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

k-Nearest Neighbor Search for Large Scale High Dimensional data

Tohoku University徳山

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

Bio-inspired Optical Flow Interpretation with Fuzzy Logic for Behavior-Based Robot Control

This paper presents a bio-inspired approach for optical flow data interpretation based on fuzzy inference decision making for visual mobile robot navigation. The interpretation results of regionally averaged optical flow patterns with pyramid segmentation of the optical flow field deliver fuzzy topological and topographic information of the surrounding environment (topological structure from motion). It allows a topological localization in a global map as well as controlled locomotion (obstacle avoidance, goal seeking) in a changing and dynamic environment. The topological optical flow processing is embedded in a behavior based mobile robot navigation system which uses only a mono-camera as primary navigation sensor. The paper discusses the optical flow processing approach as well as the rule based fuzzy inference algorithms used. The implemented algorithms have been tested successfully with synthetic image data for a first verification and parameter tuning as well as in a real office environment with real image data

The lasting effects of innovation on firm profitability: Panel evidence from a transitional economy

This study is the first to study the lasting effects of innovation on firm profitability in Vietnam. Using a unique panel dataset for the period 2005-2015, our results show that innovators achieve higher profit in comparison with non-innovating firms. The positive effects of innovation on firm profitability are observed not only in the short term but also in the longer term. The benefits of innovation for firm profitability can be seen in higher export probability, better productivity, better access to formal credit, and the ability to secure government support, but only after innovation

Shortage in Access to Basic Social Services: A Case Study of Ethnic Minority Groups in Vietnam

The research uses shortages in the multi-dimensional poverty criteria and household living standards survey data collected by the General Statistic Office to evaluate the access to basic social services by ethnic minorities. The result shows significant differences in shortages between the Kinh and ethnic minorities, among ethnic minority groups and among localities. Differences in geographical location, level of development and lacking specific policies are the main causes. A number of policy recommendations has been made to narrow the gap in access to basic social services among ethnic minority groups in Vietnam. Keywords: Ethnic minorities, basic social services, shortage in access to basic social services, Vietnam DOI: 10.7176/EJBM/12-17-15 Publication date:June 30th 202

The lasting effects of innovation on firm profitability: panel evidence from a transitional economy

This study is the first to study the lasting effects of innovation on firm profitability in Vietnam. Using a unique panel dataset for the period 2005–2015, our results show that innovators achieve higher profit in comparison with non-innovating firms. The positive effects of innovation on firm profitability are observed not only in the short term but also in the longer term. The benefits of innovation for firm profitability can be seen in higher export probability, better productivity, better access to formal credit, and the ability to secure government support, but only after innovation

Impact of Vocational Education and Training On Decent Job Opportunities Regarding Information and Communication Technology

Vocational education and training plays a crucial part in the creation of decent job. The fast growth of information and communication technology (ICT) during the past ten years has increased work opportunities for laborers. This research uses the yearly labor force survey data to study the impact of vocational education and training on decent job opportunities for employees in the context of Vietnam's information and communication technology development. The logit model results indicate that the 2020 rate of decent jobs is still low, at approximately 25.6 percent. However, vocational education and training and the advancement of ICT positively affect decent job opportunities. This long-lasting impact affects laborers in various areas, regions, and industries. These findings suggest that vocational education and training and ICT must play a significant role in transitioning from precarious to more secure employment, so laborers must acquire professional knowledge and information technology skills