Dissertations / Theses on the topic 'Concave and convex functions'
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Ghadiri, Hamid Reza. "Convex Functions." Thesis, Karlstads universitet, Fakulteten för teknik- och naturvetenskap, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-7473.
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Zagar, Susanna Maria. "Convex functions." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/986.
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Ma, Jie. "Heterogeneous nucleation on convex and concave spherical surfaces." Thesis, University of Portsmouth, 2008. http://eprints.port.ac.uk/15503/.
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Nucleation is a phenomenon of broad scientific interest and technological importance. It refers to the very early stages of the formation of a new phase, which can be solid, gaseous, and liquid, in a metastable parent phase. Most nucleation occurs heterogeneously unless the metastable parent phase from which the nuclei form is perfectly homogeneous and isolated from any catalyzing medium. This thesis project deals with heterogeneous nucleation on convex and concave spherical surfaces. In brief, the main achievements of the project are • An innovative analytical thermodynamic approach has been invented, which enabled rigorous thermodynamic formation of the energy barrier to nucleation, the critical radius and the shape factor, for nucleation on both convex and on concave surfaces. The rigorous thermodynamic analyses conducted have revealed a number of features for heterogeneous nucleation on convex and concave spherical surfaces as opposed to heterogeneous nucleation on a flat substrate surface. These are described in detail in Chapters 2 and 4. • Nucleation is the easiest on a concave spherical surface while it is the most difficult on a convex spherical surface assuming the contact angle and the critical embryo radius are the same. Nucleation on a flat substrate surface falls in between. This is determined by their shape factors. • The ratio R = 20r*, where R is the radius of the spherical substrate and r* is the critical embryo radius, (always define the symbols when they are first used. No one knows what they mean. R could be gas constant) can be regarded as a sufficiently accurate boundary that distinguishes between spherical and flat substrates for heterogeneous nucleation. • The investigation of the growth of crystal nuclei on a convex spherical surface has revealed that no growth barrier exists to the growth of a nucleus on a convex spherical substrate surface regardless of R < r* or R > r*. All nuclei formed on a convex spherical substrate surface are thus transformation nuclei. Turnbull’s transformation nucleus model or the recently developed free growth model does not apply to the growth of a spherical-cap nucleus on a convex spherical surface. • For heterogeneous nucleation in undercooled liquid metals, the cap thickness varies in a very narrow range by just a few angstroms and is typically about a few atomic layers thick according to Turnbull’s nucleation rate equation. The variations in the cap thickness are generally limited to less than 1Å when the contact angle Ɵ is varied in the range from 0° to 45°. It is anticipated that these findings will help to better understand the classical models for heterogeneous nucleation and provide new insights into the control of heterogeneous nucleation where convex or concave spherical substrate surfaces are involved.
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Choe, Byung-Tae. "Essays on concave and homothetic utility functions." Uppsala : Stockholm, Sweden : s.n. ; Distributor, Almqvist & Wiksell International, 1991. http://catalog.hathitrust.org/api/volumes/oclc/27108685.html.
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Brennan, Derek. "Convex functions, majorization properties and the convex conjugate transform." Thesis, McGill University, 2005. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=81603.
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Let Kw(t) := inf∥P∥ 22≤t 01 |1 - P(theta)|2w(theta) dtheta, where w ≥ 0 is a decreasing function and P ∈ L2([0,1], Leb.).Then we have Kw ≤ Kv ↔01 1wq +l dtheta ≥ 01 1vq+l dtheta ∀ lambda ≥ 0.
We present two similar proofs of this result, which are analogous to the well-known majorization theorem: Let v, w ≥ 0 be decreasing functions, and suppose 01 w(theta)dtheta = 01 v(theta)dtheta.
Then Fw ≤ Fv ↔01 (w(theta) - x)+ dtheta ≥ 01 (v(theta) - x)+ dtheta ∀ x ≥ 0, where F w(t) = t1 w(theta)dtheta. Since our proofs of this result rely mainly on the convex conjugate transform, or Legendre transform, we include an exposition of convex functions and convex conjugate transforms.
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Caglar, Umut. "Divergence And Entropy Inequalities For Log Concave Functions." Case Western Reserve University School of Graduate Studies / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=case1400598757.
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Saysupan, Sutthilak. "Design and Fabrication of convex and concave Lenses made of Transparent Liquids." Thesis, KTH, Skolan för elektroteknik och datavetenskap (EECS), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-295596.
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This report studies about optical convex and concavelenses, made of liquid materials. Design proposals of the liquidlenses and the required supporting structure (container), as wellas manufacturing method have been investigated. 8 lenses aredesigned: 4 convex and 4 concave. The theoretical expectationsare validated by simulation and experimental results. The methodhave both advantages and disadvantages. The materials in thelenses are water, syrup, benzyl benzoate and bromone naphtha-lene.Undersökning gällande optiska linser konvexa och konkava, bestående av flytande material. Designförslag av linser och skall, samt tillverkningsmetod har undersökts. De teoretiska förväntningarna validerades genom simulering och experimentella resultat. Metoden visas har både fördelar och nackdelar. Material i de linserna som vi har undersökt är vatten, sockerlösning, bensylbensoat och Bromonaftalen.
Kandidatexjobb i elektroteknik 2020, KTH, Stockholm
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Morales, J. M. "Structured sparsity with convex penalty functions." Thesis, University College London (University of London), 2012. http://discovery.ucl.ac.uk/1355964/.
