Showing papers on "Dual norm published in 2006"

An approximation technique for robust nonlinear optimization

Abstract: Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.

On maps that preserve orthogonality in normed spaces

Abstract: We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations to a wide class of Banach spaces.

Linear Structures in the Set of Norm-Attaining Functionals on a Banach Space

Abstract: In this note, we investigate the spaceability properties of the set NA(X) of all norm attaining functionals on a Banach space X. This study is intimately related with isometric duality theory. If X is a dual space, then NA(X) certainly contains its predual. It is also motivated by proximinality questions: indeed, an hyperplane H = kerx £ of X is proximinal in X if and only if x £ 2 NA(X). If a flnite codimensional subspace Y of X is proximinal in X, then Y ? † NA(X), and the converse also holds in some Banach spaces, but not always. We refer to [3, 19, 20, 28] for some recent progress on this question. When a Banach space X has an inflnite dimensional quotient which is isomorphic to a dual space, it is easy to construct an equivalent norm on X for which NA(X) is spaceable. We show in this paper that the converse holds for Banach spaces whose dual unit ball is

On the use of the Dual Norms in Bounded Variation Type Regularization

Abstract: Recently Y. Meyer gave a characterization of the minimizer o f the Rudin-OsherFatemi functional in terms of the -norm. In this work we generalize this result to regularization models with higher order derivatives of b unded variation. This requires us to define generalized -norms. We present some numerical experiments to support the theoretical considerations.

Smooth norms and approximation in Banach spaces of the type C(K)

Abstract: We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^infty (except at 0).

Clément Interpolation and Its Role in Adaptive Finite Element Error Control

Abstract: Several approximation operators followed Philippe Clement’s seminal paper in 1975 and are hence known as Clement-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V ⊂ W k,p(Ω) onto some finite element space V h ⊂ W k,p(Ω) and generalize nodal interpolation operators whenever W k,p(Ω) ⊄ C 0(Ω), i.e., when p ≤ n/k for a bounded Lipschitz domain Ω ⊂ ℝn. The original motivation was H 2 ⊄ C 0(Ω) for higher dimensions n ≥ 4 and hence nodal interpolation is not well defined.

The norm of the product of polynomials in infinite dimensions

Abstract: Abstract Given a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

Numerical radius Haagerup norm and square factorization through Hilbert spaces

Abstract: We study a factorization of bounded linear maps from an operator space A to its dual space A * . It is shown that $T: A \rightarrow A*$ factors through a pair of column Hilbert space ℋ c and its dual space if and only if T is a bounded linear form on A ⊗ A by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.

On the stable norm of surfaces

Abstract: Given a Riemmanian metric on a closed surface, we consider the stable norm associated with the metric on the real homology group, and study some consequences when the operator of the Poincarduality is an isometry respect to the stable norm. In analogy with the marked length spectrum problem we study, if two metrics with the same stable norm are isometric.

Negative norm least-squares spectral methods for the elliptic problem

Abstract: In this paper, we develop and analyze an negative norm first-order system least-squares spectral method for the second-order elliptic boundary value problem. We consider a least-squares functional defined by the sum of the L²- and H -1 -norm of the residual equations. We define a discrete negative norm and then define the discrete negative norm least-squares functional for spectral approximation. The spectral convergence is derived for the proposed method and we provide some numerical results.

The Gleason's problem for some polyharmonic and hyperbolic harmonic function spaces

Abstract: Let Ω ⊆ ℝn be a bounded convex domain with C 2 boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H k p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H k p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.

Antiproximinal convex bounded sets in the space c0(Γ) equipped with the day norm

Abstract: We construct a convex smooth antiproximinal set in any infinite-dimensional space c 0(Γ) equipped with the Day norm; moreover, the distance function to the set is Gâteaux differentiable at each point of the complement.