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Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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Статті в журналах з теми "Approximation of convex function"

Petrova, T. "One counterexample for convex approximation of function with fractional derivatives, r>4." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 3 (2018): 53–56. http://dx.doi.org/10.17721/1812-5409.2018/3.7.

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We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

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Zala, Vidhi, Mike Kirby, and Akil Narayan. "Structure-Preserving Function Approximation via Convex Optimization." SIAM Journal on Scientific Computing 42, no. 5 (January 2020): A3006—A3029. http://dx.doi.org/10.1137/19m130128x.

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J. J. Koliha. "Approximation of Convex Functions." Real Analysis Exchange 29, no. 1 (2004): 465. http://dx.doi.org/10.14321/realanalexch.29.1.0465.

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Tang, Wee-Kee. "Sets of differentials and smoothness of convex functions." Bulletin of the Australian Mathematical Society 52, no. 1 (August 1995): 91–96. http://dx.doi.org/10.1017/s0004972700014477.

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Approximation by smooth convex functions and questions on the Smooth Variational Principle for a given convex function f on a Banach space are studied in connection with majorising f by C1-smooth functions.

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Bosch, Paul. "A Numerical Method for Two-Stage Stochastic Programs under Uncertainty." Mathematical Problems in Engineering 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/840137.

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Motivated by problems coming from planning and operational management in power generation companies, this work extends the traditional two-stage linear stochastic program by adding probabilistic constraints in the second stage. In this work we describe, under special assumptions, how the two-stage stochastic programs with mixed probabilities can be treated computationally. We obtain a convex conservative approximations of the chance constraints defined in second stage of our model and use Monte Carlo simulation techniques for approximating the expectation function in the first stage by the average. This approach raises with another question: how to solve the linear program with the convex conservative approximation (nonlinear constrains) for each scenario?

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Chen, Xin, Houduo Qi, Liqun Qi, and Kok-Lay Teo. "Smooth Convex Approximation to the Maximum Eigenvalue Function." Journal of Global Optimization 30, no. 2-3 (November 2004): 253–70. http://dx.doi.org/10.1007/s10898-004-8271-2.

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Ubhaya, Vasant A. "Uniform approximation by quasi-convex and convex functions." Journal of Approximation Theory 55, no. 3 (December 1988): 326–36. http://dx.doi.org/10.1016/0021-9045(88)90099-8.

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Ubhaya, Vasant A. "Lp approximation by quasi-convex and convex functions." Journal of Mathematical Analysis and Applications 139, no. 2 (May 1989): 574–85. http://dx.doi.org/10.1016/0022-247x(89)90130-3.

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Zwick, D. "Best Approximation by Convex Functions." American Mathematical Monthly 94, no. 6 (June 1987): 528. http://dx.doi.org/10.2307/2322845.

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Zwick, D. "Best Approximation by Convex Functions." American Mathematical Monthly 94, no. 6 (June 1987): 528–34. http://dx.doi.org/10.1080/00029890.1987.12000679.

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Дисертації з теми "Approximation of convex function"

Azimi, Roushan Tahere. "Inequalities related to norm and numerical radius of operators." Thesis, Pau, 2020. http://www.theses.fr/2020PAUU3002.

