A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

Establishing a baseline: A social realist perspective on academic advising at a South African university prior to COVID-19

Academic advising remains an emerging practice and profession within the South African higher education sector. Although there has been an increase in literature about advising in South Africa recently, there remains a dearth of literature about the experiences of academic advisors working in this context. This article aims to make such a contribution, by focusing in particular on the experiences and insights from 15 South African advisors (from one university) about academic advising prior to the COVID-19 pandemic. The data that informs this study was collected through semi-structured interviews. The focus in this article is on advisor responses to three of the interview questions, which proved sufficient because of the richness of the data. The study draws on elements of social realist Margaret Archer’s (1995) morphogenetic framework to explicate why this perspective on advising within a South African higher education context is necessary. Archer’s (1995, 2005) work on structure, culture, and agency is then used as analytic lenses with which to analyse the advisors’ experiences and insights of advising prior to the pandemic. A phenomenographic approach (Marton 1981; Tight 2016) is adopted to explore the varying conceptions (Cibangu and Hepworth 2016) of advising offered by the academic advisors. Nine focal areas emerge from these insights, which are analysed and discussed using Archer’s (1995, 2005) structure, culture, and agency. It becomes apparent that academic advising was complex even before the pandemic. The advisors express an urgency to help others, raise concerns about entrenched inequities and resource constraints, highlight the pitfalls of inadequate help-seeking among students, and emphasise the need for better institutional integration of academic advising at the advisors’ university, among other things. It becomes clear that there are numerous structural and cultural tensions that constrain advisor and student agency, possibly to the detriment of student success. The article leaves the reader with insights into the experiences of academic advisors prior to the pandemic, thus providing a baseline against which to measure advising during and beyond the pandemic, at a time when advising in South African higher education is still being developed and defined

On the Complexity of Optimization over the Standard Simplex

We review complexity results for minimizing polynomials over the standard simplex and unit hypercube.In addition, we show that there exists a polynomial time approximation scheme (PTAS) for minimizing Lipschitz continuous functions and functions with uniformly bounded Hessians over the standard simplex.This extends an earlier result by De Klerk, Laurent and Parrilo [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science, to appear.]global optimization;standard simplex;PTAS;multivariate Bernstein approximation;semidefinite programming

Global optimization of rational functions:A semidefinite programming approach

We consider the problem of global minimization of rational functions on (unconstrained case), and on an open, connected, semi-algebraic subset of , or the (partial) closure of such a set (constrained case). We show that in the univariate case (n = 1), these problems have exact reformulations as semidefinite programming (SDP) problems, by using reformulations introduced in the PhD thesis of Jibetean [16]. This extends the analogous results by Nesterov [13] for global minimization of univariate polynomials. For the bivariate case (n = 2), we obtain a fully polynomial time approximation scheme (FPTAS) for the unconstrained problem, if an a priori lower bound on the infimum is known, by using results by De Klerk and Pasechnik [1]. For the NP-hard multivariate case, we discuss semidefinite programming-based relaxations for obtaining lower bounds on the infimum, by using results by Parrilo [15], and Lasserre [12]

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.Accepted versio

Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

Discrete Least-norm Approximation by Nonnegative (Trigonomtric) Polynomials and Rational Functions

Polynomials, trigonometric polynomials, and rational functions are widely used for the discrete approximation of functions or simulation models.Often, it is known beforehand, that the underlying unknown function has certain properties, e.g. nonnegative or increasing on a certain region.However, the approximation may not inherit these properties automatically.We present some methodology (using semidefinite programming and results from real algebraic geometry) for least-norm approximation by polynomials, trigonometric polynomials and rational functions that preserve nonnegativity.