Exact polynomial optimization strengthened with Fritz John conditions

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let $C$ be the set of critical points defined by \begin{equation} C=\{x\in\mathbb R^n\,:\,\text{rank}(\varphi(x))< m\}\quad\text{with}\quad\varphi:=\begin{bmatrix} \nabla g\\ \text{diag}(g) \end{bmatrix}\,. \end{equation} Assume that the image of $C$ under $f$, denoted by $f(C)$, is empty or finite. (Our assumption holds generically since $C$ is empty in a Zariski open set in the space of the coefficients of $g_1,\dots,g_m$ with given degrees.) We provide a sequence of values, returned by semidefinite programs, finitely converges to the minimal value attained by $f$ over the basic semi-algebraic set $S$ defined by \begin{equation} S:=\{x\in\mathbb R^n\,:\,g_j(x)\ge 0\,,\,j=1,\dots,m\}\,. \end{equation} Consequently, we can compute exactly the minimal value of any polynomial with real coefficients in $x$ over one of the following sets: the unit ball, the unit hypercube and the unit simplex. Under a slightly more general assumption, we extend this result to the minimization of any polynomial over a basic convex semi-algebraic set that has non-empty interior and is defined by the inequalities of concave polynomials.Comment: 32 pages and 2 tables, merged with arXiv:2205.0845

Practical polynomial optimization through positivity certificates with and without denominators

Les certificats de positivité ou Positivstellens"atze fournissent des représentations de polynômes positifs sur des ensembles semialgébriques de basiques, c'est-à-dire des ensembles définis par un nombre fini d'inégalités polynomiales. Le célèbre Positivstellensatz de Putinar stipule que tout polynôme positif sur un ensemble semialgébrique basique fermé $S$ peut être écrit comme une combinaison pondérée linéaire des polynômes décrivant $S$, sous une certaine condition sur $S$ légèrement plus forte que la compacité. Lorsqu'il est écrit comme ceci, il devient évident que le polynôme est positif sur $S$, et donc cette description alternative fournit un certificat de positivité sur $S$. De plus, comme les poids polynomiaux impliqués dans le Positivstellensatz de Putinar sont des sommes de carrés (SOS), de tels certificats de positivité permettent de concevoir des relaxations convexes basées sur la programmation semidéfinie pour résoudre des problèmes d'optimisation polynomiale (POP) qui surviennent dans diverses applications réelles, par exemple dans la gestion des réseaux d'énergie et l'apprentissage automatique pour n'en citer que quelques unes. Développée à l'origine par Lasserre, la hiérarchie des relaxations semidéfinies basée sur le Positivstellensatz de Putinar est appelée la emph{hiérarchie Moment-SOS}. Dans cette thèse, nous proposons des méthodes d'optimisation polynomiale basées sur des certificats de positivité impliquant des poids SOS spécifiques, sans ou avec dénominateurs.Positivity certificates or Positivstellens"atze provide representations of polynomials positive on basic semialgebraic sets, i.e., sets defined by finitely many polynomial inequalities. The famous Putinar's Positivstellensatz states that every positive polynomial on a basic closed semialgebraic set $S$ can be written as a linear weighted combination of the polynomials describing $S$, under a certain condition on $S$ slightly stronger than compactness. When written in this it becomes obvious that the polynomial is positive on $S$, and therefore this alternative description provides a certificate of positivity on $S$. Moreover, as the polynomial weights involved in Putinar's Positivstellensatz are sums of squares (SOS), such Positivity certificates enable to design convex relaxations based on semidefinite programming to solve polynomial optimization problems (POPs) that arise in various real-life applications, e.g., in management of energy networks and machine learning to cite a few. Originally developed by Lasserre, the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the emph{Moment-SOS hierarchy}. In this thesis, we provide polynomial optimization methods based on positivity certificates involving specific SOS weights, without or with denominators

A Nichtnegativstellensatz on singular varieties under the denseness of regular loci

Let $V$ be a real algebraic variety with singularities and $f$ be a real polynomial non-negative on $V$. Assume that the regular locus of $V$ is dense in $V$ by the usual topology. Using Hironaka's resolution of singularities and Demmel--Nie--Powers' Nichtnegativstellensatz, we obtain a sum of squares-based representation that characterizes the non-negativity of $f$ on $V$. This representation allows us to build up exact semidefinite relaxations for polynomial optimization problems whose optimal solutions are possibly singularities of the constraint sets.Comment: 20 pages, 1 table, fixed typo

Vietnamese EFL High School Teachers’ Self-assessment of their English Language Proficiency in Relation to the Official Assessment Test Results

The current study explored how Vietnamese EFL high school teachers self-assessed their current English proficiency. It also compared teachers’ self-ratings with the results they obtained in the Project 2010 test, which was  used to assess English level of EFL teachers across Vietnam as part of the implementation of the National Foreign Language Project 2020. The data were drawn from 15-item online questionnaire with 94 teacher participants. The findings revealed that teachers perceived themselves as being more proficient in skills of writing and reading, compared to listening, spoken production (e.g., making presentations) and spoken interaction (e.g., making conversations). Most of them rated their current English proficiency level lower than the level they achieved in the Project 2020 test.  The findings offer information  about the language aspects that teachers were least proficient, which can be used to specify what support they actually desire from in-service training courses.  Also, as teachers are at the centre of the Project 2020, their perceptions of their own English proficiency provide insightful input to enhance the effectiveness of this educational reform

Time-Frequency Nexus between Tourism Development, Economic Growth, Energy Consumption, and Ecological Footprint in Singapore

Singapore has been listed as one of the top-visited countries and has the highest ecological deficit. Despite the abundance of previous studies, the distinction between short, medium, and long term by decomposing tourism development, economic growth, energy consumption, and ecological footprint has been largely ignored. This study aims to investigate the lead-lag nexus structures between ecological footprint and Singapore's economic activities from 1978 to 2016. By adopting the wavelet analysis and scale-by-scale Granger causality test, the outcomes show that energy consumption positively impacts ecological footprint at high frequencies, while tourism and economic growth positively drive ecological footprint at high and medium frequencies. We also find that the positive impact of macroeconomic variables on ecological footprint has not been evident since 2003. Additionally, the wavelet-based Granger test confirms a bi-directional causal between economic growth and ecological footprint at all frequencies, whilst there is a bi-directional relationship between tourism, energy consumption, and ecological footprint at high frequency. Based on these findings, the research may further strengthen the belief of Singapore’s policy-makers on the promotion of tourism and suggests some helpful lessons for emerging countries

Nonparametric estimation of the fragmentation kernel based on a PDE stationary distribution approximation

We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier technics we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the non-standard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling

Talent Retention at Commercial Banks in Vietnam

This study aims to identify talent criteria at commercial banks in Vietnam and build a scale of factors affecting talent retention. By qualitative research method through expert interviews and non-parametric testing to determine the relationship between factors affecting talent retention at commercial banks in Vietnam. The research has identified the following four factors influencing talent retention: Job satisfaction, Motivation, Commitment and Loyalty. In addition, the study surveyed the Snowball sampling method and collected 877 satisfactory questionnaires, performed quantitative research and tested the linear structural model of SEM. Research results show that among the factors that affect talent retention, job satisfaction is the most influential, followed by commitment, loyalty and work motivation. The research's contribution helps bank leaders to have appropriate policies to retain their bank's talents. Keywords: Talent retention talent, Commitment; Job satisfaction; Loyalty; Motivation. JEL Classifications: J20, J28 DOI: https://doi.org/10.32479/irmm.1090