines, its order of growth is shown to be √ k.

Many analog neural network approaches for sparse recovery were based on using ?1-norm as the surrogate of ?0-norm. This paper proposes an model, namely Lagrange programming with ?p objective and quadratic constraint (LPNN-LPQC), ?0-norm sparsity measurement solving constrained basis pursuit denoise (CBPDN) problem. As is non-differentiable, we first use a differentiable ?p-norm-like function to...

Let (E,F) be a locally convex space. We denote the bounded elements of E by Eb : ={x ∈ E : ‖x‖F = sup ∈F (x)<∞}. In this paper, we prove that if BEb is relatively compact with respect to the F topology and f : I × Eb → Eb is a measurable family of F-continuous maps then for each x0 ∈ Eb there exists a norm-differentiable, (i.e. differentiable with respect to the ‖ · ‖F norm) local solution to t...

We consider optimizing a function smooth convex function f that is the average of a set of differentiable functions fi, under the assumption considered by Solodov [1998] and Tseng [1998] that the norm of each gradient f ′ i is bounded by a linear function of the norm of the average gradient f . We show that under these assumptions the basic stochastic gradient method with a sufficiently-small c...

Mixed Poisson regression models, a class of generalized linear mixed models, are commonly used to analyze count data that exhibit overdispersion. Because inference for these models can be computationally difficult, simplifying distributional assumptions are often made. We consider an influence function representing effects of infinitesimal perturbations of the mixing distribution. This function...

For Orlicz spaces endowed with the Orlicz norm and the Luxemburg norm, the criteria for uniformly nonsquare points and nonsquare points are given.

We give a characterization of K-weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions.