Showing papers on "Regular polygon published in 2021"

Reachable Set Estimation of Delayed Markovian Jump Neural Networks Based on an Improved Reciprocally Convex Inequality.

Abstract: This brief investigates the reachable set estimation problem of the delayed Markovian jump neural networks (NNs) with bounded disturbances. First, an improved reciprocally convex inequality is proposed, which contains some existing ones as its special cases. Second, an augmented Lyapunov-Krasovskii functional (LKF) tailored for delayed Markovian jump NNs is proposed. Thirdly, based on the proposed reciprocally convex inequality and the augmented LKF, an accurate ellipsoidal description of the reachable set for delayed Markovian jump NNs is obtained. Finally, simulation results are given to illustrate the effectiveness of the proposed method.

Blaschke-Santaló diagram for volume, perimeter and first Dirichlet eigenvalue

Abstract: We are interested in the study of Blaschke-Santalo diagrams describing the possible inequalities involving the first Dirichlet eigenvalue, the perimeter and the volume, for different classes of sets. We give a complete description of the diagram for the class of open sets in R d , basically showing that the isoperimetric and Faber-Krahn inequalities form a complete system of inequalities for these three quantities. We also give some qualitative results for the Blaschke-Santalo diagram for the class of planar convex domains: we prove that in this case the diagram can be described as the set of points contained between the graphs of two continuous and increasing functions. This shows in particular that the diagram is simply connected, and even horizontally and vertically convex. We also prove that the shapes that fill the upper part of the boundary of the diagram are smooth (C 1,1), while those on the lower one are polygons (except for the ball). Finally, we perform some numerical simulations in order to have an idea on the shape of the diagram; we deduce both from theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.

Policy Mirror Descent for Reinforcement Learning: Linear Convergence, New Sampling Complexity, and Generalized Problem Classes.

Abstract: We present new policy mirror descent (PMD) methods for solving reinforcement learning (RL) problems with either strongly convex or general convex regularizers. By exploring the structural properties of these overall highly nonconvex problems we show that the PMD methods exhibit fast linear rate of convergence to the global optimality. We develop stochastic counterparts of these methods, and establish an ${\cal O}(1/\epsilon)$ (resp., ${\cal O}(1/\epsilon^2)$) sampling complexity for solving these RL problems with strongly (resp., general) convex regularizers using different sampling schemes, where $\epsilon$ denote the target accuracy. We further show that the complexity for computing the gradients of these regularizers, if necessary, can be bounded by ${\cal O}\{(\log_\gamma \epsilon) [(1-\gamma)L/\mu]^{1/2}\log (1/\epsilon)\}$ (resp., ${\cal O} \{(\log_\gamma \epsilon ) (L/\epsilon)^{1/2}\}$)for problems with strongly (resp., general) convex regularizers. Here $\gamma$ denotes the discounting factor. To the best of our knowledge, these complexity bounds, along with our algorithmic developments, appear to be new in both optimization and RL literature. The introduction of these convex regularizers also greatly expands the flexibility and applicability of RL models.

Measuring the similarity between multipolygons using convex hulls and position graphs

Abstract: Polygon similarity can play an important role in geographic information retrieval, map matching and updating, and spatial data mining applications. Geographic information science (GIS) represents v...

Modeling and sensitivity analysis of HBV epidemic model with convex incidence rate

Abstract: In this article, we study the qualitative analysis of the Hepatitis B epidemic model with convex incidence rate. First, we formulate the model without control and study local and global stability. We show the global stability of disease-free equilibrium using the method of Castillo-Chavez while for disease-endemic, we use the method of geometrical approach. Furthermore, we develop the proposed model with suitable optimal control strategies. We aim to minimize the infection of Hepatitis B in the host population. In order to do this, we use three control variables. Moreover, sensitivity analysis complemented by simulations is performed to assess how parameters changes affect the dynamical behavior of the system. The numerical simulations are performed to show the feasibility of the control strategy and the effectiveness of the theoretical results.

Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization

Abstract: In this paper, we study the behavior of solutions of the ODE associated to the Heavy Ball method. Since the pioneering work of B.T. Polyak [25], it is well known that such a scheme is very efficient for C2 strongly convex functions with Lipschitz gradient. But much less is known when the C2 assumption is dropped. Depending on the geometry of the function to minimize, we obtain optimal convergence rates for the class of convex functions with some additional regularity such as quasi-strong convexity or strong convexity. We perform this analysis in continuous time for the ODE, and then we transpose these results for discrete optimization schemes. In particular, we propose a variant of the Heavy Ball algorithm which has the best state of the art convergence rate for first order methods to minimize strongly, composite non smooth convex functions.