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We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. This problem is relevant in Machine Learning, Statistics and Signal Processing. It is well known that a linear regression can benefit from knowledge that the underlying regression vector is sparse. The combinatorial problem of selecting the nonzero components of this vector can be “relaxed” by regularising the squared error with a convex penalty function like the ℓ1 norm. However, in many applications, additional conditions on the structure of the regression vector and its sparsity pattern are available. Incorporating this information into the learning method may lead to a significant decrease of the estimation error. In this thesis, we present a family of convex penalty functions, which encode prior knowledge on the structure of the vector formed by the absolute values of the regression coefficients. This family subsumes the ℓ1 norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We establish several properties of these penalty functions and discuss some examples where they can be computed explicitly. Moreover, for solving the regularised least squares problem with these penalty functions, we present a convergent optimisation algorithm and proximal method. Both algorithms are useful numerical techniques taylored for different kinds of penalties. Extensive numerical simulations highlight the benefit of structured sparsity and the advantage offered by our approach over the Lasso method and other related methods, such as using other convex optimisation penalties or greedy methods.
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Edwards, Teresa Dawn. "The box method for minimizing strictly convex functions over convex sets." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/30690.
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Semu, Mitiku Kassa. "On minimal pairs of compact convex sets and of convex functions /." [S.l. : s.n.], 2002. http://www.gbv.de/dms/zbw/36225754X.pdf.
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Wang, Yanhui. "Affine scaling algorithms for linear programs and linearly constrained convex and concave programs." Diss., Georgia Institute of Technology, 1996. http://hdl.handle.net/1853/24919.
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El-Ashwah, R. M. "On Bazilevic and close-to-convex functions." Thesis, Swansea University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.636777.
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Denote by S the class of functions f which are regular, normalised and univalent in the unit disc. The subclasses of S, consisting of functions which are Convex, Starlike, Close-to-Convex and Bazilevic, are denoted by C, S, K and B(α,β), respectively. In Chapter 1 known results for the class S and its subclasses, which are needed in the thesis, are listed. The class B1(α) is the subject of investigation in Chapter 2 in which results concerning coefficients and length-area estimates are given. Growth estimates for the logarithmic derivative and the second integral mean of the derivative are also considered. Further, a subclass of B_1(α) is introduced. Distortion theorems and other properties for this subclass are studied. The final chapter introduces two new classes of univalent functions, Starlike with respect to different points. These classes are subsets of K. Sharp coefficients results and area problems are given. Finally, other new classes of close-to-convex functions are introduced and several properties are proved including some results concerning integral operators.
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Nekooie, Batool. "Convex optimization involving matrix inequalities." Diss., Georgia Institute of Technology, 1994. http://hdl.handle.net/1853/13880.
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Burton, Andrew P. "Isoperimetric inequalities and applications of convex integral functions." Thesis, Keele University, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.277183.
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Fontaine, Xavier. "Sequential learning and stochastic optimization of convex functions." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM024.
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Dans cette thèse nous étudions plusieurs problèmes d'apprentissage automatique qui sont tous liés à la minimisation d'une fonction bruitée, qui sera souvent convexe.Du fait de leurs nombreuses applications nous nous concentrons sur des problèmes d'apprentissage séquentiel, qui consistent à traiter des données ``à la volée'', ou en ligne.La première partie de cette thèse est ainsi consacrée à l'étude de trois différents problèmes d'apprentissage séquentiel dans lesquels nous rencontrons le compromis classique ``exploration vs. exploitation''.Dans chacun de ces problèmes un agent doit prendre des décisions pour maximiser une récompense ou pour évaluer un paramètre dans un environnement incertain, dans le sens où les récompenses ou les résultats des différentes actions sont inconnus et bruités.Nous étudions tous ces problèmes à l'aide de techniques d'optimisation stochastique convexe, et nous proposons et analysons des algorithmes pour les résoudre.Dans la deuxième partie de cette thèse nous nous concentrons sur l'analyse de l'algorithme de descente de gradient stochastique qui est vraisemblablement l'un des algorithmes d'optimisation stochastique les plus utilisés en apprentissage automatique.Nous en présentons une analyse complète dans le cas convexe ainsi que dans certaines situations non convexes en étudiant le modèle continu qui lui est associé, et obtenons de nouveaux résultats de convergence optimauxIn this thesis we study several machine learning problems that are all linked with the minimization of a noisy function, which will often be convex.Inspired by real-life applications we focus on sequential learning problems which consist in treating the data ``on the fly'', or in an online manner.The first part of this thesis is thus devoted to the study of three different sequential learning problems which all face the classical ``exploration vs. exploitation'' trade-off.Each of these problems consists in a situation where a decision maker has to take actions in order to maximize a reward or to evaluate a parameter under uncertainty, meaning that the rewards or the feedback of the possible actions are unknown and noisy.We demonstrate that all of these problems can be studied under the scope of stochastic convex optimization, and we propose and analyze algorithms to solve them.In the second part of this thesis we focus on the analysis of the Stochastic Gradient Descent algorithm, which is likely one of the most used stochastic optimization algorithms in machine learning.We provide an exhaustive analysis in the convex setting and in some non-convex situations by studying the associated continuous-time model, and obtain new optimal convergence results
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Lin, Chin-Yee. "Interior point methods for convex optimization." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/15044.
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Junike, Gero Quintus Rudolf. "Advanced stock price models, concave distortion functions and liquidity risk in finance." Doctoral thesis, Universitat Autònoma de Barcelona, 2019. http://hdl.handle.net/10803/667194.
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Esta tesis consta de tres ensayos. En el primer ensayo, probamos empíricamente el desempeño de los precios de varios modelos financieros avanzados para opciones exóticas. Calibramos seis modelos avanzados para precios de acciones a una serie de datos de mercado reales de opciones europeas en el DAX, el índice de referencia de la Bolsa alemana. A través de una simulación de Monte Carlo, calculamos precios de opciones de barrera para todos los modelos y comparamos los precios modelados con los precios del mercado de las opciones de barrera. El modelo Bates reproduce bien los precios de las opciones de barrera. El modelo BNS sobrevalora y los modelos Lévy con cambio temporal estocástico y con efecto de palanca subestiman los precios de las opciones exóticas. Un análisis heurístico sugiere que el diferente grado de fluctuación de las trayectorias aleatorias de los modelos es el responsable de producir diferentes precios para las opciones de barrera.
En el segundo ensayo de esta tesis se examinan medidas de riesgo coherentes y funciones de distorsión cóncavas. Una familia de funciones cóncavas de distorsión es un conjunto de funciones cóncavas y crecientes, con dominio e imagen igual al intervalo unitario. Se usan las funciones de distorsión para definir medidas de riesgo coherentes. Demostramos que cualquier familia de funciones de distorsión que cumpla una ecuación de traslación, puede ser representada por una función de distribución. Una aplicación se puede encontrar en la ciencia actuarial: los principios de primas basados en los momentos son fáciles de entender, pero en general no son monótonos y no se pueden utilizar para comparar los riesgos de diferentes contratos de seguros entre sí. Nuestro teorema de representación permite comparar dos riesgos de seguros entre sí de acuerdo con un principio de primas basado en un momento, definiendo adecuadamente una medida de riesgo coherente.
En el último ensayo de esta tesis, investigamos los mercados financieros con fricciones, donde los precios de compra y venta de instrumentos financieros se describen mediante funciones de precios sublineares. Estas funciones pueden definirse recursivamente utilizando medidas de riesgo coherentes. En un modelo binomial y en la presencia de costes de transacción, demostramos la convergencia de los precios de compra y venta para varias opciones europeas y americanas, en particular opciones plain vanillas, asiáticas, lookback y de barrera. Realizamos varios experimentos numéricos para confirmar los hallazgos teóricos. Aplicamos los resultados a los datos de mercado reales de las opciones plain vanilla europeas y americanas y calculamos una liquidez implícita para describir la diferencia de precios de compra y venta. Este método describe muy bien la liquidez en comparación con el enfoque clásico de describir la diferencia entre los precios de compra y venta con las volatilidades implícitas de dichos precios.This thesis consists of three essays. In the first essay, we test empirically the pricing performance of several advanced financial models. We calibrate six advanced stock price models to a time series of real market data of European options on the DAX, a German blue chip index. Via a Monte Carlo simulation, we price barrier down-and-out call options for all models and compare the modelled prices to given real market data of the barrier options. The Bates model reproduces barrier option prices well. The BNS model overvalues and Lévy models with stochastic time-change and leverage undervalue the exotic options. A heuristic analysis suggests that the different degree of fluctuation of the random paths of the models are responsible of producing different prices for the barrier options. The second essay of this thesis discusses the relationship between coherent risk measures and concave distortion functions. A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment based premium principle by defining an appropriate coherent risk measure. In the last essay of this thesis, we investigate financial markets with frictions, where bid and ask prices of financial intruments are described by sublinear pricing functionals. Such functionals can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model in the presence of transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of European and American plain vanilla options and compute an implied liquidity to describe the bid-ask spread. This method describes liquidity over time very well, compared to the classical approach of describing the bid-ask spread by quoting bid and ask implied volatilities.
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Trienis, Michael Joseph. "Computational convex analysis : from continuous deformation to finite convex integration." Thesis, University of British Columbia, 2007. http://hdl.handle.net/2429/2799.
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After introducing concepts from convex analysis, we study how to continuously transform one convex
function into another. A natural choice is the arithmetic average, as it is pointwise continuous;
however, this choice fails to average functions with different domains. On the contrary, the proximal
average is not only continuous (in the epi-topology) but can actually average functions with
disjoint domains. In fact, the proximal average not only inherits strict convexity (like the arithmetic
average) but also inherits smoothness and differentiability (unlike the arithmetic average).
Then we introduce a computational framework for computer-aided convex analysis. Motivated
by the proximal average, we notice that the class of piecewise linear-quadratic (PLQ) functions is
closed under (positive) scalar multiplication, addition, Fenchel conjugation, and Moreau envelope.
As a result, the PLQ framework gives rise to linear-time and linear-space algorithms for convex
PLQ functions. We extend this framework to nonconvex PLQ functions and present an explicit
convex hull algorithm.
Finally, we discuss a method to find primal-dual symmetric antiderivatives from cyclically monotone
operators. As these antiderivatives depend on the minimal and maximal Rockafellar functions
[5, Theorem 3.5, Corollary 3.10], it turns out that the minimal and maximal function in [12,
p.132,p.136] are indeed the same functions. Algorithms used to compute these antiderivatives can
be formulated as shortest path problems.
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Lan, Guanghui. "Convex optimization under inexact first-order information." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29732.
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Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2009.Committee Chair: Arkadi Nemirovski; Committee Co-Chair: Alexander Shapiro; Committee Co-Chair: Renato D. C. Monteiro; Committee Member: Anatoli Jouditski; Committee Member: Shabbir Ahmed. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Potaptchik, Marina. "Portfolio Selection Under Nonsmooth Convex Transaction Costs." Thesis, University of Waterloo, 2006. http://hdl.handle.net/10012/2940.
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We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve.
We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method.
If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.
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Wattanataweekul, Hathaikarn. "Convex analysis and flows in infinite networks." Diss., Mississippi State : Mississippi State University, 2006. http://sun.library.msstate.edu/ETD-db/ETD-browse/browse.
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SAMPAIO, RAIMUNDO JOSE B. DE. "A CONTRIBUITION TO THE STUDY OF D.C.: DIFFERENCE OF TWO CONVEX FUNCTIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1990. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8617@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORUNIVERSIDADE FEDERAL DO PARANÁ
Este trabalho está dividido em duas partes. A primeira parte trata das relações entre o problema de otimização d.c. (diferença de duas funções convexas) e o problema de otimização d.c. regularizado por inf-convolução, com núcleo (2 lambda)-1 l l . l l 2 , lambda > 0. Neste sentido se generaliza a relação de TOLAND (1979): inf { g(x) - h(x) } = inf { h(asterístico (y) - g (asterístico(y) }, H H E a relação de GABAY (1982): inf { g(x) - h(x) } = inf { g lambda (x) - h lambda (x) } H H Onde g, h , são funções convexas próprias e semicontínuas inferiormente, g(asterístico), h(asterístico), são conjugadas de g e h, respectivamente, H é um espaço de Hilbert real, e g (lambda), h lambda , são as funções regularizadas respectivas de g e h, por inf-convolução com núcleo (2 lambda)-1 l l . l l 2 , lambda > 0. A segunda parte deste trabalho apresenta um algoritmo novo para tratar com o problema de otimização d.c.. Trata-se de um método de descida do tipo proximal, onde se leva em consideração separadamente as propriedades de convexidade das duas funções convexas.
The work is divided in two parts. The first part is concerned with the relationship between the d.c. optimization problem. In this sence we geralize the TOLAND´s relation (1979): inf { g(x) - h(x) } = inf { h(asteristic)(y) - g (asteristic)(y) }, H H And the GABAY´s relation (1982): inf { g(x) - h(x) } = inf { g lambda (x) - h lambda (x) } H H Where g, h, are l.s.c. convex functions, g(asteristic) and h(asteristic) are their conjugates, H is a real Hilbert space, and g lambda, h lambda, are the inf-convolution of g and h respectively, with the núcleos 8( . ) = (2 lambda)- 1 l l . l l 2 , lambda > 0. In the second part we present a new algorithm for dealing with d.c. functions. It is a descent method of proximal kind which takes in consideration the convex properties of the two convex functions separately
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LIFSCHITZ, SERGIO. "AN ALGORITHM FOR THE COMPUTATION OF SOME DISTANCE FUNCTIONS BETWEEN CONVEX POLYGONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1990. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=9412@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORApresenta-se nesta dissertação um novo algoritmo para o cálculo de algumas funções distância entre polígonos convexos, no caso geral em que os polígonos podem se interseptar, cuja complexidade linear de pior caso é melhor do que a dos algoritmos até então conhecidos na literatura. O algoritmo é baseado em um algoritmo de complexidade linear originalmente proposto para determinação da distância de Hausdorff entre polígonos convexos disjuntos e utiliza como sua principal componente um algoritmo linear para o cálculo da interseção entre polígonos convexos. A motivação para o estudo de algoritmos eficientes para este problema de cálculo de distâncias decorre de aplicações em reconhecimento de formas e superposição ótima de contornos. Resultados computacionais também são apresentados.
We present in this dissertation a new algorithm for the computation of some distance functions between convex polygons, in the general case where they can intersect, whose worst case time complexity is better than of the previously known algorithms. The algorthm is based on an algorithm originally proposed for the computation of the Hausdorff distance between disjoint polygons and uses as its main component a linear time algorithm for finding the intersection of convex polygons. The motivation for the study of efficient algorithms for this distance computation problem comes from applications in pattern recognition and contour fitting. Computatioal results are also presented.
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Rodriguez-Mancilla, Jose Ramon. "Investment under risk tolerance constraints and non-concave utility functions: implicit risks, incentives and optimal strategies." Thesis, University of British Columbia, 2007. http://hdl.handle.net/2429/31468.
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The objective of this thesis is to contribute in the understanding of both the induced
behavior and the underlying risks of a decision maker who is rewarded through option-like
compensation schemes or who is subject to risk tolerance constraints.
In the first part of the thesis we consider a risk averse investor who maximizes his
expected utility subject to a risk tolerance constraint expressed in terms of the risk measure
known as Conditional Value-at-Risk. We study some of the implicit risks associated with the
optimal strategies followed by this investor. In particular, embedded probability measures
are uncovered using duality theory and used to assess the probability of surpassing a loss
threshold defined by the risk measure known as Value-at-Risk. Using one of these embedded
probabilities, we derive a measure of the financial cost of hedging the loss exposure associated
to the optimal strategies, and we show that, under certain assumptions, it is a coherent
measure of risk.
In the second part of the thesis, we analyze the investment decisions that managers
undertake when they are paid with option-like compensation packages. We consider two
particular cases:
• We study the optimal risk taking strategies followed by a fund manager who is paid
through a relatively general option-like compensation scheme. Our analysis is developed
in a continuous-time framework that permits to obtain explicit formulas. These
are used first to analyze the incentives induced by this type of compensation schemes
and, second, to establish criterions to determine appropriate parameter values for these
compensation packages in order to induce specific manager's behaviors.
• We consider a hedge fund manager who is paid through a simple option-like compensation
scheme and whose investment universe includes options. We analyze the nature of the optimal investment strategies followed by this manager. In particular, we establish
explicit optimal conditions for option investments in terms of embedded martingale
measures that are derived using duality theory. Our analysis in this case is developed
in a discrete-time framework, which allows to consider incomplete markets and fat-tailed
distributions -such as option return distributions- in a much simpler manner
than in a continuous-time framework.Business, Sauder School of
Graduate
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Hodrea, Ioan Bogdan. "Farkas - type results for convex and non - convex inequality systems." Doctoral thesis, [S.l. : s.n.], 2008. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200800075.
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Schlee, Glen A. (Glen Alan). "Borel sets with convex sections and extreme point selectors." Thesis, University of North Texas, 1991. https://digital.library.unt.edu/ark:/67531/metadc332694/.
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In this dissertation separation and selection theorems are presented. It begins by presenting a detailed proof of the Inductive Definability Theorem of D. Cenzer and R.D. Mauldin, including their boundedness principle for monotone coanalytic operators.
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Cox, Bruce. "Applications of accuracy certificates for problems with convex structure." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/39489.
