TL;DR: Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively.

Abstract: We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in $${\mathbb {R}}^d$$
, and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a $$(1-\varepsilon )$$
-approximation to the minimum of any polynomial of degree not exceeding n by $${\mathcal {O}}\left( (n/\sqrt{\varepsilon })^{\alpha d}\right) $$
samples, with $$\alpha =2$$
in the general case, and $$\alpha =1$$
in the smooth case. Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively.

TL;DR: This work will review some results on inner and inner conic approximations of the convex cone of positive Borel measures, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximation.

Abstract: The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, option pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, due to J.B. Lasserre, is to approximate the convex cone of positive Borel measures by finite dimensional outer and inner conic approximations. We will review some results on these approximations, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximations.

TL;DR: It is shown that Lasserre measure-based hierarchies for polynomial optimization can be implemented by directly computing the discrete minimum at a suitable set of algebraic quadrature nodes.

Abstract: We show that Lasserre measure-based hierarchies for polynomial optimization can be implemented by directly computing the discrete minimum at a suitable set of algebraic quadrature nodes. The sampling cardinality can be much lower than in other approaches based on grids or norming meshes. All the vast literature on multivariate algebraic quadrature becomes in such a way relevant to polynomial optimization.

TL;DR: It is shown that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs forPolynomial regression.

Abstract: We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measure concentration via Caratheodory-Tchakaloff compression.

6 citations

Cites background from "Markov inequalities, Dubiner distan..."

...This result is stated in the following
Lemma 2 Let K ⊂ Rd be a compact set of the form (21)-(22)....

[...]

...By Lemma 1 we can now prove the following proposition on near G-optimality by polynomial meshes constructed via the Dubiner distance....

[...]

...Lemma 1 Let Yn = Yn(α), n ≥ 1, be a sequence of finite sets of a compact set K ⊂ Rd, whose covering radius with respect to the Dubiner distance does not exceed α/n, where α ∈ (0, π/2), i.e.
r(Yn) = max x∈K dubK(x, Yn) = max x∈K min y∈Yn
dubK(x, y) ≤ α
n ....

[...]

...By Lemma 2 we get immediately the following proposition....

[...]

...(19)
The proof follows essentially the lines of that of Proposition 1, with Y2n(π/(2m)) replacing X2mn, observing that by Lemma 1 for every p ∈ Pd2n(K) we have ‖p‖K ≤ cm ‖p‖Y2n(π/(2m))....

TL;DR: This work connects the approximation theoretic notions of polynomial norming mesh and Tchakaloff-like quadrature to the statistical theory of optimal designs, obtaining near optimal polynometric regression at a near optimal number of sampling locations on domains with different shapes.

Abstract: We connect the approximation theoretic notions of polynomial norming mesh and Tchakaloff-like quadrature to the statistical theory of optimal designs, obtaining near optimal polynomial regression at a near optimal number of sampling locations on domains with different shapes

Abstract: We construct polynomial meshes from Tchakaloff quadrature points for measures on compact domains with certain boundary regularity, via estimates of the Christoffel function and its reciprocal.

TL;DR: This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators and is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun.

Abstract: Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.

Abstract: This considerably enriched new edition provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is the shape or the structure of a geometric object. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field. A series of generic examples has been added to the introduction and special emphasis has been put on the construction of important metrics. Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new version of the book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization. This area of research is very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, shape and geometric identification, image processing, and design of endoprotheses in interventional cardiology. Audience: This book is intended for applied mathematicians and advanced engineers and scientists, but the book is also structured as an initiation to shape analysis and calculus techniques for a broader audience of mathematicians. Some chapters are self-contained and can be used as lecture notes for a minicourse. The material at the beginning of each chapter is accessible to a broad audience, while the subsequent sections may sometimes require more mathematical maturity. Contents: List of Figures; Preface; Chapter 1: Introduction: Examples, Background, and Perspectives; Chapter 2: Classical Descriptions of Geometries and Their Properties; Chapter 3: Courant Metrics on Images of a Set; Chapter 4: Transformations Generated by Velocities; Chapter 5: Metrics via Characteristic Functions; Chapter 6: Metrics via Distance Functions; Chapter 7: Metrics via Oriented Distance Functions; Chapter 8: Shape Continuity and Optimization; Chapter 9: Shape and Tangential Differential Calculuses; Chapter 10: Shape Gradients under a State Equation Constraint; Elements of Bibliography; Index of Notation; Index.

TL;DR: Silverlight’s 2-D drawing support is the basic foundation for many of its more sophisticated features, such as custom-drawn controls, interactive graphics, and animation, so even if you don’t plan to create customized art for your application, you need to have a solid understanding of Silverlight's drawing fundamentals.

Abstract: Silverlight’s 2-D drawing support is the basic foundation for many of its more sophisticated features, such as custom-drawn controls, interactive graphics, and animation Even if you don’t plan to create customized art for your application, you need to have a solid understanding of Silverlight’s drawing fundamentals You’ll use it to add professional yet straightforward touches, like reflection effects You’ll also need it to add interactivity to your graphics—for example, to make shapes move or change in response to user actions

TL;DR: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities.