We address the problem of identifying the neighborhood structure of an undirected graph, whose nodes are labeled with the elements of a multivariate normal (MVN) random vector. A semi-definite program is given for estimating the information matrix under arbitrary constraints on its elements. More importantly, a closed-form expression is given for the maximum likelihood (ML) estimator of the information matrix, under the constraint that the information matrix has pre-specified elements in a given pattern (e.g., in a principal submatrix). The results apply to the identification of dependency labels in a graphical model with neighborhood constraints. This neighborhood structure excludes nodes which are conditionally independent of a given node and the graph is determined by the non- zero elements in the information matrix for the random vector. A cross-validation principle is given for determining whether the constrained information matrix returned from this procedure is an acceptable model for the information matrix, and as a consequence for the neighborhood structure of the Markov Random Field (MRF) that is identified with the MVN random vector.

The facility layout design problem is a fundamental optimization problem encountered in many manufacturing and service organizations that was originally formulated in 1963 by Armour & Buffa. This thesis derives a convex programming model, IBIMODEL, that is designed to improve upon the ModCoAR model of Anjos & Vannelli for the facility layout problem with unequal areas. The purpose of IBIMODEL is to find 'good' initial locations for the departments that a second model then uses to produce a detailed solution to the facility layout problem. The proposed model has four ideas behind it: unlike ModCoAR, it does not improve the objective function as the departments start overlapping, it takes into account the aspect ratio requirements, it introduces a systematic approach to making parameter choices, and it uses a new second stage recently proposed by Luo, Anjos & Vannelli to obtain the actual facility layouts. In this way, IBIMODEL efficiently generates a reasonably diverse set of superior solutions that allow the second stage to provide a wide variety of layouts with relatively low aspect ratios and total cost. The proposed methodology was implemented and numerical results are presented on well-known large layout problems from the literature. To demonstrate the potential of the combination of IBIMODEL with Luo, Anjos & Vannelli's model, our results are compared with the best layouts found to date for these well-known large facility layout problems. The results support the conclusion that the propose a methodology consistently produces competitive, and often improved, layouts for large instances when compared with other approaches in the literature.

The facility layout design problem is a fundamental optimization problem encountered in many manufacturing and service organizations that was originally formulated in 1963 by Armour & Buffa. This thesis derives a convex programming model, IBIMODEL, that is designed to improve upon the ModCoAR model of Anjos & Vannelli for the facility layout problem with unequal areas. The purpose of IBIMODEL is to find 'good' initial locations for the departments that a second model then uses to produce a detailed solution to the facility layout problem. The proposed model has four ideas behind it: unlike ModCoAR, it does not improve the objective function as the departments start overlapping, it takes into account the aspect ratio requirements, it introduces a systematic approach to making parameter choices, and it uses a new second stage recently proposed by Luo, Anjos & Vannelli to obtain the actual facility layouts. In this way, IBIMODEL efficiently generates a reasonably diverse set of superior solutions that allow the second stage to provide a wide variety of layouts with relatively low aspect ratios and total cost. The proposed methodology was implemented and numerical results are presented on well-known large layout problems from the literature. To demonstrate the potential of the combination of IBIMODEL with Luo, Anjos & Vannelli's model, our results are compared with the best layouts found to date for these well-known large facility layout problems. The results support the conclusion that the propose a methodology consistently produces competitive, and often improved, layouts for large instances when compared with other approaches in the literature.

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

The aim of this work is to propose viscosity-like methods for finding a specific common fixed point of a finite family T={ T_{i} }_{i=1}^{N} of nonexpansive self-mappings of a closed convex subset C of a Hilbert space H.We propose two schemes: one implicit and the other explicit.The implicit scheme determines a set {x_{t} : 0 < t < 1} through the fixed point equation x_{t}= tf (x_{t} ) + (1− t)Tx_{t}, where f : C¡÷C is a contraction.The explicit scheme is the discretization of the implicit scheme and de defines a sequence {x_{n} } by the recursion x_{n+1}=£\\_{n}f(x_{n}) +(1−£\\_{n})Tx_{n} for n ≥ 0, where {£\\_{n} }⊂ (0,1) It has been shown in the literature that both implicit and explicit schemes converge in norm to a fixed point of T (with additional conditions imposed on the sequence {£\ _{n} } in the explicit scheme).We will extend both schemes to the case of a finite family of nonexpansive mappings. Our proposed schemes converge in norm to a common fixed point of the family which in addition solves a variational inequality.

Modern organizations (e.g., hospitals, banks, social networks, search engines) hold large volumes of personal information, and rely heavily on auditing for enforcement of privacy policies. These audit mechanisms combine automated methods with human input to detect and punish violators. Since human audit resources are limited, and often not sufficient to investigate all potential violations, current state-of-the -art audit tools provide heuristics to guide human effort. However, numerous reports of privacy breaches caused by malicious insiders bring to question the effectiveness of these audit mechanisms. Our thesis is that effective audit resource allocation and punishment levels can be efficiently computed by modeling the audit process as a game between a rational auditor and a rational or worst-case auditee. We present several results in support of the thesis. In the worst-case adversary setting, we design a game model taking into account organizational cost of auditing and loss from violations. We propose the notion of low regret as a desired audit property and provide a regret minimizing audit algorithm that outputs an optimal audit resource allocation strategy. The algorithm improves upon prior regret bounds in the partial information setting. In the rational adversary setting, we enable punishments by the auditor, and model the adversary's utility as a trade-off between the benefit from violations and loss due to punishment when detected. Our Stackelberg game model generalizes an existing deployed security game model with punishment parameters. It applies to natural auditing settings with multiple auditors where each auditor is restricted to audit a subset of the potential violations. We provide novel polynomial time algorithms to approximate the non-convex optimization problem used to compute the Stackelberg equilibrium. The algorithms output optimal audit resource allocation strategy and punishment levels. We also provide a method to reduce the optimization problem size, achieving up to 5x speedup for realistic instances of the audit problem, and for the related security game instances.

In this thesis, algorithms for blind and non-blind motion deblurring of digital images are proposed. The non-blind algorithm is based on a convex program consisting of a data fitting term and a sparsity-promoting regularization term. The data fitting term is the squared l_2 norm of the residual between the blurred image and the latent image convolved with a known blur kernel. The regularization term is the l_1 norm of the latent image under a wavelet frame (framelet) decomposition. This convex program is solved with the first-order primal-dual algorithm proposed by Chambolle and Pock. The proposed blind deblurring algorithm is based on the work of Cai, Ji, Liu, and Shen. It works by embedding the proposed non-blind algorithm in an alternating minimization scheme and imposing additional constraints in order to deal with the challenging non-convex nature of the blind deblurring problem. Numerical experiments are performed on artificially and naturally blurred images, and both proposed algorithms are found to be competitive with recent deblurring methods. / Graduate / 0544 / tdanniels@gmail.com

A clique of a graph G is a set of pairwise adjacent nodes of G. Similarly, a biclique (U, V ) of a bipartite graph G is a pair of disjoint, independent vertex sets such that each node in U is adjacent to every node in V in G. We consider the problems of identifying the maximum clique of a graph, known as the maximum clique problem, and identifying the biclique (U, V ) of a bipartite graph that maximizes the product |U | · |V |, known as the maximum edge biclique problem. We show that finding a clique or biclique of a given size in a graph is equivalent to finding a rank one matrix satisfying a particular set of linear constraints. These problems can be formulated as rank minimization problems and relaxed to convex programming by replacing rank with its convex envelope, the nuclear norm. Both problems are NP-hard yet we show that our relaxation is exact in the case that the input graph contains a large clique or biclique plus additional nodes and edges. For each problem, we provide two analyses of when our relaxation is exact. In the first, the diversionary edges are added deterministically by an adversary. In the second, each potential edge is added to the graph independently at random with fixed probability p. In the random case, our bounds match the earlier bounds of Alon, Krivelevich, and Sudakov, as well as Feige and Krauthgamer for the maximum clique problem. We extend these results and techniques to the k-disjoint-clique problem. The maximum node k-disjoint-clique problem is to find a set of k disjoint cliques of a given input graph containing the maximum number of nodes. Given input graph G and nonnegative edge weights w, the maximum mean weight k-disjoint-clique problem seeks to identify the set of k disjoint cliques of G that maximizes the sum of the average weights of the edges, with respect to w, of the complete subgraphs of G induced by the cliques. These problems may be considered as a way to pose the clustering problem. In clustering, one wants to partition a given data set so that the data items in each partition or cluster are similar and the items in different clusters are dissimilar. For the graph G such that the set of nodes represents a given data set and any two nodes are adjacent if and only if the corresponding items are similar, clustering the data into k disjoint clusters is equivalent to partitioning G into k-disjoint cliques. Similarly, given a complete graph with nodes corresponding to a given data set and edge weights indicating similarity between each pair of items, the data may be clustered by solving the maximum mean weight k-disjoint-clique problem. We show that both instances of the k-disjoint-clique problem can be formulated as rank constrained optimization problems and relaxed to semidefinite programs using the nuclear norm relaxation of rank. We also show that when the input instance corresponds to a collection of k disjoint planted cliques plus additional edges and nodes, this semidefinite relaxation is exact for both problems. We provide theoretical bounds that guarantee exactness of our relaxation and provide empirical examples of successful applications of our algorithm to synthetic data sets, as well as data sets from clustering applications.