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Applications of accuracy certificates for problems with convex structure This dissertation addresses the efficient generation and potential applications of accuracy certificates in the framework of “black-box-represented” convex optimization problems - convex problems where the objective and the constraints are represented by “black boxes” which, given on input a value x of the argument, somehow (perhaps in a fashion unknown to the user) provide on output the values and the derivatives of the objective and the constraints at x. The main body of the dissertation can be split into three parts. In the first part, we provide our background --- state of the art of the theory of accuracy certificates for black-box-represented convex optimization. In the second part, we extend the toolbox of black-box-oriented convex optimization algorithms with accuracy certificates by equipping with these certificates a state-of-the-art algorithm for large-scale nonsmooth black-box-represented problems with convex structure, specifically, the Non-Euclidean Restricted Memory Level (NERML) method. In the third part, we present several novel academic applications of accuracy certificates. The dissertation is organized as follows: In Chapter 1, we motivate our research goals and present a detailed summary of our results. In Chapter 2, we outline the relevant background, specifically, describe four generic black-box-represented generic problems with convex structure (Convex Minimization, Convex-Concave Saddle Point, Convex Nash Equilibrium, and Variational Inequality with Monotone Operator), and outline the existing theory of accuracy certificates for these problems. In Chapter 3, we develop techniques for equipping with on-line accuracy certificates the state-of-the-art NERML algorithm for large-scale nonsmooth problems with convex structure, both in the cases when the domain of the problem is a simple solid and in the case when the domain is given by Separation oracle. In Chapter 4, we develop several novel academic applications of accuracy certificates, primarily to (a) efficient certifying emptiness of the intersection of finitely many solids given by Separation oracles, and (b) building efficient algorithms for convex minimization over solids given by Linear Optimization oracles (both precise and approximate). In Chapter 5, we apply accuracy certificates to efficient decomposition of “well structured” convex-concave saddle point problems, with applications to computationally attractive decomposition of a large-scale LP program with the constraint matrix which becomes block-diagonal after eliminating a relatively small number of possibly dense columns (corresponding to “linking variables”) and possibly dense rows (corresponding to “linking constraints”).
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Backlund, Ulf. "Envelopes of holomorphy for bounded holomorphic functions." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 1992. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-141155.
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Some problems concerning holomorphic continuation of the class of bounded holomorphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Gleason’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves. If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) : Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated by (z1 -p1) , ... , (zn - pn) ? A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines. Certain properties of some open sets defined by global plurisubharmonic functions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) < 0} and Eh = {{z,w) e Cn X C : h(z,w) < 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of discontinuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°. A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions.digitalisering@umu.se
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Hefer, Torsten. "Regularität von Randwerten der kanonischen Lösung der [mean partial differential operative]-Gleichung auf streng pseudokonkaven Gebieten." Bonn [Germany] : Rheinische Friedrich-Wilhelms-Universität, 1999. http://catalog.hathitrust.org/api/volumes/oclc/45761314.html.
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Knörr, Jonas [Verfasser], Andreas [Gutachter] Bernig, Raman [Gutachter] Sanyal, and Andrea [Gutachter] Colesanti. "Smooth valuations on convex functions / Jonas Knörr ; Gutachter: Andreas Bernig, Raman Sanyal, Andrea Colesanti." Frankfurt am Main : Universitätsbibliothek Johann Christian Senckenberg, 2021. http://d-nb.info/1228432546/34.
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Vargyas, Emese Tünde. "Duality for convex composed programming problems." Doctoral thesis, Universitätsbibliothek Chemnitz, 2004. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200401793.
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The goal of this work is to present a conjugate duality treatment of composed programming as well as to give an overview of some recent developments in both scalar and multiobjective optimization. In order to do this, first we study a single-objective optimization problem, in which the objective function as well as the constraints are given by composed functions. By means of the conjugacy approach based on the perturbation theory, we provide different kinds of dual problems to it and examine the relations between the optimal objective values of the duals. Given some additional assumptions, we verify the equality between the optimal objective values of the duals and strong duality between the primal and the dual problems, respectively. Having proved the strong duality, we derive the optimality conditions for each of these duals. As special cases of the original problem, we study the duality for the classical optimization problem with inequality constraints and the optimization problem without constraints. The second part of this work is devoted to location analysis. Considering first the location model with monotonic gauges, it turns out that the same conjugate duality principle can be used also for solving this kind of problems. Taking in the objective function instead of the monotonic gauges several norms, investigations concerning duality for different location problems are made. We finish our investigations with the study of composed multiobjective optimization problems. In doing like this, first we scalarize this problem and study the scalarized one by using the conjugacy approach developed before. The optimality conditions which we obtain in this case allow us to construct a multiobjective dual problem to the primal one. Additionally the weak and strong duality are proved. In conclusion, some special cases of the composed multiobjective optimization problem are considered. Once the general problem has been treated, particularizing the results, we construct a multiobjective dual for each of them and verify the weak and strong dualitiesIn dieser Arbeit wird, anhand der sogenannten konjugierten Dualitätstheorie, ein allgemeines Dualitätsverfahren für die Untersuchung verschiedener Optimierungsaufgaben dargestellt. Um dieses Ziel zu erreichen wird zuerst eine allgemeine Optimierungsaufgabe betrachtet, wobei sowohl die Zielfunktion als auch die Nebenbedingungen zusammengesetzte Funktionen sind. Mit Hilfe der konjugierten Dualitätstheorie, die auf der sogenannten Störungstheorie basiert, werden für die primale Aufgabe drei verschiedene duale Aufgaben konstruiert und weiterhin die Beziehungen zwischen deren optimalen Zielfunktionswerten untersucht. Unter geeigneten Konvexitäts- und Monotonievoraussetzungen wird die Gleichheit dieser optimalen Zielfunktionswerte und zusätzlich die Existenz der starken Dualität zwischen der primalen und den entsprechenden dualen Aufgaben bewiesen. In Zusammenhang mit der starken Dualität werden Optimalitätsbedingungen hergeleitet. Die Ergebnisse werden abgerundet durch die Betrachtung zweier Spezialfälle, nämlich die klassische restringierte bzw. unrestringierte Optimierungsaufgabe, für welche sich die aus der Literatur bekannten Dualitätsergebnisse ergeben. Der zweite Teil der Arbeit ist der Dualität bei Standortproblemen gewidmet. Dazu wird ein sehr allgemeines Standortproblem mit konvexer zusammengesetzter Zielfunktion in Form eines Gauges formuliert, für das die entsprechenden Dualitätsaussagen abgeleitet werden. Als Spezialfälle werden Optimierungsaufgaben mit monotonen Normen betrachtet. Insbesondere lassen sich Dualitätsaussagen und Optimalitätsbedingungen für das klassische Weber und Minmax Standortproblem mit Gauges als Zielfunktion herleiten. Das letzte Kapitel verallgemeinert die Dualitätsaussagen, die im zweiten Kapitel erhalten wurden, auf multikriterielle Optimierungsprobleme. Mit Hilfe geeigneter Skalarisierungen betrachten wir zuerst ein zu der multikriteriellen Optimierungsaufgabe zugeordnetes skalares Problem. Anhand der in diesem Fall erhaltenen Optimalitätsbedingungen formulieren wir das multikriterielle Dualproblem. Weiterhin beweisen wir die schwache und, unter bestimmten Annahmen, die starke Dualität. Durch Spezialisierung der Zielfunktionen bzw. Nebenbedingungen resultieren die klassischen konvexen Mehrzielprobleme mit Ungleichungs- und Mengenrestriktionen. Als weitere Anwendungen werden vektorielle Standortprobleme betrachtet, zu denen wir entsprechende duale Aufgaben formulieren
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Gu, Guohua. "Distance-two constrained labellings of graphs and related problems." HKBU Institutional Repository, 2005. http://repository.hkbu.edu.hk/etd_ra/590.
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Goebel, Rafal. "Convexity, convergence and feedback in optimal control /." Thesis, Connect to this title online; UW restricted, 2000. http://hdl.handle.net/1773/5792.
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White, Edward C. Jr. "Polar - legendre duality in convex geometry and geometric flows." Thesis, Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24689.
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Chiou, Ing-Jang, and 邱英璋. "Concave-convex functions and Lie groups." Thesis, 1998. http://ndltd.ncl.edu.tw/handle/27090919040418382430.
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碩士國立交通大學
應用數學系
87
Let V be an n-dimensional vector space, equipped with a non-degenerate symmetic bilinear form. Let G be the group of all linear mappings on V which preserves this bilinear form. In this thesis, we study the topological properties of G and its orbits, as well as the Lie algebra of G.
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Wang, Hung-Ta, and 王鴻達. "A note on integral inequalities of Hadamard type for log-convex and log-concave functions." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/y5mnn7.
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Bourn, Stephen. "Probabilistic shoot-look-shoot combat models." Thesis, 2012. http://hdl.handle.net/2440/73043.
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In military operations research the term shoot-look-shoot (SLS) describes repetitive shots at a target until the target is hit. A many-on-many SLS engagement involves multiple targets. The expected number of targets hit is of interest when the maximum number of shots is limited. For the homogeneous case an algebraic expression for expected hits is known. The expression was derived indirectly as a limited expected value function applied to a binomial distribution. For the case when shots are heterogeneous expected hits can be calculated from a known set of recursive equations. This thesis explicitly constructs a homogeneous SLS probability space using a hybrid of the binomial and negative binomial distributions. Expected hits is then
calculated directly as the expected number of successes. Similarly an explicit heterogeneous SLS probability space is constructed and used to derive an algebraic expression for expected hits. The many-on-many SLS model is then enhanced to explicitly include weapons, where each weapon is characterised by its maximum number of shots and stochastic availability rate in addition to the single shot probability of a hit. Both the homogeneous and heterogeneous cases are considered. A generalised result concerning constrained optimisation of concave
functions was proved and applied to show that in the homogeneous case the expected number of hits is maximised when shots are evenly distributed amongst weapons. A similar tendency for the heterogeneous case has been successfully applied in the Air Defence Command Post Automation (ADCPA) software package to optimise the deployment of surface-to-air missile fire units.
Three other noteworthy results are as follows. A continuous function is derived that coincides with expected hits for homogenous SLS distributions as the number of targets and maximum number of shots varies. Secondly for any distribution based on a sequence of Bernoulli trials it is shown that the expected number of successes, failures and trials have common ratios determined by the single trial probability of success. Finally a hybrid of the gamma and Poisson distributions is presented as a limiting case of the homogeneous SLS distribution.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2012
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Pivovarov, Peter. "Volume distribution and the geometry of high-dimensional random polytopes." Phd thesis, 2010. http://hdl.handle.net/10048/1170.
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Thesis (Ph. D.)--University of Alberta, 2010.Title from pdf file main screen (viewed on July 13, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.
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Carretero, G. Juan Antonio. "Distance determination algorithms for convex and concave objects." Thesis, 2001. https://dspace.library.uvic.ca//handle/1828/10297.
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Determining the minimum distance between two objects is a problem that has been solved using many different approaches. Most methods proposed so far are, in essence, limited to solve the problem amongst convex polyhedra. Thus, to deal with concave objects, these methods partition concave objects into convex sub-objects and solve the convex problem between all possible sub-object combinations. This adds a large computational expense, especially when the concave objects in the scene are complicated, or when concave quadratically bound objects are to be linearized.
In this work, two optimization-based formulations are proposed to solve the minimum distance problem without the need for partitioning concave objects into convex sub-objects. The first one, referred to as the continuous approach, uses concepts of computational solid geometry in order to represent objects with concavities. On the other hand, in the second formulation, referred to as the combinatorial approach, the geometries of the objects are replaced by large sets of points arranged in surface meshes.
Since the optimization problem is not unimodal (i.e., has more than one local minimum point), global optimization techniques are used. Simulated Annealing and Genetic Algorithms, with constraint handling techniques such as penalty and repair strategies are used in the continuous approach. In order to eliminate the computational expense of determining the feasibility of every trial point, the combinatorial approach replaces the objects' geometry by a set of points on the surface of each object. This reduces the minimum distance problem to an unconstrained combinatorial optimization problem where the combination of points (one on each object) that minimizes the distance between objects is the solution.
Additionally, Genetic Algorithms with niche formation techniques were developed in order to allow the distance algorithm to track multiple minima.
