With the rapid development of wireless technologies, the demand for positioning services has grown dramatically over the past three decades. The Global Positioning System (GPS) is widely used in wireless devices for positioning purposes. However, in addition to having bulky and expensive equipment, GPS receivers do not operate properly in dense and indoor environments. Difficulties in using GPS lead us to use sensor localization in which the position information is obtained from the measurements collected within the network without the aid of external resources. Sensor localization has been a great topic of interest during past decades. Although many positioning algorithms have been developed previously in the literature, positioning is still a challenging task. There are many factors that can affect the positioning performance if they are neglected or not treated properly. These factors introduce many nuisance parameters which need to be either estimated or considered when the location is estimated. In this work, we exploit cooperative localization as a recent and trending technology and semidefinite programming (SDP) as a powerful tool in our research. Cooperative localization has several advantages over the traditional noncooperative localization in terms of positioning accuracy and localizability. Cooperation is also highly beneficial for networks with few anchor nodes and low communication range. On the other hand, SDP provides an alternative solution to the optimal maximum-likelihood (ML) estimation. Unlike in the ML estimator, convergence to the global minimum is guaranteed in SDP. It also has significantly lower complexity especially for cooperative networks in exchange for small performance degradation. Using these two concepts, four open problems within the area of cooperative localization and tracking in the presence of nuisance parameters are addressed. In particular, we focus on cooperative received signal strength-based localization when the propagation parameters including path-loss exponent and transmit powers are unknown. Cooperative time-of-arrival-based localization in harsh environments in the presence of severe non-line-of-sight (NLOS) propagation is also investigated. Cooperative localization in asynchronous networks is studied where the clock parameters are considered as nuisance parameters and the focus is on a joint synchronization and localization approach. Lastly, source tracking in NLOS environments is studied where source nodes are mobile and their status changes rapidly from LOS to NLOS and vice versa. / Ph. D.

Sze, Kam Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 49-53). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

"Experts in information theory have long been interested in the maximal size, A(n, d), of a binary error-correcting code of length n and minimum distance d, The problem of determining A(n, d) involves both the construction of good codes and the search for good upper bounds. For quite some time now, Delsarte's linear programming approach has been the dominant approach to obtaining the strongest general purpose upper bounds on the efficiency of error-correcting codes. From 1973 forward, the linear programming bound found many applications, but there were few significant theoretical advances until Schrijver proposed a new code upper bound via semidefinite programming in 2003. Using the Terwilliger algebra, a recently introduced extension of the Bose-Mesner algebra, Schrijver formulated a new SDP strengthening of the LP approach. In this project we look at the dual solutions of the semidefinite programming bound for binary error-correcting codes. We explore the combinatorial meaning of these variables for small n and d, such as n = 4 and d = 2. To obtain information like this, we wrote a computer program with both Matlab and CVX modules to get solution of our primal SDP formulation. Our program efficiently generates the primal solutions with corresponding constraints for any n and d. We also wrote a program in C++ to parse the output of the primal SDP problem, and another Matlab script to generate the dual SDP problem, which could be used in assigning combinatorial meaning to the values given in the dual optimal solution. Our code not only computes both the primal and dual optimal variable values, but allows the researcher to display them in meaningful ways and to explore their relationship and dependence on arameters. These values are expected to be useful for later study of the combinatorial meaning of such solutions."

Digital filter banks have found in a wide variety of applications in data compression, digital communications, and adaptive signal processing. The common objectives of the filter bank design consist of frequency selectivity of the individual filters and perfect reconstruction of the filter banks. The design problems of filter banks are intrinsically challenging because their natural formulations are nonconvex constrained optimization problems. Therefore, there is a strong motivation to cast the design problems into convex optimization problems whose globally optimal solutions can be efficiently obtained. The main contributions of this dissertation are to exploit the convex optimization algorithms to design several classes of the filter banks. First, the two-channel orthogonal symmetric complex-valued filter banks are investigated. A key contribution is to derive the necessary and sufficient condition for the existence of complex-valued symmetric spectral factors. Moreover, this condition can be expressed as linear matrix inequalities (LMIs), and hence semi-definite programming (SDP) is applicable. Secondly, for two-channel symmetric real-valued filter banks, a more general and efficient method for designing the optimal triplet halfband filter banks with regularity is developed. By exploiting the LMI characterization of nonnegative cosine polynomials, the semi-infinite constraints can be efficiently handled. Consequently, the filter bank design is cast as an SDP problem. Furthermore, it is demonstrated that the resulting filter banks are applied to image coding with improved performance. It is not straightforward to extend the proposed design methods for two-channel filter banks to M-channel filter banks. However, it is investigated that the design problem of M-channel cosine-modulated filter banks is a nonconvex optimization problem with the low degree of nonconvexity. Therefore, the efficient semidefinite relaxation technique is proposed to design optimal prototype filters. Additionally, a cheap iterative algorithm is developed to further improve the performance of the filter banks. Finally, the application of filter banks to multicarrier systems is considered. The condition on the transmit filter bank and channel for the existence of zero-forcing filter bank equalizers is obtained. A closed-form expression of the optimal equalizer is then derived. The proposed filter bank transceivers are shown to outperform the orthogonal frequency-division multiplexing (OFDM) systems.

The facility location problem is the task of optimally placing a given number of facilities in a certain subset of the plane. In this thesis, we present various mathematical programming formulations of the planar facility location problem, where potential facility locations are not specified. We first consider mixed-integer programming formulations of the planar facility locations problems with squared Euclidean and rectangular distance metrics to solve this problem to provable optimality. We also investigate a heuristic approach to solving the problem by extending the $K$-means clustering algorithm and formulating the facility location problem as a variant of a semidefinite programming problem, leading to a relaxation algorithm. We present computational results for the mixed-integer formulations, as well as compare the objective values resulting from the relaxation algorithm and the modified $K$-means heuristic. In addition, we briefly discuss some of the practical issues related to the facility location model under the continuous customer distribution.

The facility location problem is the task of optimally placing a given number of facilities in a certain subset of the plane. In this thesis, we present various mathematical programming formulations of the planar facility location problem, where potential facility locations are not specified. We first consider mixed-integer programming formulations of the planar facility locations problems with squared Euclidean and rectangular distance metrics to solve this problem to provable optimality. We also investigate a heuristic approach to solving the problem by extending the $K$-means clustering algorithm and formulating the facility location problem as a variant of a semidefinite programming problem, leading to a relaxation algorithm. We present computational results for the mixed-integer formulations, as well as compare the objective values resulting from the relaxation algorithm and the modified $K$-means heuristic. In addition, we briefly discuss some of the practical issues related to the facility location model under the continuous customer distribution.

The continuous facility layout problem consists of arranging a set of facilities so that no pair overlaps and the total sum of the pairwise connection costs (proportional to the center-to-center rectilinear distance) is minimized. This thesis presents a completely mixed integer semidefinite programming (MISDP) model for the continuous facility layout problem. To begin we describe the problem in detail; discuss the conditions required for a feasible layout; and define quaternary variables. These variables are the basis of the MISDP model. We prove that the model is an exact formulation and a distinction is made between the constraints that semidefinite programming (SDP) optimization software can solve and those that must be relaxed. The latter are called exactness constraints and three possible exactness constraints are shown to be equivalent. The main contribution of this thesis is the theoretical development of a MISDP model that is based on quaternary, as oppose to binary, variables; nevertheless preliminary computational results will be presented for problems with 5 to 20 facilities. The optimal solution is found for problems with 5 and 6 facilities, confirming the validity of the model; and the potential of the model is revealed as a new upper bound is found for an 11-facility problem.

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The continuous facility layout problem consists of arranging a set of facilities so that no pair overlaps and the total sum of the pairwise connection costs (proportional to the center-to-center rectilinear distance) is minimized. This thesis presents a completely mixed integer semidefinite programming (MISDP) model for the continuous facility layout problem. To begin we describe the problem in detail; discuss the conditions required for a feasible layout; and define quaternary variables. These variables are the basis of the MISDP model. We prove that the model is an exact formulation and a distinction is made between the constraints that semidefinite programming (SDP) optimization software can solve and those that must be relaxed. The latter are called exactness constraints and three possible exactness constraints are shown to be equivalent. The main contribution of this thesis is the theoretical development of a MISDP model that is based on quaternary, as oppose to binary, variables; nevertheless preliminary computational results will be presented for problems with 5 to 20 facilities. The optimal solution is found for problems with 5 and 6 facilities, confirming the validity of the model; and the potential of the model is revealed as a new upper bound is found for an 11-facility problem.

Digital filter banks have found in a wide variety of applications in data compression, digital communications, and adaptive signal processing. The common objectives of the filter bank design consist of frequency selectivity of the individual filters and perfect reconstruction of the filter banks. The design problems of filter banks are intrinsically challenging because their natural formulations are nonconvex constrained optimization problems. Therefore, there is a strong motivation to cast the design problems into convex optimization problems whose globally optimal solutions can be efficiently obtained. The main contributions of this dissertation are to exploit the convex optimization algorithms to design several classes of the filter banks. First, the two-channel orthogonal symmetric complex-valued filter banks are investigated. A key contribution is to derive the necessary and sufficient condition for the existence of complex-valued symmetric spectral factors. Moreover, this condition can be expressed as linear matrix inequalities (LMIs), and hence semi-definite programming (SDP) is applicable. Secondly, for two-channel symmetric real-valued filter banks, a more general and efficient method for designing the optimal triplet halfband filter banks with regularity is developed. By exploiting the LMI characterization of nonnegative cosine polynomials, the semi-infinite constraints can be efficiently handled. Consequently, the filter bank design is cast as an SDP problem. Furthermore, it is demonstrated that the resulting filter banks are applied to image coding with improved performance. It is not straightforward to extend the proposed design methods for two-channel filter banks to M-channel filter banks. However, it is investigated that the design problem of M-channel cosine-modulated filter banks is a nonconvex optimization problem with the low degree of nonconvexity. Therefore, the efficient semidefinite relaxation technique is proposed to design optimal prototype filters. Additionally, a cheap iterative algorithm is developed to further improve the performance of the filter banks. Finally, the application of filter banks to multicarrier systems is considered. The condition on the transmit filter bank and channel for the existence of zero-forcing filter bank equalizers is obtained. A closed-form expression of the optimal equalizer is then derived. The proposed filter bank transceivers are shown to outperform the orthogonal frequency-division multiplexing (OFDM) systems.