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Dans cette thèse, après la présentation des notions et des introductions nécessaires, nous étudionslinégalité Hermite-Hadamard pour les fonctions convexes géométriques. Après, nousdéveloppons les résultats en introduisant la fonction convexe géométrique opérationnelle etnous confirmons linégalité Hermite-Hadamard pour ces sortes de fonctions. Ensuite, nousmontrons certaines améliorations du cas normatif de certaines inégalités opérationnelles célèbres,en montrant le rôle convexe logarithmique de quelques fonctions classées selon lanorme stable et aussi en considérant le lien entre les fonctions convexes géométriques et lesfonctions logarithmiques. De plus, nous confirmons les résultats numériques obtenus pourapprocher une catégorie des fonctions convexes pour leur version opérationnelle et nousaméliorons linégalitéHermite-Hadamard pour certaines fonctions convexes opérationnellesen tant quune utilisation des résultats obtenus. Enfin, nous discutons à propos du rayon numériquedun opérateur qui est équivalent de sa norme opérationnelle et nous présentonsdes résultats concernés. Nous terminons cette thèse en obtenant les bornes supérieures dunombre Berezin dun opérateur
In this thesis, after expressing concepts and necessary preconditions, we investigate Hermite-Hadamard inequality for geometrically convex functions. Then, by introducing operator geometricallyconvex functions, we extend the results and prove Hermite-Hadamard type inequalityfor these kind of functions. In the following, by proving the log-convexity of somefunctions which are based on the unitarily invariant norm and considering the relation betweengeometrically convex functions and log-convex functions, we present several refinementsfor some well-known operator norm inequalities. Also, we prove operator version ofsome numerical results, which were obtained for approximating a class of convex functions,as an application,we refine Hermite-Hadamard inequality for a class of operator convex functions.Finally, we discuss about the numerical radius of an operator which is equivalent withthe operator norm and we state some related results, and by obtaining some upper boundsfor the Berezin number of an operator which is contained in the numerical range of that operator, we finish the thesis
در این رساله، پس از بیان مفاهیم و مقدمات لازم به بررسی نامساوی هرمیت-هادامار برای توابع محدب هندسی پرداخته سپس با معرفی تابع محدب هندسی عملگری، نتایج را توسیع داده و نامساوی هرمیت-هادامار گونه را برای این دست توابع اثبات می کنیم. در ادامه با نشان دادن محدب لگاریتمی بودن چند تابع که براساس نرم پایای یکانی تعریف شده اند، و با در نظر گرفتن ارتباط بین توابع محدب هندسی و توابع محدب لگاریتمی بهبودهایی از حالت نرمی چند نامساوی عملگری معروف ارائه می دهیم. هم چنین نتایج عددی بدست آمده جهت تقریب رده ای از توابع محدب را برای نسخه عملگری آن ها اثبات نموده و به عنوان کاربردی از نتایج حاصل، نامساوی هرمیت-هادامار را برای برخی توابع محدب عملگری بهبود می بخشیم. در نهایت، در مورد شعاع عددی یک عملگر، که معادل با نرم عملگری آن می باشد بحث نموده و به بیان برخی از نتایج مرتبط پرداخته، و با بدست آوردن کران های بالایی از عدد برزین یک عملگر که زیر مجموعه ای از برد عددی آن عملگر می باشد، رساله را به پایان می بریم

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Висоцька, Марія Андріївна. "Модель оптимального податку". Bachelor's thesis, КПІ ім. Ігоря Сікорського, 2021. https://ela.kpi.ua/handle/123456789/45204.

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Дипломна робота містить 95 с., 16 рис., 8 табл., 1 додатки, 11 джерел. Тема дослідження: модель оптимального податку Мета дослідження: проаналізувати існуючі модель оптимального податку, запропонувати альтернативну моделі та зробити дослідження на основі деяких демонстраційних даних. Результатом даної роботи є програмний продукт з користувацьким інтерфейсом, який допомагає знайти оптимальну моддель оподаткування для деяких даних та перевіряє її на коректність. На вхід ми отримуємо дані про розмір податку, ставку від податку та ще їх графіки.
The diploma thesis contains 95 p., 16 fig., 8 tabl, 2 appendices, 11 sources. Theme: optimal tax model. The purpose: to analyze the existing model of the optimal tax, to propose an alternative model and to make a study based on some demonstration data. Objective: analyze the existing model of the optimal tax, propose an alternative model and do research based on some demonstration data. The result of this work is a software product with a user interface that helps to find the optimal tax model for some data and checks it for correctness. At the entrance we receive data on the amount of tax, tax rate and their schedules.

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Bose, Gibin. "Approximation H infini, interpolation analytique et optimisation convexe : application à l’adaptation d’impédance large bande." Thesis, Université Côte d'Azur, 2021. http://www.theses.fr/2021COAZ4007.