In a series of numerical examples, a preliminary implementation of the proposed algorithms has proven to be robust and equivalent, in terms of computational efficiency, to some conventional approaches.Graduate
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Ferland, Alane Susan. "Isoperimetric inequalities and concave functions." 1999. https://scholarworks.umass.edu/dissertations/AAI9932309.
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In this thesis we prove a one parameter family of Bonnesen-style and Osserman-style discrete Sobolev inequalities which hold for a large class of concave functions. We will show that this class of concave functions include solutions of some well known second order differential equations. In particular, for the sine and cosine functions we will apply these discrete Sobolev inequalities to obtain uncountably many new Bonnesen-style and Osserman-style isoperimetric inequalities for the generalized star polygon Pn,m of type {[special characters omitted]}. We will also see that for certain classes of polygons P n we are able to give a complete solution for the existence of Bonnesen-style and Osserman-style isoperimetric inequalities. Our inequalities will be extended further to include the surfaces of constant curvature and prove global Bonnesen-style and Osserman-style inequalities for the generalized star polygon on the unit 2-sphere, hyperbolic plane and Euclidean plane. Finally, derived from the geometry of the surface, we develop Hyperbolic and Elliptic lengths and areas to prove additional interesting inequalities on the unit 2-sphere and hyperbolic plane.
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Jian-Da, Huang. "On Generalized Convex Functions." 2000. http://www.cetd.com.tw/ec/thesisdetail.aspx?etdun=U0009-0112200611284837.
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Huang, Jian-Da, and 黃建達. "On Generalized Convex Functions." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/05369104825051284346.
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碩士元智大學
工業工程研究所
88
A class of functions called weakly convex and weakly quasiconvex functions is introduced by relaxing the definitions of convex and quasiconvex functions. Necessary and sufficient conditions, under which a lower semi-continuous function on a nonempty closed convex subsets of the n-dimensional Euclidean space is convex or quasiconvex are established. Another class of functions, called $-convex and $-B-vex, is defined. Some properties for these functions are presented.
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Lin, Wen-Hua, and 林文化. "(a,b)-Convex Functions." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/15757279667918426000.
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碩士中原大學
應用數學研究所
96
Abstract The theory of (a,b)-convex functions was introduced by Norber Kuhn in 1987【1】.Kuhn focused mainly on the structure and the properties of (a,b)-convex functions.And we generalize a result raised by Zygfryd Kominek in 1992【2】.He would like to know on what conditions under which an (a,b)-convex function is a constant function. Given a function f：I→[－∞ , ∞），we define the following three sets： (Ⅰ) K’(f)＝﹛(a,b)€(0,1)×(0,1) | f is an (a,b)-convex function﹜ (Ⅱ) K(f)＝﹛a€(0,1) | f is an a-convex function﹜ (Ⅲ) A’(f)＝﹛(a,b)€(0,1)×(0,1) | f is an (a,b)-affine function﹜ We proceed to discuss the properties of these sets K’(f)、K(f) and A’(f).Then we show that，if f：I→[－∞ , ∞）is a continuous (a,b)-convex function，f is a convex function. Finally we prove that，if (a,b)€K’(f)，a≠b and a€Q ，then f is a constant function.
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"Coordinate ascent for maximizing nondifferentiable concave functions." Massachusetts Institute of Technology, Laboratory for Information and Decision Systems], 1988. http://hdl.handle.net/1721.1/3107.
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by Paul Tseng.Cover title.
Includes bibliographical references.
Partially supported by the U.S. Army Research Office (Center for Intelligent Control Systems) DAAL03-86-K-0171 Partially supported by the National Science Foundation. NSF-ECS-8519058
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Yang, Yuan-Ju, and 楊媛茹. "Study on the Convex- and Concave-Type Parallax-Barrier Autostereoscopic Displays." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/61211650459918354219.
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碩士國立臺灣大學
光電工程學研究所
104
Among the 3D imaging technology, the auto-stereoscopic 3D display technology has generally accepted because of its advantages about no need to wear glasses and various use. In recent years, the rise of flexible display make wearable displays and curved displays spring up, such as smart watch and curved smart phone and the research of the curved display is more important. Based on the bending direction, Curved displays are divided into concave-type displays and convex-type displays and provide the more natural and comfortable visual experience. The concave-type displays provide the effect of Panorama image and the convex-type provide the effect of Surrounding image. Therefore, the curved auto-stereoscopic display is highly expected in the future. For an auto-stereoscopic display, viewing position and the movable range for observers strongly affect image quality. Viewing zone describes the range that an observer could move while experiencing 3D images in front of the display. Regarding a curved auto-stereoscopic display, radius of curvature and width of the screen determine the relative degree of bending, by which the spacial light distribution is determined. In addition, this is a key factor influencing image quality, especially for the display which directs the views for left and right eyes based on binocular parallax. In this thesis, we investigate the effect of viewing zones with concave-type and convex-type in small size displays(such as mobile devices), medium size displays(such as desktop monitors) and large size displays(such as televisions) by modulating the radius of curvature. According to the results, both concave-type and convex-type can be applied for small size displays and medium size displays by different use, and the concave-type is better than convex-type for large size displays. Meanwhile, the limitation factors of the concave-type and convex-type are studied in this thesis. The purpose of this thesis is to provide a reference for designing the parameters of curved auto-stereoscopic displays in the future.
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Shu-Chi, Hsu, and 許淑琪. "On (a,b)-convex functions." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/49436620216066290040.
