A relaxed cutting plane method for semi-infinite semi-definite programming

In this paper, we develop two discretization algorithms with a cutting plane scheme for solving combined semi-infinite and semi-definite programming problems, i.e., a general algorithm when the parameter set is a compact set and a typical algorithm when the parameter set is a box set in the m-dimensional space. We prove that the accumulation point of the sequence points generated by the two algorithms is an optimal solution of the combined semi-infinite and semi-definite programming problem under suitable assumption conditions. Two examples are given to illustrate the effectiveness of the typical algorithm.     

On stochastic generation of ultrametrics in high-dimensional Euclidean spaces

We present a proof of the theorem which states that a matrix of Euclidean distances on a set of specially distributed random points in the n-dimensional Euclidean space R ⁿ converges in probability to an ultrametric matrix as n → ∞. Values of the elements of an ultrametric distance matrix are completely determined by variances of coordinates of random points. Also we present a probabilistic algorithm for generation of finite ultrametric structures of any topology in high-dimensional Euclidean space. Validity of the algorithm is demonstrated by explicit calculations of distance matrices and ultrametricity indexes for various dimensions n.     

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88-91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.     

Finiteness property of pairs of 2×2 sign-matrices via real extremal polytope norms

This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product.Here we address the problem of establishing the finiteness property of pairs of $2\times 2$ sign-matrices. Such problem is related to the conjecture that pairs of sign-matrices fulfil the finiteness property for any dimension. This would imply, by a recent result by Blondel and Jungers, that finite sets of rational matrices fulfil the finiteness property, which would be very important in terms of the computation of the joint spectral radius. The technique used in this paper could suggest an extension of the analysis to $n\times n$ sign-matrices, which still remains an open problem.As a main tool of our proof we make use of a procedure to find a so-called real extremal polytope norm for the set. In particular, we present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing.For pairs of sign-matrices we develop the computations exactly and hence are able to prove analytically the finiteness property. On the other hand, the algorithm can be used in a floating point arithmetic and provide a general tool for approximating the joint spectral radius of a set of matrices.     

Some combinatorial properties of centrosymmetric matrices

The elements in the group of centrosymmetric n×n permutation matrices are the extreme points of a convex subset of n²-dimensional Euclidean space, which we characterize by a very simple set of linear inequalities, thereby providing an interesting solvable case of a difficult problem posed by L. Mirsky, as well as a new analogue of the famous theorem on doubly stochastic matrices due to G. Birkhoff. Several further theorems of a related nature also are included.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

On an Efficient Use of Gradient Information for Accelerating Interval Global Optimization Algorithms

This paper analyzes and evaluates an efficient n-dimensional global optimization algorithm. It is a natural n-dimensional extension of the algorithm of Casado et al. [1]. This algorithm takes advantage of all available information to estimate better bounds of the function. Numerical comparison made on a wide set of multiextremal test functions has shown that on average the new algorithm works faster than a traditional interval analysis global optimization method.     

Counting polycubes without the dimensionality curse

d-dimensional polycubes are the generalization of planar polyominoes to higher dimensions. That is, a d-D polycube of size n is a connected set of n cells of a d-dimensional hypercubic lattice, where connectivity is through (d−1)-dimensional faces of the cells. Computing A_d(n), the number of distinct d-dimensional polycubes of size n, is a long-standing elusive problem in discrete geometry. In a previous work we described the generalization from two to higher dimensions of a polyomino-counting algorithm of Redelmeier [D.H. Redelmeier, Counting polyominoes: Yet another attack, Discrete Math. 36 (1981) 191-203]. The main deficiency of the algorithm is that it keeps the entire set of cells that appear in any possible polycube in memory at all times. Thus, the amount of required memory grows exponentially with the dimension. In this paper we present an improved version of the same method, whose order of memory consumption is a (very low) polynomial in both n and d. We also describe how we parallelized the algorithm and ran it through the Internet on dozens of computers simultaneously.     

Hull properties in location problems

Some properties of the solution set for single and multifacility continuous location problems with l_p distances are given. A set reduction algorithm is developed for problems in k-dimensional space having rectangular distances.     

