The configuration polytope of ℓ-line configurations in Steiner triple systems

It has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≤ m ≤ ℓ; full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in v, the order of the Steiner triple system. Moreover, the counts of full configurations, together with v, form a linear basis for all of the configuration counts when ℓ ≤ 7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored. To Alex Rosa on the Occasion of his Seventieth Birthday     

Combined ℓ<Subscript>2</Subscript> data and gradient fitting in conjunction with ℓ<Subscript>1</Subscript> regularization

We are interested in minimizing functionals with ℓ₂ data and gradient fitting term and ℓ₁ regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ₁ norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ₂ gradient fitting term can be used to avoid both edge blurring and staircasing effects.      

Absorbing Angles,Steiner Minimal Trees,and Antipodality

We give a new proof that a star {op _i :i=1,…,k} in a normed plane is a Steiner minimal tree of vertices {o,p ₁,…,p _k } if and only if all angles formed by the edges at o are absorbing (Swanepoel in Networks 36: 104–113, 2000). The proof is simpler and yet more conceptual than the original one. We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star {op _i :i=1,…,k} in any CL-space is a Steiner minimal tree of vertices {o,p ₁,…,p _k } if and only if all angles are absorbing, which in turn holds if and only if all distances between the normalizations \frac1||p_i||p_i\frac{1}{\Vert p_{i}\Vert}p_{i} equal 2. CL-spaces include the mixed ℓ ₁ and ℓ _∞ sum of finitely many copies of ℝ.     

Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞

We study hypercyclicity and supercyclicity of weighted shifts on ℓ^∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c₀ or ℓ^p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ^∞ is norm nowhere dense. Finally, we show that ℓ^∞ supports an isometry that is weak * sequentially supercyclic.     

Norms on unitizations of banach algebras revisited

Let A be an algebra without unit. If ∥ ∥ is a complete regular norm on A it is known that among the regular extensions of ∥ ∥ to the unitization of A there exists a minimal (operator extension) and maximal (ℓ₁-extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer. The second author was partially supported by the grant No. 201/03/0041 of GAČR.     

Vertex Degrees of Steiner Minimal Trees in ℓ p d and Other Smooth Minkowski Spaces

We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees (SMTs) in the d -dimensional Banach spaces _p ^d independent of d . This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d [19]. Our upper bounds follow from characterizations of singularities of SMTs due to Lawlor and Morgan [14], which we extend, and certain _p -inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d -dimensional Banach space (i.e. Minkowski space); the same upper bound for Steiner points having been found by Lawlor and Morgan. We obtain a second upper bound for the degrees of vertices in terms of 1 -summing norms. Received April 22, 1997, and in revised form October 1, 1997.     

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ _p → ℓ _q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ _p → ℓ _q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.     

Banach Spaces with Property (M) and their Szlenk Indices

We show that a separable Banach space with property (M*) has a Szlenk index equal to ω₀, and a norm with an optimal modulus of asymptotic uniform smoothness. From this we derive a condition on the Szlenk functions of the space and its dual which characterizes embeddability into c ₀ or an ℓ _p -sum of finite dimensional spaces. We also prove that two Lipschitz-isomorphic Orlicz sequence spaces contain the same ℓ _p -spaces.      

A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties

We present a null-space primal-dual interior-point algorithm for solving nonlinear optimization problems with general inequality and equality constraints. The algorithm approximately solves a sequence of equality constrained barrier subproblems by computing a range-space step and a null-space step in every iteration. The ℓ₂ penalty function is taken as the merit function. Under very mild conditions on range-space steps and approximate Hessians, without assuming any regularity, it is proved that either every limit point of the iterate sequence is a Karush-Kuhn-Tucker point of the barrier subproblem and the penalty parameter remains bounded, or there exists a limit point that is either an infeasible stationary point of minimizing the ℓ ₂ norm of violations of constraints of the original problem, or a Fritz-John point of the original problem. In addition, we analyze the local convergence properties of the algorithm, and prove that by suitably controlling the exactness of range-space steps and selecting the barrier parameter and Hessian approximation, the algorithm generates a superlinearly or quadratically convergent step. The conditions on guaranteeing that all slack variables are still positive for a full step are presented.     

On a problem of Namioka on norm-attaining functionals

Isaac Namioka conjectured that every nonreflexive Banach space can be renormed is such a way that, in the new norm, the set of norm attaining functionals has an empty interior in the norm topology. We prove the rightness of this conjecture for spaces containing an isomorphic copy of ℓ₁. As a consequence, we prove also that the same result holds for a wide class of Banach spaces containing, for example, the weakly compactly generated ones.     

Stability of kernel-based interpolation

It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L ₂ or L _∞ norms of interpolants can be bounded above by discrete ℓ₂ and ℓ_∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance.     

A variational problem with lack of compactness for H(S;S) maps of prescribed degree

We consider, for maps in H^1/2(S¹;S¹), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H^1/2(S¹;S¹) of prescribed topological degree equal to 1 and minimal norm. In general, the answer is no, due to concentration phenomena. The existence of a minimal map is sensitive to small perturbations of the norm. We derive a sufficient condition for the existence of minimal maps. In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained. In case there is no minimizer, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a point of S¹. We describe this concentration in terms of bubbling-off of circles.     

The point of continuity property in Banach spaces not containing ℓ<Subscript>1</Subscript>

We obtain a local characterization of the point of continuity property for bounded subsets in Banach spaces not containing basic sequences equivalent to the standard basis of ℓ₁ and, as a consequence, we deduce that, in Banach spaces with a separable dual, every closed, bounded, convex and nonempty subset failing the point of continuity property contains a further subset which can be seen inside the set of Borel regular probability measures on the Cantor set in a weak-star dense way. Also, we characterize in terms of trees the point of continuity property of Banach spaces not containing ℓ₁, by proving that a Banach space not containing ℓ₁ satis- fies the point of continuity property if, and only if, every seminormalized weakly null tree has a boundedly complete branch.     

A coordinate gradient descent method for <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-regularized convex minimization

In applications such as signal processing and statistics, many problems involve finding sparse solutions to under-determined linear systems of equations. These problems can be formulated as a structured nonsmooth optimization problems, i.e., the problem of minimizing ℓ ₁-regularized linear least squares problems. In this paper, we propose a block coordinate gradient descent method (abbreviated as CGD) to solve the more general ℓ ₁-regularized convex minimization problems, i.e., the problem of minimizing an ℓ ₁-regularized convex smooth function. We establish a Q-linear convergence rate for our method when the coordinate block is chosen by a Gauss-Southwell-type rule to ensure sufficient descent. We propose efficient implementations of the CGD method and report numerical results for solving large-scale ℓ ₁-regularized linear least squares problems arising in compressed sensing and image deconvolution as well as large-scale ℓ ₁-regularized logistic regression problems for feature selection in data classification. Comparison with several state-of-the-art algorithms specifically designed for solving large-scale ℓ ₁-regularized linear least squares or logistic regression problems suggests that an efficiently implemented CGD method may outperform these algorithms despite the fact that the CGD method is not specifically designed just to solve these special classes of problems.     

Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

We give some new examples of bounded multilinear forms on the Hilbert spaces ℓ₂ and L₂ (0, ∞). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m ≥ 3) on ℓ₂ which can be regarded as the multilinear analogue of the famous Hilbert matrix. We also determine the norm of the permanent on where      

On semigroups in which every Rees one-sided congruence is a congruence

The purpose of this paper is to examine the structure of those semigroups which satisfy one or both of the following conditions: A_r(A_ℓ): The Rees right (left) congruence associated with any right (left) ideal is a congruence. The conditions A_r and A_ℓ are generalizations of commutativity for semigroups. This paper is a continuation of the work of Oehmke [5] and Jordan [4] on H-semigroups (H for hamiltonian, a semigroup is called an H-semigroup if every one-sided congruence is a two-sided congruence). In fact the results of section 2 of Oehmke [5] are proved here under the condition A_r and/or A_ℓ and not the stronger hamiltonian condition. Section 1 of this paper is essentially a summary of the known results of Oehmke. In section 2 we examine the structure of irreducible semigroups satisfying the condition A_r and/or A_ℓ. In particular we determine all regular (torsion) irreducible semigroups satisfying both the conditions A_r and A_ℓ. This research has been supported by Grant A7877 of the National Research Council of Canada.     

Packing Steiner trees: polyhedral investigations

LetG=(V, E) be a graph andT⊆V be a node set. We call an edge setS a Steiner tree forT ifS connects all pairs of nodes inT. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graphG=(V, E) with edge weightsw _e , edge capacitiesc _e ,e∈E, and node setT ₁,…,T _N , find edge setsS ₁,…,S _N such that eachS _k is a Steiner tree forT _k , at mostc _e of these edge sets use edgee for eache∈E, and the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from a routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing problem from a polyhedral point of view and define an associated polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch & cut algorithm. This algorithm is described in our companion paper (in this issue).     

On equivalent strictly G-convex renormings of Banach spaces

We study strictly G-convex renormings and extensions of strictly G-convex norms on Banach spaces. We prove that ℓ_ω(Γ) space cannot be strictly G-convex renormed given Γ is uncountable and G is bounded and separable.     

Monotone curves

We estimate the rate of convergence of products of projections on K finite dimensional or finite codimensional subspaces in ℓ ₂ by the rate of convergence of the decreasing sequence of the squares of the norms of the iterates.