This thesis discusses mathematical optimization techniques for waveform design in cognitive radars. These techniques have been designed with an increasing level of sophistication, starting from a bistatic model (i.e. two transmitters and a single receiver) and ending with a cognitive network (i.e. multiple transmitting and multiple receiving radars). The environment under investigation always features strong signal-dependent clutter and noise. All algorithms are based on an iterative waveform-filter optimization. The waveform optimization is based on convex optimization techniques and the exploitation of initial radar waveforms characterized by desired auto and cross-correlation properties. Finally, robust optimization techniques are introduced to account for the assumptions made by cognitive radars on certain second order statistics such as the covariance matrix of the clutter. More specifically, initial optimization techniques were proposed for the case of bistatic radars. By maximizing the signal to interference and noise ratio (SINR) under certain constraints on the transmitted signals, it was possible to iteratively optimize both the orthogonal transmission waveforms and the receiver filter. Subsequently, the above work was extended to a convex optimization framework for a waveform design technique for bistatic radars where both radars transmit and receive to detect targets. The method exploited prior knowledge of the environment to maximize the accumulated target return signal power while keeping the disturbance power to unity at both radar receivers. The thesis further proposes convex optimization based waveform designs for multiple input multiple output (MIMO) based cognitive radars. All radars within the system are able to both transmit and receive signals for detecting targets. The proposed model investigated two complementary optimization techniques. The first one aims at optimizing the signal to interference and noise ratio (SINR) of a specific radar while keeping the SINR of the remaining radars at desired levels. The second approach optimizes the SINR of all radars using a max-min optimization criterion. To account for possible mismatches between actual parameters and estimated ones, this thesis includes robust optimization techniques. Initially, the multistatic, signal-dependent model was tested against existing worst-case and probabilistic methods. These methods appeared to be over conservative and generic for the considered signal-dependent clutter scenario. Therefore a new approach was derived where uncertainty was assumed directly on the radar cross-section and Doppler parameters of the clutters. Approximations based on Taylor series were invoked to make the optimization problem convex and {subsequently} determine robust waveforms with specific SINR outage constraints. Finally, this thesis introduces robust optimization techniques for through-the-wall radars. These are also cognitive but rely on different optimization techniques than the ones previously discussed. By noticing the similarities between the minimum variance distortionless response (MVDR) problem and the matched-illumination one, this thesis introduces robust optimization techniques that consider uncertainty on environment-related parameters. Various performance analyses demonstrate the effectiveness of all the above algorithms in providing a significant increase in SINR in an environment affected by very strong clutter and noise.

Motivated by recent developments in wireless communication, this thesis aims to characterize the secrecy performance in several types of typical wireless networks. Advanced techniques are designed and evaluated to enhance physical layer security in these networks with realistic assumptions, such as signal propagation loss, random node distribution and non-instantaneous channel state information (CSI). The first part of the thesis investigates secret communication through relay-assisted cognitive interference channel. The primary and secondary base stations (PBS and SBS) communicate with the primary and secondary receivers (PR and SR) respectively in the presence of multiple eavesdroppers. The SBS is allowed to transmit simultaneously with the PBS over the same spectrum instead of waiting for an idle channel. To improve security, cognitive relays transmit cooperative jamming (CJ) signals to create additional interferences in the direction of the eavesdroppers. Two CJ schemes are proposed to improve the secrecy rate of cognitive interference channels depending on the structure of cooperative relays. In the scheme where the multiple-antenna relay transmits weighted jamming signals, the combined approach of CJ and beamforming is investigated. In the scheme with multiple relays transmitting weighted jamming signals, the combined approach of CJ and relay selection is analyzed. Numerical results show that both these two schemes are effective in improving physical layer security of cognitive interference channel. In the second part, the focus is shifted to physical layer security in a random wireless network where both legitimate and eavesdropping nodes are randomly distributed. Three scenarios are analyzed to investigate the impact of various factors on security. In scenario one, the basic scheme is studied without a protected zone and interference. The probability distribution function (PDF) of channel gain with both fading and path loss has been derived and further applied to derive secrecy connectivity and ergodic secrecy capacity. In the second scenario, we studied using a protected zone surrounding the source node to enhance security where interference is absent. Both the cases that eavesdroppers are aware and unaware of the protected zone boundary are investigated. Based on the above scenarios, further deployment of the protected zones at legitimate receivers is designed to convert detrimental interference into a beneficial factor. Numerical results are investigated to check the reliability of the PDF for reciprocal of channel gain and to analyze the impact of protected zones on secrecy performance. In the third part, physical layer security in the downlink transmission of cellular network is studied. To model the repulsive property of the cellular network planning, we assume that the base stations (BSs) follow the Mat´ern hard-core point process (HCPP), while the eavesdroppers are deployed as an independent Poisson point process (PPP). The distribution function of the distances from a typical point to the nodes of the HCPP is derived. The noise-limited and interference-limited cellular networks are investigated by applying the fractional frequency reuse (FFR) in the system. For the noise-limited network, we derive the secrecy outage probability with two different strategies, i.e. the best BS serve and the nearest BS serve, by analyzing the statistics of channel gains. For the interference-limited network with the nearest BS serve, two transmission schemes are analyzed, i.e., transmission with and without the FFR. Numerical results reveal that both the schemes of transmitting with the best BS and the application of the FFR are beneficial for physical layer security in the downlink cellular networks, while the improvement due to the application of the FFR is limited by the capacity of the legitimate channel.