En este trabajo se estudia el problema de optimización mín xES f(x) donde S es un subconjunto convexo en un espacio normado X f : X (flecha funcional) R. Asimismo, se presenta una extensión del teorema de Kuhn-Tucker que resuelve el problema de minimización sobre el conjunto S = {x E S/g(x) E -C donde C ∧ h(x) = 0Z}es un cono de orden y h, g dos funcionales Fréchet diferenciables. / Tesis

This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.

Es werden verschiedene Verfahren zur Lösung verallgemeinerter Nash-Gleichgewichtsprobleme mit dem Schwerpunkt auf deren globaler Konvergenz entwickelt. Ein globalisiertes Newton-Verfahren zur Berechnung normalisierter Lösungen, ein nichtglattes Optimierungsverfahren basierend auf einer unrestringierten Umformulierung des spieltheoretischen Problems, und ein Minimierungsansatz sowei eine Innere-Punkte-Methode zur Lösung der gemeinsamen Karush-Kuhn-Tucker-Bedingungen der Spieler werden theoretisch untersucht und numerisch getestet. Insbesondere das Innere-Punkte Verfahren erweist sich als das zur Zeit wohl beste Verfahren zur Lösung verallgemeinerter Nash-Gleichgewichtsprobleme. / In this thesis different algorithms for the solution of generalized Nash equilibrium problems with the focus on global convergence properties are developed. A globalized Newton method for the computation of normalized solutions, a nonsmooth algorithm based on an optimization reformulation of the game-theoretic problem, and a merit function approach and an interior point method for the solution of the concatenated Karush-Kuhn-Tucker-system are analyzed theoretically and numerically. The interior point method turns out to be one of the best existing methods for the solution of generalized Nash equilibrium problems.

In conventional wireless channel models, there is no control on the gains of different subchannels. In such channels, the transmitted signal undergoes attenuation and phase shift and is subject to multi-path propagation effects. We herein refer to such channels as passive channels. In this dissertation, we study the problem of joint power allocation and channel design for a parallel channel which conveys information from a source to a destination through multiple orthogonal subchannels. In such a link, the power over each subchannel can be adjusted not only at the source but also at each subchannel. We refer to this link as an active parallel channel. For such a channel, we study the problem of sum-rate maximization under the assumption that the source power as well as the energy of the active channel are constrained. This problem is investigated for equal and unequal noise power at different subchannels. For equal noise power over different subchannels, although the sum-rate maximization problem is not convex, we propose a closed-form solution to this maximization problem. An interesting aspect of this solution is that it requires only a subset of the subchannels to be active and the remaining subchannels should be switched off. This is in contrast with passive parallel channels with equal subchannel signal-tonoise- ratios (SNRs), where water-filling solution to the sum-rate maximization under a total source power constraint leads to an equal power allocation among all subchannels. Furthermore, we prove that the number of active channels depends on the product of the source and channel powers. We also prove that if the total power available to the source and to the channel is limited, then in order to maximize the sum-rate via optimal power allocation to the source and to the active channel, half viii ix of the total available power should be allocated to the source and the remaining half should be allocated to the active channel. We extend our analysis to the case where the noise powers are unequal over different subchannels. we show that the sum-rate maximization problem is not convex. Nevertheless, with the aid of Karush-Kuhn-Tucker (KKT) conditions, we propose a computationally efficient algorithm for optimal source and channel power allocation. To this end, first, we obtain the feasible number of active subchannels. Then, we show that the optimal solution can be obtained by comparing a finite number of points in the feasible set and by choosing the best point which yields the best sum-rate performance. The worst-case computational complexity of this solution is linear in terms of number of subchannels. / UOIT

No description available.