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ C d and iteration theory

Abstract: We study the homeomorphic extension of biholomorphisms between convex domains in $${\mathbb {C}}^d$$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.

Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings

Abstract: This paper aims at providing further studies of the notion of quasi-relative interior for convex sets. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.

An optimal gradient method for smooth strongly convex minimization

Abstract: We present an optimal gradient method for smooth strongly convex optimization. The method is optimal in the sense that its worst-case bound on the distance to an optimal point exactly matches the lower bound on the oracle complexity for the class of problems, meaning that no black-box first-order method can have a better worst-case guarantee without further assumptions on the class of problems at hand. In addition, we provide a constructive recipe for obtaining the algorithmic parameters of the method and illustrate that it can be used for deriving methods for other optimality criteria as well.

Vortex Patches Choreography for Active Scalar Equations

Abstract: This paper deals with the existence of N vortex patches located at the vertex of a regular polygon with N sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and $$\text {(SQG)}_\beta $$ equations, with $$\beta \in (0,1)$$ , but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the N point vortex system, that is, N point vortices located at the vertex of a regular polygon with N sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite-dimensional implicit function theorem and the well-chosen of the function spaces.

Optimal Model-Based Trajectory Planning With Static Polygonal Constraints

Abstract: The main contribution of this article is a novel method for planning globally optimal trajectories for dynamical systems subject to polygonal constraints. The proposed method is a hybrid trajectory planning approach, which combines graph search, i.e., a discrete roadmap method, with convex optimization, i.e., a complete path method. Contrary to past approaches, which have focused on using simple obstacle approximations, or suboptimal spatial discretizations, our approach is able to use the exact geometry of polygonal constraints in order to plan optimal trajectories. The performance and flexibility of the proposed method are evaluated via simulations by planning distance-optimal trajectories for a Dubins car model, as well as time-, distance-, and energy-optimal trajectories for a marine vehicle.

Analysis of temperature data by using innovative polygon trend analysis and trend polygon star concept methods: a case study for Susurluk Basin, Turkey

Abstract: Climate change is an event that has significant effects as direct or indirect on ecosystem and living things. In order to be prepared for the effect of climate change, it is necessary to anticipate these changes and take measures for this change. Therefore, many studies have been carried out on changes in climate parameters in recent years. The most common method used in these studies is trend methods. Innovative Polygon Trend Analysis (IPTA) and Trend Polygon Star Concept are trend analysis methods. IPTA Method divides data series into two as first and second data set and analyzes these two data sets by comparing them with each other. Trend Polygon Star Concept analyzes distance between two months in data set in graph, which is result of IPTA, and shows analysis result by dividing it into four regions. Therefore, in this study, monthly average temperature data are analyzed by using this two-polygon method. This data set is for 22 years (1996–2017). Polygon graphics were created as a result of study. Besides, trend slopes and lengths of temperature data with IPTA Method were calculated. The values of graphs created with Trend Polygon Star Concept Method on x- and y-axis were given in a table. When the results of both analysis methods were examined for a station, the following results were observed. For example, a regular polygon was not seen in arithmetic mean and standard deviation graphs of IPTA Method of Bandirma Station. Besides, when general evaluation of arithmetic mean analysis results was examined an increasing trend in most months. When arithmetic average graph created by Trend Polygon Star Concept Method of Bandirma Station was examined, transition between two months was seen first and third region. When standard deviation graph was examined, transitions between two months were seen in all four regions.

A limited memory q-BFGS algorithm for unconstrained optimization problems

Abstract: A limited memory q-BFGS (Broyden–Fletcher–Goldfarb–Shanno) method is presented for solving unconstrained optimization problems. It is derived from a modified BFGS-type update using q-derivative (quantum derivative). The use of Jackson’s derivative is an effective mechanism for escaping from local minima. The q-gradient method is complemented to generate the parameter q for computing the step length in such a way that the search process gradually shifts from global in the beginning to almost local search in the end. Further, the global convergence is established under Armijo-Wolfe conditions even if the objective function is not convex. The numerical experiments show that proposed method is potentially efficient.

