The problem of distributed beamforming is considered for a network which consists of a transmitter, a receiver, and r relay nodes. Assuming that the second order statistics of the channel coefficients are available, w...
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ISBN:
(纸本)9781424414833
The problem of distributed beamforming is considered for a network which consists of a transmitter, a receiver, and r relay nodes. Assuming that the second order statistics of the channel coefficients are available, we design a distributed beamforming technique via maximization of the receiver signal-to-noise ratio (SNR) subject to individual relay power constraints. We show that using semi-definite relaxation, this SNR maximization can be turned into a convex feasibility semi-definite programming problem, and therefore, it can be efficiently solved using interior point methods. We also obtain a performance bound for the semi-definite relaxation and show that the semi-definite relaxation approach provides a c-approximation to the (nonconvex) SNR maximization problems where c = O((log r)(-1)) and r is the number of relays.
Moment Problems with Applications to Value-At-Risk and Portfolio ManagementByRuilin TianMay 2008Committee Chair:Dr. Samuel H. CoxMajor Department:Risk Management and Insurance My dissertation provides new applic...
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Moment Problems with Applications to Value-At-Risk and Portfolio ManagementByRuilin TianMay 2008Committee Chair: Dr. Samuel H. CoxMajor Department: Risk Management and Insurance My dissertation provides new applications of moment theory and optimization to financial and insurance risk management. In the investment and managerial areas, one often needs to determine some measure of risk, especially the risk of extreme events. However, complete information of the underlying outcomes is usually unavailable; instead one has access to partial information such as the mean, variance, mode, or range. In Chapters 2 and 3, we find the semiparametric upper and lower bounds for the value-at-risk (VaR) with incomplete information, that is, moments of the underlying distribution. When a single variable is concerned, bounds on VaR are computed to obtain a 100% confidence interval. When the sample financial data have a global maximum, we show that unimodal assumption tightens the optimal bounds. Next we further analyze a function of two correlated random variables. Specifically, we find bounds on the probability of two joint extreme events. When three or more variables are involved, the multivariate problem can sometimes be converted to a single variable problem. In all cases, we use the physical measure rather than the commonly used equivalent pricing probability measure. In addition to solving these problems using the traditional approach based on the geometry of a moment problem, a more efficient method is proposed to solve a general class of moment bounds via semidefinite programming. In the last part of the thesis, we apply optimization techniques to improve financial portfolio risk management. Instead of considering VaR, we work with a coherent risk measure, the conditional VaR (CVaR). As an extension of Krokhmal et al. (2002), we impose CVaR-related functions to the portfolio selection problem. The CVaR approach sets a β-level CVaR as the objective function an
The design of multivariable PID controllers which guarantee the stability of closed loop systems, have a fixed attenuation degree, and robust performance specifications is described. The design of such a multivariable...
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The design of multivariable PID controllers which guarantee the stability of closed loop systems, have a fixed attenuation degree, and robust performance specifications is described. The design of such a multivariable PID controller is transformed to a static output feedback (SOF) problem. The problem of system control is converted into a semi-definite programming problem with LMI constraints, and then by using linear matrix inequalities (LMI) and bilinear matrix inequalities (BMI), the required algorithm is derived and the parameters of the PID controllers can be obtained by further transforms. The validity and robustness of the algorithm is demonstrated by simulation of a numerical example in chemical industry process control.
A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice a...
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A wide variety of problems involving analysis of systems can be rewritten as a semidefinite program. When solving these problems optimization algorithms are used. Large size makes the problems unsolvable in practice and computationally more effective solvers are needed. This paper investigates how to exploit structure and problem knowledge in some control applications. It is shown that inexact search directions are useful to reduce the computational burden and that operator formalism can be utilized to derive tailored calculations.
This paper presents a proof that the use of polynomial Lyapunov functions is not conservative for studying exponential stability properties of nonlinear ordinary differential equations on bounded regions. The main res...
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This paper presents a proof that the use of polynomial Lyapunov functions is not conservative for studying exponential stability properties of nonlinear ordinary differential equations on bounded regions. The main result implies that if there exists an n-times continuously differentiable Lyapunov function which proves exponential decay on a bounded subset of ℝ n , then there exists a polynomial Lyapunov function which proves that same rate of decay on the same region. Our investigation is motivated by the use of semidefinite programming to construct polynomial Lyapunov functions for delayed and nonlinear systems of differential equations.
Fundamental matrix estimation is a central problem in computer vision and forms the basis of tasks such as stereo imaging and structure from motion. A new method for the estimation of the fundamental matrix from point...
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ISBN:
(纸本)9781424421138
Fundamental matrix estimation is a central problem in computer vision and forms the basis of tasks such as stereo imaging and structure from motion. A new method for the estimation of the fundamental matrix from point correspondences is presented. The minimization of an objective function closer to the geometric distance is performed based L. minimization framework. The fundamental matrix is optimally computed with taking into account the rank-two constraint, and the method is no need for normalization of the image coordinates. It is shown how this nonlinearly estimating the fundamental matrix can be solved avoiding local minima by using semidefinite programming. Experiments on real images show that this method provides a more accurate estimate of the fundamental matrix and superior to previous approaches.
semidefinite programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the l...
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ISBN:
(纸本)9781605580470
semidefinite programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of semidefinite *** this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite *** this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every ***, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.
We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an *** introduce a new analysis of the standard SDP in this case that involves correlations among dist...
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ISBN:
(纸本)9781605580470
We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an *** introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.
We describe an SDP relaxation based method for the position estimation problem in wireless sensor networks. The optimization problem is set up so as to minimize the error in sensor positions to fit distance measures. ...
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ISBN:
(纸本)1581138466
We describe an SDP relaxation based method for the position estimation problem in wireless sensor networks. The optimization problem is set up so as to minimize the error in sensor positions to fit distance measures. Observable gauges are developed to check the quality of the point estimation of sensors or to detect erroneous sensors. The performance of this technique is highly satisfactory compared to other techniques. Very few anchor nodes are required to accurately estimate the position of all the unknown nodes in a network. Also the estimation errors are minimal even when the anchor nodes are not suitably placed within the network or the distance measurements are noisy.
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