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碩士中原大學
數學系
88
This thesis discusses so-called (a, b)-convex functions which was first raised by Norbert Kuhn in 1987 (see[1]). Generally speaking, "good"(a, b)-convex functions are almost constant. On the other hand, it is difficult to construct a "sickly" (a, b)-convex function. For a function f:I to [-\infty, \infty), we define the sets K'(f),K(f) and A'(f) as follows : K'(f)={(a,b) in (0,1)*(0,1): f is an (a,b)-convex function} K(f)={a in (0,1) : f is an a-convex function} A'(f)={(a,b) in (0,1)*(0,1):f is an (a,b)-affine function} First, we will discuss the structure of these sets K'(f) and A'(f), and give some suitable conditions under which an (a,b)-convex function is a constant.Also the structure of a (a, b)- convex function will be understood by means of K'(f) and A'(f). We present a result which was raised by Norbert Kuhn in 1987, he used the result concerning (a,b)-affine functions to investigate (a,b)-convex functions. And prove a result : "under some suitable conditions, K'(f) is almost a field ". We generalize a result raised by Zygfrvd Kominek in 1992 : " If (a, b) in K'(f), a not equal b and a in Q then f is a constant function " .We will have the same result if we change rational number into algebraic number and b is not algebraic number. Furthermore, we will consider the set Omega_{a, b}={f:I to [-infty, infty) :f is (a, b)-convex function }. We discuss the structure of this set.In the previous sections, the function f was given and then consider the set K'(f) consisting of all ordered pairs (a, b) which makes f (a, b)-convex. Conversely, we fix (a, b) and consider the set Omega_{a, b} consisting of all (a, b) functions on I.We find Omega_{a, b} is a convex cone. And Omega_{a, b}=Omega_{1-a, 1-b}. If a function f:I to [-infty, infty) is an (a,b)-convex function, (a, b) in K'(f), a not equal b and f is a continuous function then f is a constant function.In other words, if we want to find an example of an (a, b)-convex function then f must be discontinuous in Int I.
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Tseng, Kuei-Lin, and 曾貴麟. "On Inequalities Concern Convex Functions." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/26389526707935709103.
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博士淡江大學
數學學系
88
This dissertation consists of three chapters. The first chapter, we introduce some definitions and some foundational theorems which will be used thereafter. In the second chapter, we classify inequalities of means in discrete form derived by Ky Fan, Wang-Wang, Wang-Yang, Chong, Mitrinovic-Pečarić, Slater and Alzer into two forms. That is Form1: If M is a real linear space, U a nonempty convex set in M, ■is a real-valued function with ■ (J is an interval in R) and ■is a continuous strictly increasing function such that■is convex, then ■, (1) where ■, ■■with■. Form2: Let ■ be defined as in Form 1, and let M=R, U be an interval in R. Suppose ■ is differentiable on U with ■. Then ■ (2) We construct a continuous increasing convex function F on [0,1] in Form 1 such that ■ (3) and we also construct a continuous increasing convex function E on [0,1] in Form 2 such that ■. (4) Hence the inequalities (3) and (4) refine (1) and (2), respectively. Under different special M, U, J, f, g, (3) and (4) can refine and generalize some inequalities which found by Ky Fan, Wang-Wang, Wang-Yang, Chong, Mitrinovic-Pečarić, Slater and Alzer, respectively. In chapter 3, we investigate the inequalities of means in integral form. We start with Hermite-Hadamard inequalities and Fejér inequalities; i.e. ■ (5) and ■ (6) where■is a convex function and■is nonnegative integrable function such that g is symmetric to■. Some refinements and generalizations of (5) and (6) are found by Brenner-Alzer, Dragomir, Yang-Hong and Yang- Wang, respectively. Now we define P, Q on [0,1] such that ■ ■. (7) The inequalities (7) refine the inequalities (6) and the inequalities (5). Also, the inequalities (7) generalize some results established by Brenner-Alzer, Dragomir, Yang-Hong and Yang- Wang, respectively. 摘要 і 摘要(英文) ііі PartⅠ中文 第一章 定義與前言 1 §1‧1 加權與非加權平均數 1 §1‧2 單變數之凸函數 3 §1‧3 實線性空間上之凸函數 6 第二章 有關平均數之不等式 9 §2‧1 Ky Fan與Wang-Wang不等式 9 §2‧2 與平均數相關之不等式 14 §2‧3 有關Ky Fan與Wang-Wang不等式的一般化與細分 16 第三章 有關Hermite-Hadamard不等式之積分型態不等式 34 §3‧1 Hermite-Hadamard不等式與其細分 34 §3‧2 Hermite-Hadamard不等式之進一步細分 36 §3‧3 有關重積分的Hermite-Hadamard不等式 43 參考文獻為 47 PartⅡ英文 Contents Chapter 1. Definitions and Preliminaries 51 §1.1. Weighted and Unweighted Means 51 §1.2. One-Variable Convex Functions 53 §1.3. Convex Functions on a Normed Linear Space 56 Chapter 2. Inequalities Related To Means 59 §2.1. Ky Fan and Wang-Wang’s Inequalities 59 §2.2. Some Inequalities Related To Means 65 §2.3. Generalization and Refinement of Ky Fan and Wang-Wang’s Inequalities 67 Chapter 3. On Certain Integral Inequalities Related to Hermite- Hadamard’s Inequalities 85 §3.1. Hermite-Hadamard’s Inequalities and Their Refinements 85 §3.2. Further Refinements of Hermite-Hadamard’s Inequalities 87 §3.3. Hermite-Hadamard’s Inequalities for Multiple Integrals 95 References 99
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LIN, LAI-JU, and 林來居. "Optimization for convex set functions." Thesis, 1987. http://ndltd.ncl.edu.tw/handle/02933557435255019699.
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Huang, Hung-Yi, and 黃弘毅. "Some Inequalities For Convex Functions." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/99108769118578037817.
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碩士淡江大學
中等學校教師在職進修數學教學碩士學位班
102
Hermite-Hadamard’s inequality(or Hadamard’s inequality).Since it’s discovery in 1883,Hadamard’s inequality [3] has proven to be one of the most useful inequality in mathematical analysis. A number of papers have been written on this inequality providing new proofs, noteworthy generalizations and numerous application, see [1-4] and the references cited there in. The main purpose of this paper is to establish some generalizations of Theorem P.
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YU, CHUN HSIUNG, and 游俊雄. "Some Inequalities For Convex Functions." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/17797741379061409536.
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