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La thèse étudie en profondeur l'un des problèmes classiques de la conception de circuits RF, le problème de l'adaptation d'impédance. L’adaptation d’impédance consiste à maximiser le transfert de puissance d'une source à une charge dans une bande de fréquences. Les antennes sont l'un des dispositifs classiques dans lesquels l'adaptation d'impédance joue un rôle important. La conception d'un circuit d'adaptation pour une charge donnée revient principalement à trouver une matrice de diffusion sans perte qui, lorsqu'elle est enchaînée à la charge, minimise la réflexion de la puissance dans l'ensemble du système.Dans ce travail, les aspects théoriques du problème de l'adaptation et l'applicabilité pratique des approches développées sont dûment pris en compte. La partie I de la thèse couvre deux approches différentes mais étroitement liées du problème de l'adaptation large bande. Le cadre développé dans la première approche consiste à trouver la meilleure approximation H infini d'une fonction L infini, Փ via la théorie de Nehari. Cela revient à réduire le problème à un problème généralisé de valeurs propres basé sur un opérateur défini sur H2, l'opérateur de Hankel, HՓ. La réalisabilité d'un gain donné est fournie par la contrainte, opérateur norme de HՓ inférieure ou égale à un. La seconde approche formule le problème de l'adaptation comme un problème d'optimisation convexe où une plus grande flexibilité est fournie aux profils de gain par rapport à l'approche précédente. Il est basé sur deux théories riches, à savoir la théorie de l'adaptation de Fano-Youla et l'interpolation analytique. La réalisabilité d'un gain donné est basée sur les conditions de dé-chaînage de Fano-Youla qui se réduisent à la positivité d'une matrice classique en théorie d'interpolation analytique, la matrice de Pick. La concavité de la matrice de Pick concernée permet de trouver la solution au problème au moyen de l'implémentation d'un problème de programmation semi-défini non linéaire. Ainsi, nous estimons des limites inférieures nettes au niveau d'adaptation pour les circuits d'adaptation de degré fini et fournissons des circuits atteignant ces limites.La partie II de la thèse vise à réaliser les circuits d'adaptation sous forme de réseaux en échelle constitués d'inductances et de condensateurs et aborde également certaines contraintes importantes de réalisabilité. Les circuits d'adaptation sont conçus pour plusieurs antennes non-adaptées, testant la robustesse de l'approche développée. La théorie développée dans la première partie de la thèse offre un moyen efficace de comparer le niveau d'adaptation atteint aux limites théoriques
The thesis makes an in-depth study of one of the classical problems in RF circuit design,the problem of impedance matching. Matching problem addresses the issue of transmitting the maximum available power from a source to a load within a frequency band. Antennas are one of the classical devices in which impedance matching plays an important role. The design of a matching circuit for a given load primarily amounts to find a lossless scattering matrix which when chained to the load minimize the reflection of power in the total system.In this work, both the theoretical aspects of the broadband matching problem and thepractical applicability of the developed approaches are given due importance. Part I of the thesis covers two different yet closely related approaches to the matching problem. These are based on the classical approaches developed by Helton and Fano-Youla to study the broadband matching problems. The framework established in the first approach entails in finding the best H infinity approximation to an L infinity function, Փ via Nehari's theory. This amounts to reduce the problem to a generalized eigen value problem based on an operator defined on H2, the Hankel operator, HՓ. The realizability of a given gain is provided by the constraint, operator norm of HՓ less than or equal to one. The second approach formulates the matching problem as a convex optimisation problem where in further flexibility is provided to the gain profiles compared to the previous approach. It is based on two rich theories, namely Fano-Youla matching theory and analytic interpolation. The realizabilty of a given gain is based on the Fano-Youla de-embedding conditions which reduces to the positivity of a classical matrix in analytic interpolation theory, the Pick matrix. The concavity of the concerned Pick matrix allows finding the solution to the problem by means of implementing a non-linear semi-definite programming problem. Most importantly, we estimate sharp lower bounds to the matching criterion for finite degree matching circuits and furnish circuits attaining those bounds.Part II of the thesis aims at realizing the matching circuits as ladder networks consisting of inductors and capacitors and discusses some important realizability constraints as well. Matching circuits are designed for several mismatched antennas, testing the robustness of the developed approach. The theory developed in the first part of the thesis provides an efficient way of comparing the matching criterion obtained to the theoretical limits

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Lopez, Mario A., Shlomo Reisner, and reisner@math haifa ac il. "Linear Time Approximation of 3D Convex Polytopes." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1005.ps.

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Fung, Ping-yuen, and 馮秉遠. "Approximation for minimum triangulations of convex polyhedra." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2001. http://hub.hku.hk/bib/B29809964.

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Fung, Ping-yuen. "Approximation for minimum triangulations of convex polyhedra." Hong Kong : University of Hong Kong, 2001. http://sunzi.lib.hku.hk/hkuto/record.jsp?B23273197.

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Verschueren, Robin [Verfasser], and Moritz [Akademischer Betreuer] Diehl. "Convex approximation methods for nonlinear model predictive control." Freiburg : Universität, 2018. http://d-nb.info/1192660641/34.

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Boiger, Wolfgang Josef. "Stabilised finite element approximation for degenerate convex minimisation problems." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16790.

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Infimalfolgen nichtkonvexer Variationsprobleme haben aufgrund feiner Oszillationen häufig keinen starken Grenzwert in Sobolevräumen. Diese Oszillationen haben eine physikalische Bedeutung; Finite-Element-Approximationen können sie jedoch im Allgemeinen nicht auflösen. Relaxationsmethoden ersetzen die nichtkonvexe Energie durch ihre (semi)konvexe Hülle. Das entstehende makroskopische Modell ist degeneriert: es ist nicht strikt konvex und hat eventuell mehrere Minimalstellen. Die fehlende Kontrolle der primalen Variablen führt zu Schwierigkeiten bei der a priori und a posteriori Fehlerschätzung, wie der Zuverlässigkeits- Effizienz-Lücke und fehlender starker Konvergenz. Zur Überwindung dieser Schwierigkeiten erweitern Stabilisierungstechniken die relaxierte Energie um einen diskreten, positiv definiten Term. Bartels et al. (IFB, 2004) wenden Stabilisierung auf zweidimensionale Probleme an und beweisen dabei starke Konvergenz der Gradienten. Dieses Ergebnis ist auf glatte Lösungen und quasi-uniforme Netze beschränkt, was adaptive Netzverfeinerungen ausschließt. Die vorliegende Arbeit behandelt einen modifizierten Stabilisierungsterm und beweist auf unstrukturierten Netzen sowohl Konvergenz der Spannungstensoren, als auch starke Konvergenz der Gradienten für glatte Lösungen. Ferner wird der sogenannte Fluss-Fehlerschätzer hergeleitet und dessen Zuverlässigkeit und Effizienz gezeigt. Für Interface-Probleme mit stückweise glatter Lösung wird eine Verfeinerung des Fehlerschätzers entwickelt, die den Fehler der primalen Variablen und ihres Gradienten beschränkt und so starke Konvergenz der Gradienten sichert. Der verfeinerte Fehlerschätzer konvergiert schneller als der Fluss- Fehlerschätzer, und verringert so die Zuverlässigkeits-Effizienz-Lücke. Numerische Experimente mit fünf Benchmark-Tests der Mikrostruktursimulation und Topologieoptimierung ergänzen und bestätigen die theoretischen Ergebnisse.
Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element approximations, however, fail to resolve them in general. Relaxation methods replace the nonconvex energy with its (semi)convex hull. This leads to a macroscopic model which is degenerate in the sense that it is not strictly convex and possibly admits multiple minimisers. The lack of control on the primal variable leads to difficulties in the a priori and a posteriori finite element error analysis, such as the reliability-efficiency gap and no strong convergence. To overcome these difficulties, stabilisation techniques add a discrete positive definite term to the relaxed energy. Bartels et al. (IFB, 2004) apply stabilisation to two-dimensional problems and thereby prove strong convergence of gradients. This result is restricted to smooth solutions and quasi-uniform meshes, which prohibit adaptive mesh refinements. This thesis concerns a modified stabilisation term and proves convergence of the stress and, for smooth solutions, strong convergence of gradients, even on unstructured meshes. Furthermore, the thesis derives the so-called flux error estimator and proves its reliability and efficiency. For interface problems with piecewise smooth solutions, a refined version of this error estimator is developed, which provides control of the error of the primal variable and its gradient and thus yields strong convergence of gradients. The refined error estimator converges faster than the flux error estimator and therefore narrows the reliability-efficiency gap. Numerical experiments with five benchmark examples from computational microstructure and topology optimisation complement and confirm the theoretical results.