The United States (US) witnessed remarkable growth in annual long distance travel over the past few decades. Over half of the long distance travel in the US is made for pleasure, including visiting friends and relatives (VFR) and leisure activities. This trend could continue with increased use of information and communication technologies for socialization, and enhanced mobility being achieved using fuel-efficient (electric/hybrid) and technology enhanced vehicles. Despite these developments, and recent interest to implement alternate mass transit options to serve this market, not much exists on the measurement, analysis and modeling of long distance pleasure travel in the U.S. Statewide and national models are used to estimate long distance travel, but these are predominantly trip-based models, making it difficult to understand long distance trips as collection of household-level travel behavior. This form of travel behavior has been studied a lot in tourism, but in a piecemeal manner, such as to (from) a specific destination. Further, most of these studies are confined to analyzing leisure market, with VFR market gaining recognition only recently. In essence, annual household long distance pleasure travel behavior needs to be studied in a comprehensive manner rather than as isolated trips. This is because, most of these household travel decisions are undertaken considering their annual time and monetary budget, and their perceived cost to travel to one (or more) destination for given pleasure purpose on one (or more) occasion using a given mode of travel. Thus, the main objective of this dissertation is to develop a comprehensive behavioral model framework to analyze the above-discussed annual household long distance pleasure travel choices. To start the above effort, it is first required to collect detailed annual household travel data, last collected over two decades ago (e.g.: ATS, 1995). No such recent effort has been pursued due to the significant labor and economic resource required to undertake it. There exist recent surveys (NHTS, 2001), but collected over a shorter (four week) period, and require significant processing even to arrive at aggregate annual travel estimates. Second, besides surveys, there is a need for additional data to estimate households’ annual pleasure travel budget, and their cost to travel and stay at each of their potential destination choices, which are not readily available. Thus, as the first goal, this dissertation analyzes long distance travel reported across historical surveys (NPTS; ATS; NHTS), to understand the differences in their definition, enumeration of purpose and collection methods. The intent here is twofold, first to conceive a method to estimate annual travel from surveys with shorter collection period. Further, the second intent is to gather travel patterns from these historical datasets such that it informs the second goal of this dissertation, i.e. development of a behavioral framework to analyze annual household pleasure travel. To this effect, this research also analyzes pleasure expenditures using Consumer Expenditure Survey (CEX, BLS) data. Interestingly, the analysis reveals CEX pleasure travel expenditure pattern to be similar to the travel pattern reported for the same market segments in travel survey (ATS). Importantly, the above analysis informs the development of behavioral models, pursued as two distinct tasks to achieve the second goal. As the first task, a novel econometric model and forecasting procedure is developed to analyze a household’s annual long distance leisure travel decisions. Specifically, a households’ time spent across one (or more) destination and travel mode to such destination for leisure is modeled subject to time and money budget constraints. In this methodological framework, the destination choice is modeled as a continuous variable (time at destination) using Multiple-Discrete Continuous Extreme Value model (MDCEV). While, travel mode choice to these destination(s) are modeled as a discrete choice, through a nested Multinomial Logit Model (MNL), with price variation introduced across the above choice of destination(s) and travel modes (air/ground). This required estimating annual monetary budgets, travel cost and per night lodging cost for each sample household, with each of them having 210 potential destinations and 2 travel mode choices respectively. The second task, involved the development of a broader national model system to analyze households’ annual pleasure travel decisions such as: choice (duration) at destination(s), travel purpose (VFR or leisure), mode (airplane or auto) choice and trip frequencies to these destination(s) using the same dataset. It was modeled in two stages, with the first stage estimating households’ annual pleasure time budget using a stochastic frontier model. This budget was then used as constraint to analyze households’ annual choice of destination and purpose using a nested MDCEV-MNL model in the second stage. A log sum variable from a nested joint multinomial logit model of trip frequency and mode choice for each purpose (VFR or leisure) is also introduced as input at this stage. This model was then validated using a prediction procedure, and further applied to test a policy scenario (increase in travel cost). The above national pleasure travel demand model could be further enhanced by including monetary constraints and price variation as in the first task. Overall, the model system proposed in this dissertation forms the foundation for a national comprehensive long distance travel model. This could be achieved through inclusion of other prominent travel purpose such as business and commuting to the national travel demand model presented in this research.

Consider a lossy compression system with l distributed encoders and a centralized decoder. Each encoder compresses its observed source and forwards the compressed data to the decoder for joint reconstruction of the target signals under the mean squared error distortion constraint. It is assumed that the observed sources can be expressed as the sum of the target signals and the corruptive noises, which are generated independently from two (possibly di erent) symmetric multivariate Gaussian distributions. Depending on the parameters of such Gaussian distributions, the rate-distortion limit of this lossy compression system is characterized either completely or for a subset of distortions (including, but not necessarily limited to, those su fficiently close to the minimum distortion achievable when the observed sources are directly available at the decoder). The results are further extended to the robust distributed compression setting, where the outputs of a subset of encoders may also be used to produce a non-trivial reconstruction of the corresponding target signals. In particular, we obtain in the high-resolution regime a precise characterization of the minimum achievable reconstruction distortion based on the outputs of k + 1 or more encoders when every k out of all l encoders are operated collectively in the same mode that is greedy in the sense of minimizing the distortion incurred by the reconstruction of the corresponding k target signals with respect to the average rate of these k encoders. / Thesis / Master of Applied Science (MASc)