Employing convex shape absorber for enhancing the performance of solar still desalination system

Abstract: This paper presents an experimental study for new solar still (SS) called convex SS. The convex SS has been tested at different convex heights, different wick materials, with Nano (Ag) black paint, and with Nano phase change material (paraffin wax + Ag). The results showed that, at 15 cm convex height and jute wick the convex SS showed 54% superior productivity than conventional SS. Besides, the data illustrated that, increasing the convex height increases the daily productivity of the convex SS till reaching the peak at 15 cm. Then, the yield started to decrease again at convex height larger than 15 cm. additionally; the jute wick showed higher performance with convex SS compared with cotton wick. The thermal efficiency of convex SS at 15 cm height showed about 41.2% and 40.8% for jute and cotton wicks, respectively. Furthermore, adding Ag nanoparticles to the black paint used to paint the convex surface increased the daily productivity to be 72% over conventional SS. As well, using thermal energy storage material (paraffin wax) mixed with Ag nanoparticles (2.5 wt.%) and Nano black paint enhanced the daily productivity of the convex SS to be 112% higher than conventional SS. So, it can be concluded that using Nano PCM enhanced the productivity of convex SS by about 40%. The economic analysis indicated that, the price per liter of fresh water is estimated to be 0.028 and 0.025 $/ l for conventional SS and convex SS, respectively.

Approximating Maximum Independent Set for Rectangles in the Plane.

Abstract: We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(\log\log n)$. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.

Some remarks about the maximal perimeter of convex sets with respect to probability measures

Abstract: In this note, we study the maximal perimeter of a convex set in ℝn with respect to various classes of measures. Firstly, we show that for a probability measure μ on ℝn, satisfying very mild assumpt...

A fully-nonlinear flow and quermassintegral inequalities in the sphere

Abstract: This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows. We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de-/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem.

Geometric inequalities for static convex domains in hyperbolic space

Abstract: We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.

On the regular-convexity of Ricci shrinker limit spaces

Abstract: In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions.

Potential Function-based Framework for Making the Gradients Small in Convex and Min-Max Optimization.

Abstract: Making the gradients small is a fundamental optimization problem that has eluded unifying and simple convergence arguments in first-order optimization, so far primarily reserved for other convergence criteria, such as reducing the optimality gap. We introduce a novel potential function-based framework to study the convergence of standard methods for making the gradients small in smooth convex optimization and convex-concave min-max optimization. Our framework is intuitive and it provides a lens for viewing algorithms that make the gradients small as being driven by a trade-off between reducing either the gradient norm or a certain notion of an optimality gap. On the lower bounds side, we discuss tightness of the obtained convergence results for the convex setup and provide a new lower bound for minimizing norm of cocoercive operators that allows us to argue about optimality of methods in the min-max setup.

Tailored nematic and magnetization profiles on two-dimensional polygons.

Abstract: We study dilute suspensions of magnetic nanoparticles in a nematic host, on two-dimensional polygons. These systems are described by a nematic order parameter and a spontaneous magnetization, in the absence of any external fields. We study the stable states in terms of stable critical points of an appropriately defined free energy, with a nemato-magnetic coupling energy. We numerically study the interplay between the shape of the regular polygon, the size of the polygon, and the strength of the nemato-magnetic coupling for the multistability of this prototype system. Our notable results include (1) the coexistence of stable states with domain walls and stable interior and boundary defects, (2) the suppression of multistability for positive nemato-magnetic coupling, and (3) the enhancement of multistability for negative nemato-magnetic coupling.

Maximum principle for discrete-time stochastic optimal control problem and stochastic game

Abstract: This paper is first concerned with one kind of discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. The results are then extended to two kinds of discrete-time stochastic games. Two illustrative examples are studied, for which the explicit optimal strategies are given. This paper establishes a rigorous version of discrete-time stochastic maximum principle in a clear and concise way and paves a road for further related topics.

Higher Convexity and Iterated Second Moment Estimates

Abstract: We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive doubling under a convex function with sufficiently many strictly monotone derivatives. We show, roughly, that every time the number of terms in the equation is doubled, an additional saving of $1$ in the exponent of the trivial bound $N^{2k-1}$ is made, starting from the trivial case $k=1$. In the context of near-convex sets we also provide explicit dependencies on the additive doubling parameters. Higher convexity is necessary for such bounds to hold, as evinced by sets of perfect powers of consecutive integers. We exploit these stronger assumptions using an idea of Garaev, rather than the ubiquitous Szemeredi-Trotter theorem, which has not been adapted in earlier results to embrace higher convexity. As an application we prove small improvements for the best known bounds for sumsets of convex sets under additional convexity assumptions.

Anisotropic and crystalline mean curvature flow of mean-convex sets

Abstract: We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit.

A flow approach to the Musielak-Orlicz-Gauss image problem

Abstract: In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

Applications of convex geometry to Minkowski sums of m ellipsoids in ℝN: Closed-form parametric equations and volume bounds

Abstract: General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of m arbitrary ellipsoids in N-dimensional Euclidean space. E...