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Schulz, Henrik. "Polyhedral Surface Approximation of Non-Convex Voxel Sets and Improvements to the Convex Hull Computing Method." Forschungszentrum Dresden, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:d120-qucosa-27865.

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In this paper we introduce an algorithm for the creation of polyhedral approximations for objects represented as strongly connected sets of voxels in three-dimensional binary images. The algorithm generates the convex hull of a given object and modifies the hull afterwards by recursive repetitions of generating convex hulls of subsets of the given voxel set or subsets of the background voxels. The result of this method is a polyhedron which separates object voxels from background voxels. The objects processed by this algorithm and also the background voxel components inside the convex hull of the objects are restricted to have genus 0. The second aim of this paper is to present some improvements to our convex hull algorithm to reduce computation time.

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Schulz, Henrik. "Polyhedral Surface Approximation of Non-Convex Voxel Sets and Improvements to the Convex Hull Computing Method." Forschungszentrum Dresden-Rossendorf, 2009. https://hzdr.qucosa.de/id/qucosa%3A21613.

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Книги з теми "Approximation of convex function"

Duality in nonconvex approximation and optimization. New York: Springer, 2005.

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L, Combettes Patrick, and SpringerLink (Online service), eds. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York, NY: Springer Science+Business Media, LLC, 2011.

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Vasile, Postolică, ed. The best approximation and optimization in locally convex spaces. Frankfurt am Main: P. Lang, 1993.

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Kuhn, Daniel. Generalized bounds for convex multistage stochastic programs. Berlin: Springer, 2005.

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Geometric approximation algorithms. Providence, R.I: American Mathematical Society, 2011.

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Nikolʹskiĭ, S. M. Izbrannye trudy: V trekh tomakh. Moskva: Nauka, 2006.

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Domich, P. D. A near-optimal starting solution for polynomial approximation of a continuous function in the L. [Washington, D.C.]: U.S. Dept. of Commerce, National Bureau of Standards, 1986.

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Domich, P. D. A near-optimal starting solution for polynomial approximation of a continuous function in the L. [Washington, D.C.]: U.S. Dept. of Commerce, National Bureau of Standards, 1986.

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Hedberg, Lars Inge. An axiomatic approach to function spaces, spectral synthesis, and Luzin approximation. Providence, RI: American Mathematical Society, 2007.

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Domich, P. D. A near-optimal starting solution for polynomial approximation of a continuous function in the Lb1s norm. [Washington, D.C.]: U.S. Dept. of Commerce, National Bureau of Standards, 1986.

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Частини книг з теми "Approximation of convex function"

Neamtu, Marian. "On Approximation and Interpolation of Convex Functions." In Approximation Theory, Spline Functions and Applications, 411–18. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-011-2634-2_28.

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Awan, Muhammad Uzair, Muhammad Aslam Noor, Khalida Inayat Noor, and Themistocles M. Rassias. "Two-Dimensional Trapezium Inequalities via pq-Convex Functions." In Approximation Theory and Analytic Inequalities, 21–34. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-60622-0_2.