On orthostochastic, unistochastic and qustochastic matrices

We introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm. This is the quaternionic analogue of the unistochastic matrices studied by physicists. We also introduce quaternionic Hadamard matrices and quaternionic mutually unbiased bases (MUB). In particular we show that the number of MUB in an n-dimensional quaternionic Hilbert space is at most 2n+1. The bound is attained for n=2. We also determine all quaternionic Hadamard matrices of size n?4.     

Proximal point algorithm with Schur decomposition on the cone of symmetric semidefinite positive matrices

In this work, we propose a proximal algorithm for unconstrained optimization on the cone of symmetric semidefinite positive matrices. It appears to be the first in the proximal class on the set of methods that convert a Symmetric Definite Positive Optimization in Nonlinear Optimization. It replaces the main iteration of the conceptual proximal point algorithm by a sequence of nonlinear programming problems on the cone of diagonal definite positive matrices that has the structure of the positive orthant of the Euclidian vector space. We are motivated by results of the classical proximal algorithm extended to Riemannian manifolds with nonpositive sectional curvature. An important example of such a manifold is the space of symmetric definite positive matrices, where the metrics is given by the Hessian of the standard barrier function −lndet(X). Observing the obvious fact that proximal algorithms do not depend on the geodesics, we apply those ideas to develop a proximal point algorithm for convex functions in this Riemannian metric.     

Covering compact metric spaces greedily

A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvátal’s analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of the n-dimensional sphere and n-dimensional Euclidean space to give short and transparent proofs of several best known bounds obtained from constructions in the literature on sphere coverings.     

d-Orthogonality via generating functions

In this paper, starting from a suitable generating function of a polynomial set, we show how to decide whether the considered polynomial set is d-orthogonal and, if it is so, how to determine the corresponding d-dimensional functional vector. Then, we apply the obtained results to some known and new d-orthogonal polynomial sets. For the known ones, we give new proofs for some already obtained results.     

On linear systems with quasiperiodic coefficients and bounded solutions

For a discrete dynamical system ω _n =ω₀+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx _n+1=A(ω₀+αn)x _n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.     

On the partial order polytope of a digraph

We introduce the partial order polytope of a digraphD, defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofD. For this polytope we prove some classes of inequalities to be facet-defining and show that there is a polynomial separation algorithm for each of these classes. The results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two-chorded odd cycle inequalities. The polyhedral results concerning the partial order polytope are of interest since a cutting plane based algorithm to solve the maximum weighted transitive acyclic subdigraph problem can be used to solve the maximum weighted acyclic subdigraph problem, the maximum weighted linear ordering problem and a flexible manufacturing problem. For the acyclic subdigraph polytope we show that the separation of simplet-reinforcedk-fence-inequalities is -complete.     

Isotropical linear spaces and valuated Delta-matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n×n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Δ-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.     

Computing the Ehrhart quasi-polynomial of a rational simplex

We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of a fixed dimension. The algorithm is based on the formula relating the th coefficient of the Ehrhart quasi-polynomial of a rational polytope to volumes of sections of the polytope by affine lattice subspaces parallel to -dimensional faces of the polytope. We discuss possible extensions and open questions.

     

Determination of the element numbers of the regular polytopes

Let ?? be a set of n-dimensional polytopes. A set ?? of n-dimensional polytopes is said to be an element set for ?? if each polytope in ?? is the union of a finite number of polytopes in ?? identified along (n ? 1)-dimensional faces. The element number of the set ?? of polyhedra, denoted by e(??), is the minimum cardinality of the element sets for ??, where the minimum is taken over all possible element sets ${\Omega \in \mathcal{E}(\Sigma)}$ . It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ?? 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ?? 2.     

The Algebra of Matrices in N-dimensional Systems

The N-dimensional Roesser matrix is defined and shown to beessential to the formal solution of the N-dimensional Roesserequations. An extended Cayley-Hamilton theorem and a Leverrier-Fadeevtype algorithm are dervied for N-dminesional systems. One formof decomposition of the matrices is considered, and two particularcases are evaluated in terms of one-dimensional matrics.