This work is concerned with the construction of preconditioners for indefinite linear systems. The systems under investigation arise in the numerical solution of quadratic programming problems, for example in the form of Karush--Kuhn--Tucker (KKT) optimality conditions or in interior--point methods. Therefore, the system matrix is referred to as a KKT matrix. It is not the purpose of this thesis to investigate systems arising from general quadratic programming problems, but to study systems arising in linear quadratic control problems governed by partial differential equations. The KKT matrix is symmetric, nonsingular, and indefinite. For the solution of the linear systems generalizations of the conjugate gradient method, MINRES and SYMMLQ, are used. The performance of these iterative solution methods depends on the eigenvalue distribution of the matrix and of the cost of the multiplication of the system matrix with a vector. To increase the performance of these methods, one tries to transform the system to favorably change its eigenvalue distribution. This is called preconditioning and the nonsingular transformation matrices are called preconditioners. Since the overall performance of the iterative methods also depends on the cost of matrix--vector multiplications, the preconditioner has to be constructed so that it can be applied efficiently. The preconditioners designed in this thesis are positive definite and they maintain the symmetry of the system. For the construction of the preconditioners we strongly exploit the structure of the underlying system. The preconditioners are composed of preconditioners for the submatrices in the KKT system. Therefore, known efficient preconditioners can be readily adapted to this context. The derivation of the preconditioners is motivated by the properties of the KKT matrices arising in optimal control problems. An analysis of the preconditioners is given and various cases which are important for interior point methods are treated separately. The preconditioners are tested on a typical problem, a Neumann boundary control for an elliptic equation. In many important situations the preconditioners substantially reduce the number of iterations needed by the solvers. In some cases, it can even be shown that the number of iterations for the preconditioned system is independent of the refinement of the discretization of the partial differential equation. / Master of Science

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work some classical methods for constrained nonlinear optimization are studied. The mathematical formulations for the optimization problem with equality and inequality constrained, convergence properties and algorithms are presented. Furthermore, optimality conditions of rst order (Karush-Kuhn-Tucker conditions) and of second order. These conditions are essential for the demonstration of many results. Among the methods studied, some techniques transform the original problem into an unconstrained problem (Penalty Methods, Augmented Lagrange Multipliers Method). In others methods, the original problem is modeled as one or as a sequence of quadratic subproblems subject to linear constraints (Quadratic Programming Method, Sequential Quadratic Programming Method). In order to illustrate and compare the performance of the methods studied, two nonlinear optimization problems are considered: a bi-dimensional problem and a problem of mass minimization of a coil spring. The obtained results are analyzed and confronted with each other. / Neste trabalho são estudados alguns métodos clássicos de otimização restrita não linear. São abordadas a formulação matemática para o problema de otimização com restrições de igualdade e desigualdade, propriedades de convergência e algoritmos. Além disso, são relatadas as condições de otimalidade de primeira ordem (condições de Karush-Kuhn-Tucker) e de segunda ordem. Estas condições são essenciais para a demonstração de muitos resultados. Dentre os métodos estudados, algumas técnicas transformam o problema original em um problema irrestrito (Métodos de Penalidade, Método dos Multiplicadores de Lagrange Aumentado). Em outros métodos, o problema original é modelado como um ou uma seqüência de subproblemas quadráticos sujeito _a restrições lineares (Método de Programação Quadrática, Método de Programação Quadrática Seqüencial). A fim de ilustrar e comparar o desempenho dos métodos estudados são considerados dois problemas de otimização não linear: um problema bidimensional e o problema de minimização da massa de uma mola helicoidal. Os resultados obtidos são examinados e confrontados entre si. / Mestre em Matemática

Orientador: Valeriano Antunes de Oliveira / Resumo: Neste trabalho, primeiramente, serão apresentados problemas de otimização nos quais a função objetivo é de múltiplas variáveis e de valor intervalar e as restrições de desigualdade são dadas por funcionais clássicos, isto é, de valor real. Serão dadas as condições de otimalidade usando a E−diferenciabilidade e, depois, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade usando a gH−diferenciabilidade total são do tipo KKT e as suficientes são do tipo de convexidade generalizada. Em seguida, serão estabelecidos problemas de controle ótimo nos quais a funçãao objetivo também é com valor intervalar de múltiplas variáveis e as restrições estão na forma de desigualdades e igualdades clássicas. Serão fornecidas as condições de otimalidade usando o conceito de Lipschitz para funções intervalares de várias variáveis e, logo, a gH−diferenciabilidade total das funções com valor intervalar de várias variáveis. As condições necessárias de otimalidade, usando a gH−diferenciabilidade total, estão na forma do célebre Princípio do Máximo de Pontryagin, mas desta vez na versão intervalar. / Abstract: In this work, firstly, it will be presented optimization problems in which the objective function is interval−valued of multiple variables and the inequality constraints are given by classical functionals, that is, real−valued ones. It will be given the optimality conditions using the E−differentiability and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability are of KKT−type and the sufficient ones are of generalized convexity type. Next, it will be established optimal control problems in which the objective function is also interval−valued of multiple variables and the constraints are in the form of classical inequalities and equalities. It will be furnished the optimality conditions using the Lipschitz concept for interval−valued functions of several variables and then the total gH−differentiability of interval−valued functions of several variables. The necessary optimality conditions using the total gH−differentiability is in the form of the celebrated local Pontryagin Maximum Principle, but this time in the intervalar version. / Doutor