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Mennicken, Reinhard. "Perturbations of Semi-Fredholm Operators in Locally Convex Spaces." In Functional Analysis, Holomorphy, and Approximation Theory, 233–304. Boca Raton: CRC Press, 2020. http://dx.doi.org/10.1201/9781003072577-12.

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Jichang, Kuang. "New Trapezoid Type Inequalities for Generalized Exponentially Strongly Convex Functions." In Approximation Theory and Analytic Inequalities, 273–308. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-60622-0_15.

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Colombeau, J. F., and B. Perrot. "Convolution Equations in Spaces of Polynomials on Locally Convex Spaces." In Functional Analysis, Holomorphy, and Approximation Theory, 21–31. Boca Raton: CRC Press, 2020. http://dx.doi.org/10.1201/9781003072577-3.

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Kitahara, Kazuaki. "Approximation by vector-valued monotone increasing or convex functions." In Spaces of Approximating Functions with Haar-like Conditions, 58–77. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/bfb0091389.

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Fox, Kyle, Sungjin Im, Janardhan Kulkarni, and Benjamin Moseley. "Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions." In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 142–57. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-40328-6_11.

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Gass, Saul I., and Carl M. Harris. "Convex function." In Encyclopedia of Operations Research and Management Science, 147. New York, NY: Springer US, 2001. http://dx.doi.org/10.1007/1-4020-0611-x_166.

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Peterson, James K. "Function Approximation." In Calculus for Cognitive Scientists, 279–99. Singapore: Springer Singapore, 2016. http://dx.doi.org/10.1007/978-981-287-874-8_14.

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Abe, Shigeo. "Function Approximation." In Support Vector Machines for Pattern Classification, 395–442. London: Springer London, 2010. http://dx.doi.org/10.1007/978-1-84996-098-4_11.

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Тези доповідей конференцій з теми "Approximation of convex function"

Chickermane, Hemant, and Hae Chang Gea. "Structural Optimization Using a Generalized Convex Approximation." In ASME 1994 Design Technical Conferences collocated with the ASME 1994 International Computers in Engineering Conference and Exhibition and the ASME 1994 8th Annual Database Symposium. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/detc1994-0135.

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Abstract To reduce the computational cost of structural optimization problems, a common procedure is to generate a sequence of convex, approximate subproblems and solve them in an iterative fashion. In this paper, a new local function approximation algorithm is proposed to formulate the subproblems. This new algorithm, called Generalized Convex Approximation (GCA), uses the sensitivity information of the current and previous design points to generate a sequence of convex, separable subproblems. This algorithm gives very good local approximations and leads to faster convergence for structural optimization problems. Several numerical results of structural optimization problems are presented.

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Xu, Yi, Yilin Zhu, Zhongfei Zhang, Yaqing Zhang, and Philip S. Yu. "Convex Approximation to the Integral Mixture Models Using Step Functions." In 2015 IEEE International Conference on Data Mining (ICDM). IEEE, 2015. http://dx.doi.org/10.1109/icdm.2015.48.

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Al-Muhja, Malik Saad, Habibulla Akhadkulov, and Nazihah Ahmad. "On weighted approximation of (Co)convex functions with polynomials of varying degrees." In INTERNATIONAL UZBEKISTAN-MALAYSIA CONFERENCE ON “COMPUTATIONAL MODELS AND TECHNOLOGIES (CMT2020)”: CMT2020. AIP Publishing, 2021. http://dx.doi.org/10.1063/5.0059023.

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Caruntu, Dumitru I. "On Non-Axisymmetrical Transverse Vibrations of Circular Plates of Convex Parabolic Thickness Variation." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-62020.

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This paper presents an approach for finding the solution of the partial differential equation of motion of the non-axisymmetrical transverse vibrations of axisymmetrical circular plates of convex parabolical thickness. This approach employed both the method of multiple scales and the factorization method for solving the governing partial differential equation. The solution has been assumed to be harmonic angular-dependent. Using the method of multiple scales, the partial differential equation has been reduced to two simpler partial differential equations which can be analytically solved and which represent two levels of approximation. Solving them, the solution resulted as first-order approximation of the exact solution. Using the factorization method, the first differential equation, homogeneous and consisting of fourth-order spatial-dependent and second-order time-dependent operators, led to a general solution in terms of hypergeometric functions. Along with given boundary conditions, the first differential equation and the second differential equation, which was nonhomogeneous, gave respectively so-called zero-order and first-order approximations of the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes. Any boundary conditions could be considered. The influence of Poisson’s ratio on the natural frequencies and mode shapes could be further studied using the first-order approximations reported here. This approach can be extended to nonlinear, and/or forced vibrations.

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Dürr, Christoph, Nguyen Kim Thang, Abhinav Srivastav, and Léo Tible. "Non-monotone DR-submodular Maximization over General Convex Sets." In Twenty-Ninth International Joint Conference on Artificial Intelligence and Seventeenth Pacific Rim International Conference on Artificial Intelligence {IJCAI-PRICAI-20}. California: International Joint Conferences on Artificial Intelligence Organization, 2020. http://dx.doi.org/10.24963/ijcai.2020/297.

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Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.

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Kumar, Ashok V., and David C. Gossard. "A Sequential Approximation Method for Structural Optimization Using Logarithmic Barriers." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0366.

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Abstract A sequential approximation technique for non-linear programming is presented here that is particularly suited for problems in engineering design and structural optimization, where the number of variables are very large and function and sensitivity evaluations are computationally expensive. A sequence of sub-problems are iteratively generated using a linear approximation for the objective function and setting move limits on the variables using a barrier method. These sub-problems are strictly convex. Computation per iteration is significantly reduced by not solving the sub-problems exactly. Instead at each iteration, a few Newton-steps are taken for the sub-problem. A criteria for moving the move limit, is described that reduces or eliminates stepsize reduction during line search. The method was found to perform well for unconstrained and linearly constrained optimization problems. It requires very few function evaluations, does not require the hessian of the objective function and evaluates its gradient only once per iteration.

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Xiao, Yichi, Zhe Li, Tianbao Yang, and Lijun Zhang. "SVD-free Convex-Concave Approaches for Nuclear Norm Regularization." In Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/436.

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Minimizing a convex function of matrices regularized by the nuclear norm arises in many applications such as collaborative filtering and multi-task learning. In this paper, we study the general setting where the convex function could be non-smooth. When the size of the data matrix, denoted by m x n, is very large, existing optimization methods are inefficient because in each iteration, they need to perform a singular value decomposition (SVD) which takes O(m^2 n) time. To reduce the computation cost, we exploit the dual characterization of the nuclear norm to introduce a convex-concave optimization problem and design a subgradient-based algorithm without performing SVD. In each iteration, the proposed algorithm only computes the largest singular vector, reducing the time complexity from O(m^2 n) to O(mn). To the best of our knowledge, this is the first SVD-free convex optimization approach for nuclear-norm regularized problems that does not rely on the smoothness assumption. Theoretical analysis shows that the proposed algorithm converges at an optimal O(1/\sqrt{T}) rate where T is the number of iterations. We also extend our algorithm to the stochastic case where only stochastic subgradients of the convex function are available and a special case that contains an additional non-smooth regularizer (e.g., L1 norm regularizer). We conduct experiments on robust low-rank matrix approximation and link prediction to demonstrate the efficiency of our algorithms.

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Imamoto, Alyson, and Benjamim Tang. "A Recursive Descent Algorithm for Finding the Optimal Minimax Piecewise Linear Approximation of Convex Functions." In Advances in Electrical and Electronics Engineering - IAENG Special Edition of the World Congress on Engineering and Computer Science (WCECS). IEEE, 2008. http://dx.doi.org/10.1109/wcecs.2008.42.

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Ivanenko, Yevhen, and Sven Norde. "Approximation of dielectric spectroscopy data with Herglotz functions on the real line and convex optimization." In 2016 International Conference on Electromagnetics in Advanced Applications (ICEAA). IEEE, 2016. http://dx.doi.org/10.1109/iceaa.2016.7731537.

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Zhang, Jinhuan, Margaret M. Wiecek, and Wei Chen. "Local Approximation of the Efficient Frontier in Robust Design." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/dac-8566.

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Abstract The multiple quality aspects of robust design have brought more and more attention in the advancement of robust design methods. Neither the Taguchi’s signal-to-noise ratio nor the weighted-sum method is adequate in addressing designer’s preference in making tradeoffs between the mean and variance attributes. An interactive multiobjective robust design procedure that follows upon the developments on relating utility function optimization to a multiobjective programming method has been proposed by the authors. This paper is an extension of our previous work on this topic. It presents a formal procedure for deriving a quadratic utility function at a candidate solution as an approximation of the efficient frontier to explore alternative robust design solutions. The proposed procedure is investigated at different locations of candidate solutions, with different ranges of interest, and for efficient frontiers with both convex and nonconvex behaviors. This quadratic utility function provides a decision maker with new information regarding how to choose a most preferred Pareto solution. As an integral part of the interactive robust design procedure, the proposed method assists designers in adjusting the preference structure and exploring alternative efficient robust design solutions. It eliminates the needs of solving the original bi-objective optimization problem repeatedly using new preference structures, which is often a computationally expensive task for problems in a complex domain. Though demonstrated for robust design problems, the principle is also applicable to any bi-objective optimization problems.

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Звіти організацій з теми "Approximation of convex function"

Ward, Rachel A. Reliable Function Approximation and Estimation. Fort Belvoir, VA: Defense Technical Information Center, August 2016. http://dx.doi.org/10.21236/ad1013972.

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Lin, Daw-Tung, and Judith E. Dayhoff. Network Unfolding Algorithm and Universal Spatiotemporal Function Approximation. Fort Belvoir, VA: Defense Technical Information Center, January 1994. http://dx.doi.org/10.21236/ada453011.

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Tong, C. An Adaptive Derivative-based Method for Function Approximation. Office of Scientific and Technical Information (OSTI), October 2008. http://dx.doi.org/10.2172/945874.

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Mushtaq, Saima, Mohsan Raza, and Wasim ul Haq. Sufficient Conditions for a Meromorphic Function to Be p-valent Starlike or Convex. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, December 2019. http://dx.doi.org/10.7546/crabs.2019.12.01.

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Bai, Z. D., C. R. Rao, and L. C. Zhao. MANOVA Type Tests Under a Convex Discrepancy Function for the Standard Multivariate Linear Model. Fort Belvoir, VA: Defense Technical Information Center, January 1993. http://dx.doi.org/10.21236/ada271031.

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Nagayama, Shinobu, Tsutomu Sasao, and Jon T. Butler. Programmable Numerical Function Generators Based on Quadratic Approximation: Architecture and Synthesis Method. Fort Belvoir, VA: Defense Technical Information Center, January 2006. http://dx.doi.org/10.21236/ada599939.

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Potamianos, Gerasimos, and John Goutsias. Stochastic Simulation Techniques for Partition Function Approximation of Gibbs Random Field Images. Fort Belvoir, VA: Defense Technical Information Center, June 1991. http://dx.doi.org/10.21236/ada238611.

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Longcope, Donald B. ,. Jr, Thomas Lynn Warren, and Henry Duong. Aft-body loading function for penetrators based on the spherical cavity-expansion approximation. Office of Scientific and Technical Information (OSTI), December 2009. http://dx.doi.org/10.2172/986592.

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Tang, Ping Tak Peter. Strong uniqueness of best complex Chebyshev approximation to analytic perturbations of analytic function. Office of Scientific and Technical Information (OSTI), March 1988. http://dx.doi.org/10.2172/6357493.

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Kitago, Masaki, Shunsuke Ehara, and Ichiro Hagiwara. Efficient Construction of Finite Element Model by Implicit Function Approximation of CAD Model. Warrendale, PA: SAE International, May 2005. http://dx.doi.org/10.4271/2005-08-0127.

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